Diffgeom - User contributions [en]

Connection on a vector bundle

2025-05-26T16:22:27Z

Vipul: /* As an affine space */

{{elementof|the space of <math>\R</math>-bilinear maps <math>\Gamma(TM) \times \Gamma(E) \to \Gamma(E)</math> for a [[vector bundle]] <math>E</math> over a [[manifold]] <math>M</math>}}
{{basic construct on dm}}
==Definition==

===Given data===

* A connected [[differential manifold]] <math>M</math> with tangent bundle denoted by <math>TM</math>
* A [[vector bundle]] <math>E</math> over <math>M</math>

===Definition part (pointwise form)===

A '''connection''' is a smooth choice <math>\nabla</math> of the following: at each point <math>p \in M</math>, there is a map <math>{}^p\nabla: T_p(M) \times \Gamma(E) \to E(p)</math>, satisfying some conditions. The map is written as <math>{}^p\nabla_X(v)</math> where <math>X \in T_p(M)</math> and <math>v \in \Gamma(E)</math>.

* It is <math>\R</math>-linear in <math>X</math> (i.e., in the <math>T_p(M)</math> coordinate).
* It is <math>\R</math>-linear in <math>\Gamma(E)</math> (viz., the space of sections on <math>E</math>).
* It satisfies the following relation called the Leibniz rule:

{{quotation|<math>{}^p\nabla_X(fv) = (Xf)(p) (v) + f(p) ^p\nabla_X(v)</math>}}

===Definition part (global form)===

A '''connection''' is a map <math>\nabla:\Gamma(TM) \times \Gamma(E) \to \Gamma(E)</math>, satisfying the following:

* It is <math>C^\infty</math>-linear in <math>\Gamma(TM)</math> (in other words, it is [[tensorial map|tensorial]], or ''pointwise'', in the <math>\Gamma(TM)</math>-coordinate)
* it is <math>\mathbb{R}</math>-linear in <math>\Gamma(E)</math>
* It satisfies the following relation called the Leibniz rule:

{{quotation|<math>\nabla_X(fv) = (Xf) (v) + f \nabla_X(v) </math>}}

where <math>f</math> is a scalar function on the manifold and <math>fv</math> denotes scalar multiplication of <math>v</math> by <math>f</math>.

===Alternative definitions===

A connection is equivalent to the following:

* A choice of splitting of the [[first-order symbol sequence of a vector bundle]]: {{further|[[connection is splitting of first-order symbol sequence]]}}
* A module structure of the vector bundle, over the [[connection algebra]]: {{further|[[connection is module structure over connection algebra]]}}
* A connection on the corresponding bundle over the principal bundle over the [[general linear group]]. {{further|[[connection on vector bundle equals connection on principal GL-bundle]]}}

===Particular cases===

When <math>E = M \times \R</math> is the trivial one-dimensional bundle, then sections of <math>E</math> are the same as infinitely differentiable functions on <math>M</math>. For this bundle, there is a unique connection: the usual action of a vector field on a function.

When <math>E</math> is itself the tangent bundle, we call the connection a [[linear connection]].

==Terminology==

===Covariant derivative of a section===

{{further|[[covariant derivative of a section]]}}

Given a [[connection]] <math>\nabla</math> on a [[vector bundle]] <math>E</math> over a [[differential manifold]] <math>M</math>, the ''covariant derivative'' of a section <math>s \in \Gamma(E)</math> with respect to a [[vector field]] <math>X</math> is defined as the value:

<math>\nabla_X(s)</math>

The term ''covariant derivative'' can thus be used ''only'' if we already have a connection in mind.

===Absolute derivative of a section===

{{further|[[absolute derivative of a section]]}}

Given a connection <math>\nabla</math>, the absolute derivative of a section <math>s \in \Gamma(E)</math>, denoted <math>d_\nabla(s)</math>, is defined as the operator that sends a vector field <math>X</math> to <math>\nabla_X(s)</math>. In the particular case where <math>E = M \times \R</math> is the trivial one-dimensional bundle, this reduces to the [[de Rham derivative]] of a function, yielding a 1-form.

===Connection, transport along a curve===

{{further|[[connection along a curve]], [[transport along a curve]]}}

Given a connection on the manifold, we can obtain a connection along any curve on the manifold, using the [[pullback connection]]. A connection along the curve gives a [[transport along a curve|transport]]: a rule for transporting a basis for the fiber at one point, to a basis for the fiber at the other point. Thus, a connection is often thought of as a global ''transport rule''.

==Importance==

Consider a [[vector field]] <math>X</math> on <math>M</math>. We know that we can define a notion of ''directional derivatives'' for functions along this vector field: this differentiates the function at each point, along the vector at that point.
The derivative of <math>f</math> along the direction of <math>X</math> is a new function, denoted as <math>Xf</math>.

Note that at any point <math>p</math>, the value of <math>(Xf)(p)</math> depends on the ''local'' behavior of <math>f</math> but only on the ''pointwise'' behavior of <math>X</math>, that is, it only depends on the tangent vector <math>X(p)</math> and not on the behavior of <math>X</math> in the neighborhood.

The idea behind a connection is to extend this differentiation rule, not just to functions, but also to other kinds of objects. In particular, we want to be able to have a differentiation rule for sections of the tangent and cotangent bundles, along vector fields. In this definition, what we would like is:

* The derivative with respect to a vector field at a point should just depend on the value of the vector field at the point -- it should ''not'' depend on the behavior in the neighborhood. We say it is a [[tensorial map]] with respect to <math>X</math>.

* A Leibniz rule is satisfied with respect to scalar multiplication by functions, which connects differentiation for this connection with the differentiation of scalar functions along vector fields

Note that the usual differentiation along vector fields is thus the ''canonical'' connection on the trivial one-dimensional bundle, and we would like that any other connections we define should be compatible with this via the Leibniz rule.

==Existence==

{{further|[[Connections exist]]}}

Given any [[vector bundle]] over a [[differential manifold]], there exists a connection for that vector bundle.

==Constructions==

===Connection on a direct sum===

{{further|[[Direct sum of connections]]}}

Suppose we have connections <math>\nabla, \nabla'</math> on [[vector bundle]]s <math>E,E'</math> over a [[differential manifold]] <math>M</math>. Then, we can obtain a connection, that we'll denote <math>\nabla \oplus \nabla'</math>, on the direct sum <math>E \oplus E'</math>. This is defined by:

<math>(\nabla \oplus \nabla')(s,s') = \nabla(s) \oplus \nabla'(s')</math>.

===Connection on a tensor product===

{{further|[[Tensor product of connections]]}}

Suppose we have connections <math>\nabla, \nabla'</math> on [[vector bundle]]s <math>E,E'</math> over a [[differential manifold]] <math>M</math>. Then, we can obtain a connection, that we'll denote <math>\nabla \otimes \nabla'</math>, on the tensor product <math>E \otimes E'</math>. On ''pure'' tensors, it is given by the formula:

<math>(\nabla \otimes \nabla')_X(s \otimes s') = \nabla_X(s) \otimes s' + s \otimes \nabla'_X(s')</math>

In other words, the formula is chosen so that a Leibniz-like rule is satisfied for tensor products.

===Connection on the dual===

{{further|[[Dual connection]]}}

Given a connection <math>\nabla</math> on a vector bundle <math>E</math> over a differential manifold <math>M</math>, we can obtain a connection <math>\nabla^*</math>on the dual bundle <math>E^*</math> as follows:

<math>\nabla^*_X(l) = s \mapsto X(l(s)) - l(\nabla_X(s))</math>

==Particular kinds of connections==

===Metric connection===

{{further|[[metric connection]]}}

The notion of a metric connection makes sense when we have a [[metric bundle]]: a vector bundle with an inner product on every fiber that varies compatibly. A metric connection is a connection with the property that it satisfes a Leibniz-like rule with respect to the inner product of sections:

<math>X \left \langle s_1, s_2 \right \rangle = \left \langle \nabla_X s_1, s_2 \right \rangle + \left \langle s_1, \nabla_X s_2 \right \rangle</math>

A case of particular interest is a [[metric linear connection]]: this is a metric connection on the tangent bundle, for a [[Riemannian manifold]].

==The set of all connections==

===As an affine space===
{{further|[[Affine space of all connections]]}}

Given a manifold <math>M</math> and a vector bundle <math>E</math> over <math>M</math>, consider the set of all connections for <math>E</math>. Clearly, the connections live inside the space of <math>\R</math>-bilinear maps <math>\Gamma(TM) \times \Gamma(E) \to \Gamma(E)</math>. Hence, we can talk of linear combinations of connections. In general, a linear combination of connections need not be a connection. The problem arises from the Leibniz rule, which has a term <math>Xf</math> that does not scale with the connection.

It is true that the set of ''differences'' of connections (if nonempty) forms a vector subspace of the vector space of all bilinear maps. Since there is a fundamental theorem that connections exist, we conclude that the set of connections is in fact an affine space, viz a translate of a subspace, and thus any affine linear combination of connections is again a connection.

The vector space in question is the space of <math>C^\infty</math>-bilinear maps <math>\Gamma(TM) \times \Gamma(E) \to \Gamma(E)</math>.

===As the collection of module structures===

{{further|[[Connection is module structure over connection algebra]]}}

Given a vector bundle <math>E</math>, a connection on <math>E</math> makes <math>\Gamma(TM)</math> ''act'' on <math>\Gamma(E)</math>. Thus, we could view <math>\Gamma(E)</math> as a ''module'' over the free algebra generated by <math>\Gamma(TM)</math>. This action actually satisfies some extra conditions, and these conditions help us descend to an action of the [[connection algebra]] on <math>\Gamma(E)</math>.

Thus, a connection on a vector bundle <math>E</math> is equivalent to equipping <math>\Gamma(E)</math> with a module structure over the connection algebra.

==Local description==

===Connections localize===
{{further|[[Connections localize]]}}

Given a connection on the whole differential manifold <math>M</math>, we can get a connection on any open subset <math>U</math> of <math>M</math>. Note that this is not a completely trivial statement, because not every vector field on an open subset extends to a vector field on the whole manifold. However, we can express any vector field on an open subset, as the product of a function and a vector field that ''can'' be extended to the whole manifold, and we can then use the Leibniz rule.

It is also true that connections ''piece together''. In other words, to know <math>\nabla_X s</math> at a point <math>p</math>, it suffices to know the germ of <math>s</math> at <math>p</math>.
===Describing connections using coordinate charts===

{{further|[[Christoffel symbols of a connection]], [[matrix of connection forms]]}}

A connection is a bilinear map, and because connections localize and piece together, it suffices to describe what happens to the connection inside coordinate charts. However, we need to remember that while <math>\nabla_X(s)</math> depends only ''pointwise'' on <math>X</math> (so it depends only on the ''value'' of <math>X</math> at a point), it depends ''locally'' on <math>s</math> (so it depends on the germ of <math>s</math> at the point). So, to describe a connection at a point <math>p</math>, it is ''not'' enough to take a basis for <math>T_p(M)</math> and a basis for <math>E(p)</math> and describe what happens on that basis. Rather, we take a basis for <math>T_p(M)</math>, and pick a coordinate chart around <math>p</math>, and take constant vector fields corresponding to a choice of basis for that coordinate chart.

Linear connection

2025-05-26T16:03:00Z

Vipul: /* Definition part (global form) */

{{elementof|the space of <math>\R</math>-bilinear maps <math>\Gamma(TM) \times \Gamma(TM) \to \Gamma(TM)</math>}}

==Definition==

===Given data===

* A connected [[differential manifold]] <math>M</math> with tangent bundle denoted by <math>TM</math>

===Definition part (pointwise form)===

A '''linear connection''' is a smooth choice <math>\nabla</math> of the following: at each point <math>p \in M</math>, there is a map <math>{}^p\nabla: T_p(M) \times \Gamma(TM) \to T_p(M)</math>, satisfying some conditions. The map is written as <math>{}^p\nabla_X(v)</math> where <math>X \in T_p(M)</math> and <math>v \in \Gamma(TM)</math>.

* It is <math>\R</math>-linear in <math>X</math> (that is, in the <math>T_p(M)</math> coordinate).
* It is <math>\R</math>-linear in <math>\Gamma(TM)</math> (viz the space of sections on <math>TM</math>).
* It satisfies the following relation called the Leibniz rule:

<math>{}^p\nabla_X(fv) = (Xf)(p) (v) + f(p) ^p\nabla_X(v)</math>

===Definition part (global form)===

A '''linear connection''' is a map <math>\nabla:\Gamma(TM) \times \Gamma(TM) \to \Gamma(TM)</math>, satisfying the following:

* It is <math>C^\infty</math>-linear in the first <math>\Gamma(TM)</math>
* it is <math>\mathbb{R}</math>-linear in the second <math>\Gamma(TM)</math>
* It satisfies the following relation called the Leibniz rule:

<math>\nabla_X(fv) = (Xf) (v) + f \nabla_X(v) </math>

where <math>f</math> is a scalar function on the manifold and <math>fv</math> denotes scalar multiplication of <math>v</math> by <math>f</math>.

===Generalizations===

The notion of linear connection can be generalized to the more general notion of a [[connection]], where the second <math>TM</math> is replaced by an arbitrary [[vector bundle]] <math>E</math> over <math>M</math>.

==Operations on a linear connection==

===Torsion of a linear connection===

{{further|[[torsion of a linear connection]]}}

The torsion of a linear connection <math>\nabla</math> is denoted as <math>\tau(\nabla)</math>. It is a <math>(1,2)</math>-tensor defined as:

<math>\tau(\nabla)(X,Y)= \nabla_XY - \nabla_YX - [X,Y]</math>.

A connection whose torsion is zero is termed a [[torsion-free linear connection]].

Note that torsion makes sense ''only'' for linear connections.

===Curvature of a linear connection===

{{further|[[Riemann curvature tensor]]}}

The curvature of a linear connection <math>\nabla</math> is denoted as <math>R_\nabla</math>. It is defined as:

<math>R_\nabla(X,Y) = \nabla_X\nabla_Y - \nabla_Y\nabla_X - \nabla_{[X,Y]}</math>

The notion of curvature actually makes sense for any connection.

Connection on a vector bundle

2024-09-21T00:58:28Z

Vipul: /* As an affine space */

{{elementof|the space of <math>\R</math>-bilinear maps <math>\Gamma(TM) \times \Gamma(E) \to \Gamma(E)</math> for a [[vector bundle]] <math>E</math> over a [[manifold]] <math>M</math>}}
{{basic construct on dm}}
==Definition==

===Given data===

* A connected [[differential manifold]] <math>M</math> with tangent bundle denoted by <math>TM</math>
* A [[vector bundle]] <math>E</math> over <math>M</math>

===Definition part (pointwise form)===

A '''connection''' is a smooth choice <math>\nabla</math> of the following: at each point <math>p \in M</math>, there is a map <math>{}^p\nabla: T_p(M) \times \Gamma(E) \to E(p)</math>, satisfying some conditions. The map is written as <math>{}^p\nabla_X(v)</math> where <math>X \in T_p(M)</math> and <math>v \in \Gamma(E)</math>.

* It is <math>\R</math>-linear in <math>X</math> (i.e., in the <math>T_p(M)</math> coordinate).
* It is <math>\R</math>-linear in <math>\Gamma(E)</math> (viz., the space of sections on <math>E</math>).
* It satisfies the following relation called the Leibniz rule:

{{quotation|<math>{}^p\nabla_X(fv) = (Xf)(p) (v) + f(p) ^p\nabla_X(v)</math>}}

===Definition part (global form)===

A '''connection''' is a map <math>\nabla:\Gamma(TM) \times \Gamma(E) \to \Gamma(E)</math>, satisfying the following:

* It is <math>C^\infty</math>-linear in <math>\Gamma(TM)</math> (in other words, it is [[tensorial map|tensorial]], or ''pointwise'', in the <math>\Gamma(TM)</math>-coordinate)
* it is <math>\mathbb{R}</math>-linear in <math>\Gamma(E)</math>
* It satisfies the following relation called the Leibniz rule:

{{quotation|<math>\nabla_X(fv) = (Xf) (v) + f \nabla_X(v) </math>}}

where <math>f</math> is a scalar function on the manifold and <math>fv</math> denotes scalar multiplication of <math>v</math> by <math>f</math>.

===Alternative definitions===

A connection is equivalent to the following:

* A choice of splitting of the [[first-order symbol sequence of a vector bundle]]: {{further|[[connection is splitting of first-order symbol sequence]]}}
* A module structure of the vector bundle, over the [[connection algebra]]: {{further|[[connection is module structure over connection algebra]]}}
* A connection on the corresponding bundle over the principal bundle over the [[general linear group]]. {{further|[[connection on vector bundle equals connection on principal GL-bundle]]}}

===Particular cases===

When <math>E = M \times \R</math> is the trivial one-dimensional bundle, then sections of <math>E</math> are the same as infinitely differentiable functions on <math>M</math>. For this bundle, there is a unique connection: the usual action of a vector field on a function.

When <math>E</math> is itself the tangent bundle, we call the connection a [[linear connection]].

==Terminology==

===Covariant derivative of a section===

{{further|[[covariant derivative of a section]]}}

Given a [[connection]] <math>\nabla</math> on a [[vector bundle]] <math>E</math> over a [[differential manifold]] <math>M</math>, the ''covariant derivative'' of a section <math>s \in \Gamma(E)</math> with respect to a [[vector field]] <math>X</math> is defined as the value:

<math>\nabla_X(s)</math>

The term ''covariant derivative'' can thus be used ''only'' if we already have a connection in mind.

===Absolute derivative of a section===

{{further|[[absolute derivative of a section]]}}

Given a connection <math>\nabla</math>, the absolute derivative of a section <math>s \in \Gamma(E)</math>, denoted <math>d_\nabla(s)</math>, is defined as the operator that sends a vector field <math>X</math> to <math>\nabla_X(s)</math>. In the particular case where <math>E = M \times \R</math> is the trivial one-dimensional bundle, this reduces to the [[de Rham derivative]] of a function, yielding a 1-form.

===Connection, transport along a curve===

{{further|[[connection along a curve]], [[transport along a curve]]}}

Given a connection on the manifold, we can obtain a connection along any curve on the manifold, using the [[pullback connection]]. A connection along the curve gives a [[transport along a curve|transport]]: a rule for transporting a basis for the fiber at one point, to a basis for the fiber at the other point. Thus, a connection is often thought of as a global ''transport rule''.

==Importance==

Consider a [[vector field]] <math>X</math> on <math>M</math>. We know that we can define a notion of ''directional derivatives'' for functions along this vector field: this differentiates the function at each point, along the vector at that point.
The derivative of <math>f</math> along the direction of <math>X</math> is a new function, denoted as <math>Xf</math>.

Note that at any point <math>p</math>, the value of <math>(Xf)(p)</math> depends on the ''local'' behavior of <math>f</math> but only on the ''pointwise'' behavior of <math>X</math>, that is, it only depends on the tangent vector <math>X(p)</math> and not on the behavior of <math>X</math> in the neighborhood.

The idea behind a connection is to extend this differentiation rule, not just to functions, but also to other kinds of objects. In particular, we want to be able to have a differentiation rule for sections of the tangent and cotangent bundles, along vector fields. In this definition, what we would like is:

* The derivative with respect to a vector field at a point should just depend on the value of the vector field at the point -- it should ''not'' depend on the behavior in the neighborhood. We say it is a [[tensorial map]] with respect to <math>X</math>.

* A Leibniz rule is satisfied with respect to scalar multiplication by functions, which connects differentiation for this connection with the differentiation of scalar functions along vector fields

Note that the usual differentiation along vector fields is thus the ''canonical'' connection on the trivial one-dimensional bundle, and we would like that any other connections we define should be compatible with this via the Leibniz rule.

==Existence==

{{further|[[Connections exist]]}}

Given any [[vector bundle]] over a [[differential manifold]], there exists a connection for that vector bundle.

==Constructions==

===Connection on a direct sum===

{{further|[[Direct sum of connections]]}}

Suppose we have connections <math>\nabla, \nabla'</math> on [[vector bundle]]s <math>E,E'</math> over a [[differential manifold]] <math>M</math>. Then, we can obtain a connection, that we'll denote <math>\nabla \oplus \nabla'</math>, on the direct sum <math>E \oplus E'</math>. This is defined by:

<math>(\nabla \oplus \nabla')(s,s') = \nabla(s) \oplus \nabla'(s')</math>.

===Connection on a tensor product===

{{further|[[Tensor product of connections]]}}

Suppose we have connections <math>\nabla, \nabla'</math> on [[vector bundle]]s <math>E,E'</math> over a [[differential manifold]] <math>M</math>. Then, we can obtain a connection, that we'll denote <math>\nabla \otimes \nabla'</math>, on the tensor product <math>E \otimes E'</math>. On ''pure'' tensors, it is given by the formula:

<math>(\nabla \otimes \nabla')_X(s \otimes s') = \nabla_X(s) \otimes s' + s \otimes \nabla'_X(s')</math>

In other words, the formula is chosen so that a Leibniz-like rule is satisfied for tensor products.

===Connection on the dual===

{{further|[[Dual connection]]}}

Given a connection <math>\nabla</math> on a vector bundle <math>E</math> over a differential manifold <math>M</math>, we can obtain a connection <math>\nabla^*</math>on the dual bundle <math>E^*</math> as follows:

<math>\nabla^*_X(l) = s \mapsto X(l(s)) - l(\nabla_X(s))</math>

==Particular kinds of connections==

===Metric connection===

{{further|[[metric connection]]}}

The notion of a metric connection makes sense when we have a [[metric bundle]]: a vector bundle with an inner product on every fiber that varies compatibly. A metric connection is a connection with the property that it satisfes a Leibniz-like rule with respect to the inner product of sections:

<math>X \left \langle s_1, s_2 \right \rangle = \left \langle \nabla_X s_1, s_2 \right \rangle + \left \langle s_1, \nabla_X s_2 \right \rangle</math>

A case of particular interest is a [[metric linear connection]]: this is a metric connection on the tangent bundle, for a [[Riemannian manifold]].

==The set of all connections==

===As an affine space===
{{further|[[Affine space of all connections]]}}

Given a manifold <math>M</math> and a vector bundle <math>E</math> over <math>M</math>, consider the set of all connections for <math>E</math>. Clearly, the connections live inside the space of <math>\R</math>-bilinear maps <math>\Gamma(TM) \times \Gamma(E) \to \Gamma(E)</math>. Hence, we can talk of linear combinations of connections. In general, a linear combination of connections need not be a connection. The problem arises from the Leibniz rule, which has a term <math>Xf</math> that does not scale with the connection.

It is true that the set of ''differences'' of connections (if nonempty) forms a vector subspace of the vector space of all bilinear maps. Since there is a fundamental theorem that connections exist, we conclude that the set of connections is in fact an affine space, viz a translate of a subspace, and thus any affine linear combination of connections is again a connection.

===As the collection of module structures===

{{further|[[Connection is module structure over connection algebra]]}}

Given a vector bundle <math>E</math>, a connection on <math>E</math> makes <math>\Gamma(TM)</math> ''act'' on <math>\Gamma(E)</math>. Thus, we could view <math>\Gamma(E)</math> as a ''module'' over the free algebra generated by <math>\Gamma(TM)</math>. This action actually satisfies some extra conditions, and these conditions help us descend to an action of the [[connection algebra]] on <math>\Gamma(E)</math>.

Thus, a connection on a vector bundle <math>E</math> is equivalent to equipping <math>\Gamma(E)</math> with a module structure over the connection algebra.

==Local description==

===Connections localize===
{{further|[[Connections localize]]}}

Given a connection on the whole differential manifold <math>M</math>, we can get a connection on any open subset <math>U</math> of <math>M</math>. Note that this is not a completely trivial statement, because not every vector field on an open subset extends to a vector field on the whole manifold. However, we can express any vector field on an open subset, as the product of a function and a vector field that ''can'' be extended to the whole manifold, and we can then use the Leibniz rule.

It is also true that connections ''piece together''. In other words, to know <math>\nabla_X s</math> at a point <math>p</math>, it suffices to know the germ of <math>s</math> at <math>p</math>.
===Describing connections using coordinate charts===

{{further|[[Christoffel symbols of a connection]], [[matrix of connection forms]]}}

A connection is a bilinear map, and because connections localize and piece together, it suffices to describe what happens to the connection inside coordinate charts. However, we need to remember that while <math>\nabla_X(s)</math> depends only ''pointwise'' on <math>X</math> (so it depends only on the ''value'' of <math>X</math> at a point), it depends ''locally'' on <math>s</math> (so it depends on the germ of <math>s</math> at the point). So, to describe a connection at a point <math>p</math>, it is ''not'' enough to take a basis for <math>T_p(M)</math> and a basis for <math>E(p)</math> and describe what happens on that basis. Rather, we take a basis for <math>T_p(M)</math>, and pick a coordinate chart around <math>p</math>, and take constant vector fields corresponding to a choice of basis for that coordinate chart.

MediaWiki:Sitenotice

2024-09-12T06:02:22Z

Vipul:

Want site search autocompletion? See [[Project:Enabling site search autocompletion|here]]<br/>
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User:Vipul/Sandbox

2024-09-12T06:02:05Z

Vipul:

* <math>e^{\pi^2 + \sqrt{2}}</math>
* <math>2^{8 - 1} = 128</math>
* <math>2^7 - 1 = 127</math>
* <math>\sqrt{7 + 2}!! + 0 = 720</math>
* <math>(7 + 2)^{\sqrt{9}} = 729</math>
* <math>(3 + 4)^3 = 343</math>

MediaWiki:Sitenotice

2024-09-08T18:02:52Z

Vipul:

'''This site is in the process of being migrated to a new server. Edits made until this notice has been removed may be lost.'''<br/>
Want site search autocompletion? See [[Project:Enabling site search autocompletion|here]]<br/>
Encountering 429 Too Many Requests errors when browsing the site? See [[Project:429 Too Many Requests error|here]]

User:Vipul/Sandbox

2024-08-15T04:24:56Z

Vipul:

* <math>e^{\pi^2 + \sqrt{2}}</math>
* <math>2^{8 - 1} = 128</math>
* <math>2^7 - 1 = 127</math>
* <math>\sqrt{7 + 2}!! + 0 = 720</math>
* <math>(7 + 2)^{\sqrt{9}} = 729</math>

MediaWiki:Sitenotice

2024-08-15T04:14:33Z

Vipul:

File:Site search autocompletion working.png

2024-08-15T03:59:41Z

Vipul:

File:Site search autocompletion broken.png

2024-08-15T03:58:56Z

Vipul:

Diffgeom:Enabling site search autocompletion

2024-08-15T03:54:44Z

Vipul: Created page with "Content copied from Ref:Ref:Enabling site search autocompletion. Images used are specific to this site (Diffgeom). Site search autocompletion is currently broken by default on this site. This page includes details on how to get it to work, and what's going on. ==What's wrong with site search autocompletion and how to fix it== ===What's wrong=== When you start typing something in the site search bar, you'll see it stuck at "Loading search suggestions" as shown in..."

Content copied from [[Ref:Ref:Enabling site search autocompletion]]. Images used are specific to this site (Diffgeom).

Site search autocompletion is currently broken by default on this site. This page includes details on how to get it to work, and what's going on.

==What's wrong with site search autocompletion and how to fix it==

===What's wrong===

When you start typing something in the site search bar, you'll see it stuck at "Loading search suggestions" as shown in the screenshot below:

[[File:Site search autocompletion broken.png]]

Note that the actual search is still working -- you just have to hit Enter after typing the search query and it'll go to the search results page. It's the autocompletion before you hit Enter that is broken.

===How to fix it===

To fix it, you need to follow these steps:

* Write to vipulnaik1@gmail.com asking for a login to the site. Please include the following with your request: preferred username, preferred initial password (you can change it after logging in), real name (if you want it entered), email address to use (if you want an actual email address by which you can be contacted), and whether you want edit access as well. You don't need edit access for enabling site search autocompletion.
* Log in to the site. Then go to [[Special:Preferences]]. Go to the Appearance section and switch the Skin from "Vector (2022)" to "Vector legacy (2010)".
* Make sure to hit "Save" at the bottom.
* Now you can reload the page or load a new page.

Site search autocompletion should now work. Here's an example:

[[File:Site search autocompletion working.png]]

==More background==

We've recently upgraded the MediaWiki version of this wiki from 1.35.13 to 1.41.2 (see [[Special:Version]]). The upgrade allows us to migrate the wiki to a more modern operating system version running PHP 8. With the current setup for MediaWiki 1.41.2, we're in this situation:

* The "Vector legacy (2010)" skin has site search autocompletion working, but it doesn't render well on small screens. Specifically, even on small mobile screens, it still shows the left menu, and doesn't properly use the MobileFrontend extension settings.
* The "Vector (2022)" skin doesn't have site search autocompletion working (see screenshots in preceding section) but it does render fine on mobile devices.

It is possible to set only one default skin (that is applicable to all non-logged-in users and is the default for logged-in users who have not configured a skin for themselves). So, the selection of default skin comes down to whether it's more important for casual users to have the mobile experience working or to have site search autocompletion working. Based on a general understanding of user behavior, we believe that having a usable mobile experience is more important for casual users than having site search autocompletion.

However, for power users who are using the site extensively, site search autocompletion may be important. That's why we've written this page giving guidance on how to set up site search autocompletion.

Diffgeom:429 Too Many Requests error

2024-08-15T03:52:59Z

Vipul: Created page with "This content is copied from Ref:Ref:429 Too Many Requests error. If you get a 429 Too Many Requests error when browsing this site, read on. You're probably seeing this error because a large number of requests have been made from your IP address over a short period of time. That's probably a lot of requests from you or others who share your IP address (such as your home wi-fi network). Waiting a minute and then retrying should generally work. If you are an actual h..."

This content is copied from [[Ref:Ref:429 Too Many Requests error]].

If you get a 429 Too Many Requests error when browsing this site, read on.

You're probably seeing this error because a large number of requests have been made from your IP address over a short period of time. That's probably a lot of requests from you or others who share your IP address (such as your home wi-fi network). Waiting a minute and then retrying should generally work.

If you are an actual human being with a legitimate reason to be browsing the site heavily, first, thank you and sorry about this! We set rate limits to prevent bots, spiders, spammers, and malicious actors from consuming too much of our server's resources so that our server's resources can be devoted to real humans like you. Consider writing to vipulnaik1@gmail.com with your IP address to have the IP address whitelisted. You can get your IP address by [https://www.google.com/search?q=my+ip+address Googling "my IP address"] (scroll down a little bit to where Google includes the IP address in a box). NOTE: If you have both an IPv4 address and an IPv6 address, you should send both; the server supports both IPv4 and IPv6, so either may end up getting used. To check if you have an IPv6 address, try visiting [https://ipv6.google.com/ ipv6.google.com].

If your IP address changes, or you are away from your home network, then you'll get rate-limited again. So if you find yourself getting rate-limited after already having been whitelisted, check if you are on a different IP address than the one for which you requested whitelisting.

MediaWiki:Sitenotice

2024-08-15T03:50:36Z

Vipul: Blanked the page

User:Vipul/Sandbox

2024-08-15T03:47:17Z

Vipul:

* <math>e^{\pi^2 + \sqrt{2}}</math>
* <math>2^{8 - 1} = 128</math>
* <math>2^7 - 1 = 127</math>
* <math>\sqrt{7 + 2}!! + 0 = 720</math>

User:Vipul/Sandbox

2024-08-15T03:38:40Z

Vipul:

* <math>e^{\pi^2 + \sqrt{2}}</math>
* <math>2^{8 - 1} = 128</math>
* <math>2^7 - 1 = 127</math>

User:Vipul/Sandbox

2024-08-15T03:33:39Z

Vipul:

* <math>e^{\pi^2 + \sqrt{2}}</math>
* <math>2^{8 - 1} = 128</math>

MediaWiki:Sitenotice

2024-08-15T03:20:24Z

Vipul:

'''This wiki is in the process of being upgraded. The site may go down intermittently. Please try to avoid editing until this notice has been removed.'''

User:Vipul

2024-02-22T11:10:45Z

Vipul:

Math formula tests:

* <math>9^{\sqrt{7 + 2}} = 729</math>
* <math>6^{2 + 1} = 216</math>
* <math>5^{1 + 2} = 125</math>
* <math>(3 + 4)^3 = 343</math>
* <math>3! * 6 = 36</math>
* <math>2^{8 - 1} = 128</math>
* <math>(8 + 9)^2 = 289</math>
* <math>2^7 - 1 = 127</math>
* <math>(2\sqrt{4})! = 24</math>
* <math>(\sqrt{4})^6 = 64</math>
* <math>3! * 60 = 360</math>
* <math>(7 - 1)! - 5 = 715</math>
* <math>3 * 5! - 5 = 355</math>
* <math>4 * (5! - 6) = 456</math>
* <math>(\sqrt{7 + 2})!! + 0 = 720</math>
* <math>5^2 = 25</math>
* <math>(\sqrt{7 + 2})!! + 1 = 721</math>
* <math>(\sqrt{7 + 2})!! + 2 = 722</math>
* <math>(\sqrt{7 + 2})!! + 3 = 723</math>
* <math>(\sqrt{7 + 2})!! + 4 = 724</math>
* <math>(\sqrt{7 + 2})!! + 5 = 725</math>
* <math>(\sqrt{7 + 2})!! + 6 = 726</math>
* <math>(\sqrt{7 + 2})!! + 7 = 727</math>
* <math>3! * 6 = 36</math>

nice!

User:Vipul

2023-11-12T00:38:07Z

Vipul:

User:Vipul

2023-11-12T00:25:55Z

Vipul:

User:Vipul

2023-11-12T00:24:05Z

Vipul:

User:Vipul

2023-11-12T00:11:35Z

Vipul:

User:Vipul

2023-11-12T00:02:21Z

Vipul:

I am Vipul Naik, a B.Sc. (Hons) Math student at [http://www.cmi.ac.in Chennai Mathematical Institute]. This wiki (on Differential Geometry) is my brainchild, and so far, I have been the only contributor.

Much of the wiki is motivated by my [http://groupprops.wiki-site.com Group Properties wiki].

Other math-related wikis I am administering (and which are still in a stage of infancy):

* [http://commalg.wiki-site.com Commutative algebra wiki]
* [http://topospaces.wiki-site.com Topology wiki]

Learn more about me at my [http://www.cmi.ac.in/~vipul Home Page].

Math formula tests:

* <math>9^{\sqrt{7 + 2}} = 729</math>
* <math>6^{2 + 1} = 216</math>
* <math>5^{1 + 2} = 125</math>
* <math>(3 + 4)^3 = 343</math>
* <math>3! * 6 = 36</math>
* <math>2^{8 - 1} = 128</math>
* <math>(8 + 9)^2 = 289</math>
* <math>2^7 - 1 = 127</math>
* <math>(2\sqrt{4})! = 24</math>
* <math>(\sqrt{4})^6 = 64</math>
* <math>3! * 60 = 360</math>
* <math>(7 - 1)! - 5 = 715</math>
* <math>3 * 5! - 5 = 355</math>
* <math>4 * (5! - 6) = 456</math>
* <math>(\sqrt{7 + 2})!! + 0 = 720</math>
* <math>5^2 = 25</math>
* <math>(\sqrt{7 + 2})!! + 1 = 721</math>
* <math>(\sqrt{7 + 2})!! + 2 = 722</math>
* <math>(\sqrt{7 + 2})!! + 3 = 723</math>
* <math>(\sqrt{7 + 2})!! + 4 = 724</math>
* <math>(\sqrt{7 + 2})!! + 5 = 725</math>

User:Vipul

2023-11-12T00:01:40Z

Vipul:

User:Vipul

2023-11-12T00:00:54Z

Vipul:

User:Vipul

2023-11-11T23:57:28Z

Vipul:

User:Vipul

2023-11-11T23:56:29Z

Vipul:

User:Vipul

2023-11-11T23:53:28Z

Vipul:

User:Vipul

2023-11-11T23:52:20Z

Vipul:

User:Vipul

2023-11-11T23:47:49Z

Vipul:

User:Vipul

2023-11-11T23:39:46Z

Vipul:

I am Vipul Naik, a B.Sc. (Hons) Math student at [http://www.cmi.ac.in Chennai Mathematical Institute]. This wiki (on Differential Geometry) is my brainchild, and so far, I have been the only contributor.

Much of the wiki is motivated by my [http://groupprops.wiki-site.com Group Properties wiki].

Other math-related wikis I am administering (and which are still in a stage of infancy):

* [http://commalg.wiki-site.com Commutative algebra wiki]
* [http://topospaces.wiki-site.com Topology wiki]

Learn more about me at my [http://www.cmi.ac.in/~vipul Home Page].

Math formula tests:

* <math>9^{\sqrt{7 + 2}} = 729</math>
* <math>6^{2 + 1} = 216</math>
* <math>5^{1 + 2} = 125</math>
* <math>(3 + 4)^3 = 343</math>
* <math>3! * 6 = 36</math>
* <math>2^{8 - 1} = 128</math>
* <math>(8 + 9)^2 = 289</math>
* <math>2^7 - 1 = 127</math>
* <math>(2\sqrt{4})! = 24</math>
* <math>(\sqrt{4})^6 = 64</math>
* <math>3! * 60 = 360</math>
* <math>3 * 5! - 5 = 355</math>
* <math>4 * (5! - 6) = 456</math>

User:Vipul

2023-11-11T23:39:38Z

Vipul:

I am Vipul Naik, a B.Sc. (Hons) Math student at [http://www.cmi.ac.in Chennai Mathematical Institute]. This wiki (on Differential Geometry) is my brainchild, and so far, I have been the only contributor.

Much of the wiki is motivated by my [http://groupprops.wiki-site.com Group Properties wiki].

Other math-related wikis I am administering (and which are still in a stage of infancy):

* [http://commalg.wiki-site.com Commutative algebra wiki]
* [http://topospaces.wiki-site.com Topology wiki]

Learn more about me at my [http://www.cmi.ac.in/~vipul Home Page].

Math formula tests:

* <math>9^{\sqrt{7 + 2}} = 729</math>
* <math>6^{2 + 1} = 216</math>
* <math>5^{1 + 2} = 125</math>
* <math>(3 + 4)^3 = 343</math>
* <math>3! * 6 = 36</math>
* <math>2^{8 - 1} = 128</math>
* <math>(8 + 9)^2 = 289</math>
* <math>2^7 - 1 = 127</math>
* <math>(2\sqrt{4})! = 24</math>
* <math>(\sqrt{4})^6 = 64</math>
* <math>3! * 60 = 360</math>
* <math>3 * 5! - 5 = 355</math>
* 4 * (5! - 6) = 456</math>

User:Vipul

2023-11-11T23:38:44Z

Vipul:

User:Vipul

2023-11-11T23:37:19Z

Vipul:

User:Vipul

2023-11-11T23:24:24Z

Vipul:

User:Vipul

2023-11-11T23:24:00Z

Vipul:

User:Vipul

2023-11-11T23:12:25Z

Vipul:

User:Vipul

2023-11-11T23:09:00Z

Vipul:

User:Vipul

2023-11-11T23:06:16Z

Vipul:

User:Vipul

2023-11-11T22:57:31Z

Vipul:

User:Vipul

2023-11-11T22:54:58Z

Vipul:

User:Vipul

2023-11-11T22:53:05Z

Vipul:

User:Vipul

2023-11-11T22:48:40Z

Vipul:

User:Vipul

2023-11-11T22:40:32Z

Vipul:

User:Vipul

2023-11-11T22:34:30Z

Vipul:

Connection on a vector bundle

2023-09-14T21:33:59Z

Vipul: /* Metric connection */

{{elementof|the space of <math>\R</math>-bilinear maps <math>\Gamma(TM) \times \Gamma(E) \to \Gamma(E)</math> for a [[vector bundle]] <math>E</math> over a [[manifold]] <math>M</math>}}
{{basic construct on dm}}
==Definition==

===Given data===

* A connected [[differential manifold]] <math>M</math> with tangent bundle denoted by <math>TM</math>
* A [[vector bundle]] <math>E</math> over <math>M</math>

===Definition part (pointwise form)===

A '''connection''' is a smooth choice <math>\nabla</math> of the following: at each point <math>p \in M</math>, there is a map <math>{}^p\nabla: T_p(M) \times \Gamma(E) \to E(p)</math>, satisfying some conditions. The map is written as <math>{}^p\nabla_X(v)</math> where <math>X \in T_p(M)</math> and <math>v \in \Gamma(E)</math>.

* It is <math>\R</math>-linear in <math>X</math> (i.e., in the <math>T_p(M)</math> coordinate).
* It is <math>\R</math>-linear in <math>\Gamma(E)</math> (viz., the space of sections on <math>E</math>).
* It satisfies the following relation called the Leibniz rule:

{{quotation|<math>{}^p\nabla_X(fv) = (Xf)(p) (v) + f(p) ^p\nabla_X(v)</math>}}

===Definition part (global form)===

A '''connection''' is a map <math>\nabla:\Gamma(TM) \times \Gamma(E) \to \Gamma(E)</math>, satisfying the following:

* It is <math>C^\infty</math>-linear in <math>\Gamma(TM)</math> (in other words, it is [[tensorial map|tensorial]], or ''pointwise'', in the <math>\Gamma(TM)</math>-coordinate)
* it is <math>\mathbb{R}</math>-linear in <math>\Gamma(E)</math>
* It satisfies the following relation called the Leibniz rule:

{{quotation|<math>\nabla_X(fv) = (Xf) (v) + f \nabla_X(v) </math>}}

where <math>f</math> is a scalar function on the manifold and <math>fv</math> denotes scalar multiplication of <math>v</math> by <math>f</math>.

===Alternative definitions===

A connection is equivalent to the following:

* A choice of splitting of the [[first-order symbol sequence of a vector bundle]]: {{further|[[connection is splitting of first-order symbol sequence]]}}
* A module structure of the vector bundle, over the [[connection algebra]]: {{further|[[connection is module structure over connection algebra]]}}
* A connection on the corresponding bundle over the principal bundle over the [[general linear group]]. {{further|[[connection on vector bundle equals connection on principal GL-bundle]]}}

===Particular cases===

When <math>E = M \times \R</math> is the trivial one-dimensional bundle, then sections of <math>E</math> are the same as infinitely differentiable functions on <math>M</math>. For this bundle, there is a unique connection: the usual action of a vector field on a function.

When <math>E</math> is itself the tangent bundle, we call the connection a [[linear connection]].

==Terminology==

===Covariant derivative of a section===

{{further|[[covariant derivative of a section]]}}

Given a [[connection]] <math>\nabla</math> on a [[vector bundle]] <math>E</math> over a [[differential manifold]] <math>M</math>, the ''covariant derivative'' of a section <math>s \in \Gamma(E)</math> with respect to a [[vector field]] <math>X</math> is defined as the value:

<math>\nabla_X(s)</math>

The term ''covariant derivative'' can thus be used ''only'' if we already have a connection in mind.

===Absolute derivative of a section===

{{further|[[absolute derivative of a section]]}}

Given a connection <math>\nabla</math>, the absolute derivative of a section <math>s \in \Gamma(E)</math>, denoted <math>d_\nabla(s)</math>, is defined as the operator that sends a vector field <math>X</math> to <math>\nabla_X(s)</math>. In the particular case where <math>E = M \times \R</math> is the trivial one-dimensional bundle, this reduces to the [[de Rham derivative]] of a function, yielding a 1-form.

===Connection, transport along a curve===

{{further|[[connection along a curve]], [[transport along a curve]]}}

Given a connection on the manifold, we can obtain a connection along any curve on the manifold, using the [[pullback connection]]. A connection along the curve gives a [[transport along a curve|transport]]: a rule for transporting a basis for the fiber at one point, to a basis for the fiber at the other point. Thus, a connection is often thought of as a global ''transport rule''.

==Importance==

Consider a [[vector field]] <math>X</math> on <math>M</math>. We know that we can define a notion of ''directional derivatives'' for functions along this vector field: this differentiates the function at each point, along the vector at that point.
The derivative of <math>f</math> along the direction of <math>X</math> is a new function, denoted as <math>Xf</math>.

Note that at any point <math>p</math>, the value of <math>(Xf)(p)</math> depends on the ''local'' behavior of <math>f</math> but only on the ''pointwise'' behavior of <math>X</math>, that is, it only depends on the tangent vector <math>X(p)</math> and not on the behavior of <math>X</math> in the neighborhood.

The idea behind a connection is to extend this differentiation rule, not just to functions, but also to other kinds of objects. In particular, we want to be able to have a differentiation rule for sections of the tangent and cotangent bundles, along vector fields. In this definition, what we would like is:

* The derivative with respect to a vector field at a point should just depend on the value of the vector field at the point -- it should ''not'' depend on the behavior in the neighborhood. We say it is a [[tensorial map]] with respect to <math>X</math>.

* A Leibniz rule is satisfied with respect to scalar multiplication by functions, which connects differentiation for this connection with the differentiation of scalar functions along vector fields

Note that the usual differentiation along vector fields is thus the ''canonical'' connection on the trivial one-dimensional bundle, and we would like that any other connections we define should be compatible with this via the Leibniz rule.

==Existence==

{{further|[[Connections exist]]}}

Given any [[vector bundle]] over a [[differential manifold]], there exists a connection for that vector bundle.

==Constructions==

===Connection on a direct sum===

{{further|[[Direct sum of connections]]}}

Suppose we have connections <math>\nabla, \nabla'</math> on [[vector bundle]]s <math>E,E'</math> over a [[differential manifold]] <math>M</math>. Then, we can obtain a connection, that we'll denote <math>\nabla \oplus \nabla'</math>, on the direct sum <math>E \oplus E'</math>. This is defined by:

<math>(\nabla \oplus \nabla')(s,s') = \nabla(s) \oplus \nabla'(s')</math>.

===Connection on a tensor product===

{{further|[[Tensor product of connections]]}}

Suppose we have connections <math>\nabla, \nabla'</math> on [[vector bundle]]s <math>E,E'</math> over a [[differential manifold]] <math>M</math>. Then, we can obtain a connection, that we'll denote <math>\nabla \otimes \nabla'</math>, on the tensor product <math>E \otimes E'</math>. On ''pure'' tensors, it is given by the formula:

<math>(\nabla \otimes \nabla')_X(s \otimes s') = \nabla_X(s) \otimes s' + s \otimes \nabla'_X(s')</math>

In other words, the formula is chosen so that a Leibniz-like rule is satisfied for tensor products.

===Connection on the dual===

{{further|[[Dual connection]]}}

Given a connection <math>\nabla</math> on a vector bundle <math>E</math> over a differential manifold <math>M</math>, we can obtain a connection <math>\nabla^*</math>on the dual bundle <math>E^*</math> as follows:

<math>\nabla^*_X(l) = s \mapsto X(l(s)) - l(\nabla_X(s))</math>

==Particular kinds of connections==

===Metric connection===

{{further|[[metric connection]]}}

The notion of a metric connection makes sense when we have a [[metric bundle]]: a vector bundle with an inner product on every fiber that varies compatibly. A metric connection is a connection with the property that it satisfes a Leibniz-like rule with respect to the inner product of sections:

<math>X \left \langle s_1, s_2 \right \rangle = \left \langle \nabla_X s_1, s_2 \right \rangle + \left \langle s_1, \nabla_X s_2 \right \rangle</math>

A case of particular interest is a [[metric linear connection]]: this is a metric connection on the tangent bundle, for a [[Riemannian manifold]].

==The set of all connections==

===As an affine space===
{{further|[[Affine space of all connections]]}}
Given a manifold <math>M</math> and a vector bundle <math>E</math> over <math>M</math>, consider the set of all connections for <math>E</math>. Clearly, the connections live inside the space of <math>\R</math>-bilinear maps <math>\Gamma(TM) \times \Gamma(E) \to \Gamma(E)</math>. Hence, we can talk of linear combinations of connections. In general, a linear combination of connections need not be a connection. The problem arises from the Leibniz rule, which has a term <math>Xf</math> that does not scale with the connection.

It is true that the set of ''differences'' of connections (if nonempty) forms a vector subspace of the vector space of all bilinear maps. Since there is a fundamental theorem that connections exist, we conclude that the set of connections is in fact an affine space, viz a translate of a subspace, and thus any affine linear combination of connections is again a connection.

===As the collection of module structures===

{{further|[[Connection is module structure over connection algebra]]}}

Given a vector bundle <math>E</math>, a connection on <math>E</math> makes <math>\Gamma(TM)</math> ''act'' on <math>\Gamma(E)</math>. Thus, we could view <math>\Gamma(E)</math> as a ''module'' over the free algebra generated by <math>\Gamma(TM)</math>. This action actually satisfies some extra conditions, and these conditions help us descend to an action of the [[connection algebra]] on <math>\Gamma(E)</math>.

Thus, a connection on a vector bundle <math>E</math> is equivalent to equipping <math>\Gamma(E)</math> with a module structure over the connection algebra.

==Local description==

===Connections localize===
{{further|[[Connections localize]]}}

Given a connection on the whole differential manifold <math>M</math>, we can get a connection on any open subset <math>U</math> of <math>M</math>. Note that this is not a completely trivial statement, because not every vector field on an open subset extends to a vector field on the whole manifold. However, we can express any vector field on an open subset, as the product of a function and a vector field that ''can'' be extended to the whole manifold, and we can then use the Leibniz rule.

It is also true that connections ''piece together''. In other words, to know <math>\nabla_X s</math> at a point <math>p</math>, it suffices to know the germ of <math>s</math> at <math>p</math>.
===Describing connections using coordinate charts===

{{further|[[Christoffel symbols of a connection]], [[matrix of connection forms]]}}

A connection is a bilinear map, and because connections localize and piece together, it suffices to describe what happens to the connection inside coordinate charts. However, we need to remember that while <math>\nabla_X(s)</math> depends only ''pointwise'' on <math>X</math> (so it depends only on the ''value'' of <math>X</math> at a point), it depends ''locally'' on <math>s</math> (so it depends on the germ of <math>s</math> at the point). So, to describe a connection at a point <math>p</math>, it is ''not'' enough to take a basis for <math>T_p(M)</math> and a basis for <math>E(p)</math> and describe what happens on that basis. Rather, we take a basis for <math>T_p(M)</math>, and pick a coordinate chart around <math>p</math>, and take constant vector fields corresponding to a choice of basis for that coordinate chart.

Connection on a vector bundle

2023-09-14T21:33:43Z

Vipul: /* As the collection of module structures */

{{elementof|the space of <math>\R</math>-bilinear maps <math>\Gamma(TM) \times \Gamma(E) \to \Gamma(E)</math> for a [[vector bundle]] <math>E</math> over a [[manifold]] <math>M</math>}}
{{basic construct on dm}}
==Definition==

===Given data===

* A connected [[differential manifold]] <math>M</math> with tangent bundle denoted by <math>TM</math>
* A [[vector bundle]] <math>E</math> over <math>M</math>

===Definition part (pointwise form)===

A '''connection''' is a smooth choice <math>\nabla</math> of the following: at each point <math>p \in M</math>, there is a map <math>{}^p\nabla: T_p(M) \times \Gamma(E) \to E(p)</math>, satisfying some conditions. The map is written as <math>{}^p\nabla_X(v)</math> where <math>X \in T_p(M)</math> and <math>v \in \Gamma(E)</math>.

* It is <math>\R</math>-linear in <math>X</math> (i.e., in the <math>T_p(M)</math> coordinate).
* It is <math>\R</math>-linear in <math>\Gamma(E)</math> (viz., the space of sections on <math>E</math>).
* It satisfies the following relation called the Leibniz rule:

{{quotation|<math>{}^p\nabla_X(fv) = (Xf)(p) (v) + f(p) ^p\nabla_X(v)</math>}}

===Definition part (global form)===

A '''connection''' is a map <math>\nabla:\Gamma(TM) \times \Gamma(E) \to \Gamma(E)</math>, satisfying the following:

* It is <math>C^\infty</math>-linear in <math>\Gamma(TM)</math> (in other words, it is [[tensorial map|tensorial]], or ''pointwise'', in the <math>\Gamma(TM)</math>-coordinate)
* it is <math>\mathbb{R}</math>-linear in <math>\Gamma(E)</math>
* It satisfies the following relation called the Leibniz rule:

{{quotation|<math>\nabla_X(fv) = (Xf) (v) + f \nabla_X(v) </math>}}

where <math>f</math> is a scalar function on the manifold and <math>fv</math> denotes scalar multiplication of <math>v</math> by <math>f</math>.

===Alternative definitions===

A connection is equivalent to the following:

* A choice of splitting of the [[first-order symbol sequence of a vector bundle]]: {{further|[[connection is splitting of first-order symbol sequence]]}}
* A module structure of the vector bundle, over the [[connection algebra]]: {{further|[[connection is module structure over connection algebra]]}}
* A connection on the corresponding bundle over the principal bundle over the [[general linear group]]. {{further|[[connection on vector bundle equals connection on principal GL-bundle]]}}

===Particular cases===

When <math>E = M \times \R</math> is the trivial one-dimensional bundle, then sections of <math>E</math> are the same as infinitely differentiable functions on <math>M</math>. For this bundle, there is a unique connection: the usual action of a vector field on a function.

When <math>E</math> is itself the tangent bundle, we call the connection a [[linear connection]].

==Terminology==

===Covariant derivative of a section===

{{further|[[covariant derivative of a section]]}}

Given a [[connection]] <math>\nabla</math> on a [[vector bundle]] <math>E</math> over a [[differential manifold]] <math>M</math>, the ''covariant derivative'' of a section <math>s \in \Gamma(E)</math> with respect to a [[vector field]] <math>X</math> is defined as the value:

<math>\nabla_X(s)</math>

The term ''covariant derivative'' can thus be used ''only'' if we already have a connection in mind.

===Absolute derivative of a section===

{{further|[[absolute derivative of a section]]}}

Given a connection <math>\nabla</math>, the absolute derivative of a section <math>s \in \Gamma(E)</math>, denoted <math>d_\nabla(s)</math>, is defined as the operator that sends a vector field <math>X</math> to <math>\nabla_X(s)</math>. In the particular case where <math>E = M \times \R</math> is the trivial one-dimensional bundle, this reduces to the [[de Rham derivative]] of a function, yielding a 1-form.

===Connection, transport along a curve===

{{further|[[connection along a curve]], [[transport along a curve]]}}

Given a connection on the manifold, we can obtain a connection along any curve on the manifold, using the [[pullback connection]]. A connection along the curve gives a [[transport along a curve|transport]]: a rule for transporting a basis for the fiber at one point, to a basis for the fiber at the other point. Thus, a connection is often thought of as a global ''transport rule''.

==Importance==

Consider a [[vector field]] <math>X</math> on <math>M</math>. We know that we can define a notion of ''directional derivatives'' for functions along this vector field: this differentiates the function at each point, along the vector at that point.
The derivative of <math>f</math> along the direction of <math>X</math> is a new function, denoted as <math>Xf</math>.

Note that at any point <math>p</math>, the value of <math>(Xf)(p)</math> depends on the ''local'' behavior of <math>f</math> but only on the ''pointwise'' behavior of <math>X</math>, that is, it only depends on the tangent vector <math>X(p)</math> and not on the behavior of <math>X</math> in the neighborhood.

The idea behind a connection is to extend this differentiation rule, not just to functions, but also to other kinds of objects. In particular, we want to be able to have a differentiation rule for sections of the tangent and cotangent bundles, along vector fields. In this definition, what we would like is:

* The derivative with respect to a vector field at a point should just depend on the value of the vector field at the point -- it should ''not'' depend on the behavior in the neighborhood. We say it is a [[tensorial map]] with respect to <math>X</math>.

* A Leibniz rule is satisfied with respect to scalar multiplication by functions, which connects differentiation for this connection with the differentiation of scalar functions along vector fields

Note that the usual differentiation along vector fields is thus the ''canonical'' connection on the trivial one-dimensional bundle, and we would like that any other connections we define should be compatible with this via the Leibniz rule.

==Existence==

{{further|[[Connections exist]]}}

Given any [[vector bundle]] over a [[differential manifold]], there exists a connection for that vector bundle.

==Constructions==

===Connection on a direct sum===

{{further|[[Direct sum of connections]]}}

Suppose we have connections <math>\nabla, \nabla'</math> on [[vector bundle]]s <math>E,E'</math> over a [[differential manifold]] <math>M</math>. Then, we can obtain a connection, that we'll denote <math>\nabla \oplus \nabla'</math>, on the direct sum <math>E \oplus E'</math>. This is defined by:

<math>(\nabla \oplus \nabla')(s,s') = \nabla(s) \oplus \nabla'(s')</math>.

===Connection on a tensor product===

{{further|[[Tensor product of connections]]}}

Suppose we have connections <math>\nabla, \nabla'</math> on [[vector bundle]]s <math>E,E'</math> over a [[differential manifold]] <math>M</math>. Then, we can obtain a connection, that we'll denote <math>\nabla \otimes \nabla'</math>, on the tensor product <math>E \otimes E'</math>. On ''pure'' tensors, it is given by the formula:

<math>(\nabla \otimes \nabla')_X(s \otimes s') = \nabla_X(s) \otimes s' + s \otimes \nabla'_X(s')</math>

In other words, the formula is chosen so that a Leibniz-like rule is satisfied for tensor products.

===Connection on the dual===

{{further|[[Dual connection]]}}

Given a connection <math>\nabla</math> on a vector bundle <math>E</math> over a differential manifold <math>M</math>, we can obtain a connection <math>\nabla^*</math>on the dual bundle <math>E^*</math> as follows:

<math>\nabla^*_X(l) = s \mapsto X(l(s)) - l(\nabla_X(s))</math>

==Particular kinds of connections==

===Metric connection===

{{further|[[metric connection]]}}

The notion of a metric connection makes sense when we have a [[metric bundle]]: a vector bundle with an inner product on every fiber that varies compatibly. A metric connection is a connection with the property that it satisfes a Leibnixz-like rule with respect to the inner product of sections:

<math>X \left \langle s_1, s_2 \right \rangle = \left \langle \nabla_X s_1, s_2 \right \rangle + \left \langle s_1, \nabla_X s_2 \right \rangle</math>

A case of particular interest is a [[metric linear connection]]: this is a metric connection on the tangent bundle, for a [[Riemannian manifold]].

==The set of all connections==

===As an affine space===
{{further|[[Affine space of all connections]]}}
Given a manifold <math>M</math> and a vector bundle <math>E</math> over <math>M</math>, consider the set of all connections for <math>E</math>. Clearly, the connections live inside the space of <math>\R</math>-bilinear maps <math>\Gamma(TM) \times \Gamma(E) \to \Gamma(E)</math>. Hence, we can talk of linear combinations of connections. In general, a linear combination of connections need not be a connection. The problem arises from the Leibniz rule, which has a term <math>Xf</math> that does not scale with the connection.

It is true that the set of ''differences'' of connections (if nonempty) forms a vector subspace of the vector space of all bilinear maps. Since there is a fundamental theorem that connections exist, we conclude that the set of connections is in fact an affine space, viz a translate of a subspace, and thus any affine linear combination of connections is again a connection.

===As the collection of module structures===

{{further|[[Connection is module structure over connection algebra]]}}

Given a vector bundle <math>E</math>, a connection on <math>E</math> makes <math>\Gamma(TM)</math> ''act'' on <math>\Gamma(E)</math>. Thus, we could view <math>\Gamma(E)</math> as a ''module'' over the free algebra generated by <math>\Gamma(TM)</math>. This action actually satisfies some extra conditions, and these conditions help us descend to an action of the [[connection algebra]] on <math>\Gamma(E)</math>.

Thus, a connection on a vector bundle <math>E</math> is equivalent to equipping <math>\Gamma(E)</math> with a module structure over the connection algebra.

==Local description==

===Connections localize===
{{further|[[Connections localize]]}}

Given a connection on the whole differential manifold <math>M</math>, we can get a connection on any open subset <math>U</math> of <math>M</math>. Note that this is not a completely trivial statement, because not every vector field on an open subset extends to a vector field on the whole manifold. However, we can express any vector field on an open subset, as the product of a function and a vector field that ''can'' be extended to the whole manifold, and we can then use the Leibniz rule.

It is also true that connections ''piece together''. In other words, to know <math>\nabla_X s</math> at a point <math>p</math>, it suffices to know the germ of <math>s</math> at <math>p</math>.
===Describing connections using coordinate charts===

{{further|[[Christoffel symbols of a connection]], [[matrix of connection forms]]}}

A connection is a bilinear map, and because connections localize and piece together, it suffices to describe what happens to the connection inside coordinate charts. However, we need to remember that while <math>\nabla_X(s)</math> depends only ''pointwise'' on <math>X</math> (so it depends only on the ''value'' of <math>X</math> at a point), it depends ''locally'' on <math>s</math> (so it depends on the germ of <math>s</math> at the point). So, to describe a connection at a point <math>p</math>, it is ''not'' enough to take a basis for <math>T_p(M)</math> and a basis for <math>E(p)</math> and describe what happens on that basis. Rather, we take a basis for <math>T_p(M)</math>, and pick a coordinate chart around <math>p</math>, and take constant vector fields corresponding to a choice of basis for that coordinate chart.

Connection on a vector bundle

2023-09-14T21:33:25Z

Vipul: /* Describing connections using coordinate charts */

{{elementof|the space of <math>\R</math>-bilinear maps <math>\Gamma(TM) \times \Gamma(E) \to \Gamma(E)</math> for a [[vector bundle]] <math>E</math> over a [[manifold]] <math>M</math>}}
{{basic construct on dm}}
==Definition==

===Given data===

* A connected [[differential manifold]] <math>M</math> with tangent bundle denoted by <math>TM</math>
* A [[vector bundle]] <math>E</math> over <math>M</math>

===Definition part (pointwise form)===

A '''connection''' is a smooth choice <math>\nabla</math> of the following: at each point <math>p \in M</math>, there is a map <math>{}^p\nabla: T_p(M) \times \Gamma(E) \to E(p)</math>, satisfying some conditions. The map is written as <math>{}^p\nabla_X(v)</math> where <math>X \in T_p(M)</math> and <math>v \in \Gamma(E)</math>.

* It is <math>\R</math>-linear in <math>X</math> (i.e., in the <math>T_p(M)</math> coordinate).
* It is <math>\R</math>-linear in <math>\Gamma(E)</math> (viz., the space of sections on <math>E</math>).
* It satisfies the following relation called the Leibniz rule:

{{quotation|<math>{}^p\nabla_X(fv) = (Xf)(p) (v) + f(p) ^p\nabla_X(v)</math>}}

===Definition part (global form)===

A '''connection''' is a map <math>\nabla:\Gamma(TM) \times \Gamma(E) \to \Gamma(E)</math>, satisfying the following:

* It is <math>C^\infty</math>-linear in <math>\Gamma(TM)</math> (in other words, it is [[tensorial map|tensorial]], or ''pointwise'', in the <math>\Gamma(TM)</math>-coordinate)
* it is <math>\mathbb{R}</math>-linear in <math>\Gamma(E)</math>
* It satisfies the following relation called the Leibniz rule:

{{quotation|<math>\nabla_X(fv) = (Xf) (v) + f \nabla_X(v) </math>}}

where <math>f</math> is a scalar function on the manifold and <math>fv</math> denotes scalar multiplication of <math>v</math> by <math>f</math>.

===Alternative definitions===

A connection is equivalent to the following:

* A choice of splitting of the [[first-order symbol sequence of a vector bundle]]: {{further|[[connection is splitting of first-order symbol sequence]]}}
* A module structure of the vector bundle, over the [[connection algebra]]: {{further|[[connection is module structure over connection algebra]]}}
* A connection on the corresponding bundle over the principal bundle over the [[general linear group]]. {{further|[[connection on vector bundle equals connection on principal GL-bundle]]}}

===Particular cases===

When <math>E = M \times \R</math> is the trivial one-dimensional bundle, then sections of <math>E</math> are the same as infinitely differentiable functions on <math>M</math>. For this bundle, there is a unique connection: the usual action of a vector field on a function.

When <math>E</math> is itself the tangent bundle, we call the connection a [[linear connection]].

==Terminology==

===Covariant derivative of a section===

{{further|[[covariant derivative of a section]]}}

Given a [[connection]] <math>\nabla</math> on a [[vector bundle]] <math>E</math> over a [[differential manifold]] <math>M</math>, the ''covariant derivative'' of a section <math>s \in \Gamma(E)</math> with respect to a [[vector field]] <math>X</math> is defined as the value:

<math>\nabla_X(s)</math>

The term ''covariant derivative'' can thus be used ''only'' if we already have a connection in mind.

===Absolute derivative of a section===

{{further|[[absolute derivative of a section]]}}

Given a connection <math>\nabla</math>, the absolute derivative of a section <math>s \in \Gamma(E)</math>, denoted <math>d_\nabla(s)</math>, is defined as the operator that sends a vector field <math>X</math> to <math>\nabla_X(s)</math>. In the particular case where <math>E = M \times \R</math> is the trivial one-dimensional bundle, this reduces to the [[de Rham derivative]] of a function, yielding a 1-form.

===Connection, transport along a curve===

{{further|[[connection along a curve]], [[transport along a curve]]}}

Given a connection on the manifold, we can obtain a connection along any curve on the manifold, using the [[pullback connection]]. A connection along the curve gives a [[transport along a curve|transport]]: a rule for transporting a basis for the fiber at one point, to a basis for the fiber at the other point. Thus, a connection is often thought of as a global ''transport rule''.

==Importance==

Consider a [[vector field]] <math>X</math> on <math>M</math>. We know that we can define a notion of ''directional derivatives'' for functions along this vector field: this differentiates the function at each point, along the vector at that point.
The derivative of <math>f</math> along the direction of <math>X</math> is a new function, denoted as <math>Xf</math>.

Note that at any point <math>p</math>, the value of <math>(Xf)(p)</math> depends on the ''local'' behavior of <math>f</math> but only on the ''pointwise'' behavior of <math>X</math>, that is, it only depends on the tangent vector <math>X(p)</math> and not on the behavior of <math>X</math> in the neighborhood.

The idea behind a connection is to extend this differentiation rule, not just to functions, but also to other kinds of objects. In particular, we want to be able to have a differentiation rule for sections of the tangent and cotangent bundles, along vector fields. In this definition, what we would like is:

* The derivative with respect to a vector field at a point should just depend on the value of the vector field at the point -- it should ''not'' depend on the behavior in the neighborhood. We say it is a [[tensorial map]] with respect to <math>X</math>.

* A Leibniz rule is satisfied with respect to scalar multiplication by functions, which connects differentiation for this connection with the differentiation of scalar functions along vector fields

Note that the usual differentiation along vector fields is thus the ''canonical'' connection on the trivial one-dimensional bundle, and we would like that any other connections we define should be compatible with this via the Leibniz rule.

==Existence==

{{further|[[Connections exist]]}}

Given any [[vector bundle]] over a [[differential manifold]], there exists a connection for that vector bundle.

==Constructions==

===Connection on a direct sum===

{{further|[[Direct sum of connections]]}}

Suppose we have connections <math>\nabla, \nabla'</math> on [[vector bundle]]s <math>E,E'</math> over a [[differential manifold]] <math>M</math>. Then, we can obtain a connection, that we'll denote <math>\nabla \oplus \nabla'</math>, on the direct sum <math>E \oplus E'</math>. This is defined by:

<math>(\nabla \oplus \nabla')(s,s') = \nabla(s) \oplus \nabla'(s')</math>.

===Connection on a tensor product===

{{further|[[Tensor product of connections]]}}

Suppose we have connections <math>\nabla, \nabla'</math> on [[vector bundle]]s <math>E,E'</math> over a [[differential manifold]] <math>M</math>. Then, we can obtain a connection, that we'll denote <math>\nabla \otimes \nabla'</math>, on the tensor product <math>E \otimes E'</math>. On ''pure'' tensors, it is given by the formula:

<math>(\nabla \otimes \nabla')_X(s \otimes s') = \nabla_X(s) \otimes s' + s \otimes \nabla'_X(s')</math>

In other words, the formula is chosen so that a Leibniz-like rule is satisfied for tensor products.

===Connection on the dual===

{{further|[[Dual connection]]}}

Given a connection <math>\nabla</math> on a vector bundle <math>E</math> over a differential manifold <math>M</math>, we can obtain a connection <math>\nabla^*</math>on the dual bundle <math>E^*</math> as follows:

<math>\nabla^*_X(l) = s \mapsto X(l(s)) - l(\nabla_X(s))</math>

==Particular kinds of connections==

===Metric connection===

{{further|[[metric connection]]}}

The notion of a metric connection makes sense when we have a [[metric bundle]]: a vector bundle with an inner product on every fiber that varies compatibly. A metric connection is a connection with the property that it satisfes a Leibnixz-like rule with respect to the inner product of sections:

<math>X \left \langle s_1, s_2 \right \rangle = \left \langle \nabla_X s_1, s_2 \right \rangle + \left \langle s_1, \nabla_X s_2 \right \rangle</math>

A case of particular interest is a [[metric linear connection]]: this is a metric connection on the tangent bundle, for a [[Riemannian manifold]].

==The set of all connections==

===As an affine space===
{{further|[[Affine space of all connections]]}}
Given a manifold <math>M</math> and a vector bundle <math>E</math> over <math>M</math>, consider the set of all connections for <math>E</math>. Clearly, the connections live inside the space of <math>\R</math>-bilinear maps <math>\Gamma(TM) \times \Gamma(E) \to \Gamma(E)</math>. Hence, we can talk of linear combinations of connections. In general, a linear combination of connections need not be a connection. The problem arises from the Leibniz rule, which has a term <math>Xf</math> that does not scale with the connection.

It is true that the set of ''differences'' of connections (if nonempty) forms a vector subspace of the vector space of all bilinear maps. Since there is a fundamental theorem that connections exist, we conclude that the set of connections is in fact an affine space, viz a translate of a subspace, and thus any affine linear combination of connections is again a connection.

===As the collection of module structures===

{{further|[[Connection is module structure over connection algebra]]}}
Given a vector bundle <math>E</math>, a connection on <math>E</math> makes <math>\Gamma(TM)</math> ''act'' on <math>\Gamma(E)</math>. Thus, we could view <math>\Gamma(E)</math> as a ''module'' over the free algebra generated by <math>\Gamma(TM)</math>. This action actually satisfies some extra conditions, and these conditions help us descend to an action of the [[connection algebra]] on <math>\Gamma(E)</math>.

Thus, a connection on a vector bundle <math>E</math> is equivalent to equipping <math>\Gamma(E)</math> with a module structure over the connection algebra.

==Local description==

===Connections localize===
{{further|[[Connections localize]]}}

Given a connection on the whole differential manifold <math>M</math>, we can get a connection on any open subset <math>U</math> of <math>M</math>. Note that this is not a completely trivial statement, because not every vector field on an open subset extends to a vector field on the whole manifold. However, we can express any vector field on an open subset, as the product of a function and a vector field that ''can'' be extended to the whole manifold, and we can then use the Leibniz rule.

It is also true that connections ''piece together''. In other words, to know <math>\nabla_X s</math> at a point <math>p</math>, it suffices to know the germ of <math>s</math> at <math>p</math>.
===Describing connections using coordinate charts===

{{further|[[Christoffel symbols of a connection]], [[matrix of connection forms]]}}

A connection is a bilinear map, and because connections localize and piece together, it suffices to describe what happens to the connection inside coordinate charts. However, we need to remember that while <math>\nabla_X(s)</math> depends only ''pointwise'' on <math>X</math> (so it depends only on the ''value'' of <math>X</math> at a point), it depends ''locally'' on <math>s</math> (so it depends on the germ of <math>s</math> at the point). So, to describe a connection at a point <math>p</math>, it is ''not'' enough to take a basis for <math>T_p(M)</math> and a basis for <math>E(p)</math> and describe what happens on that basis. Rather, we take a basis for <math>T_p(M)</math>, and pick a coordinate chart around <math>p</math>, and take constant vector fields corresponding to a choice of basis for that coordinate chart.

Diffgeom:Privacy policy

2022-09-25T15:32:16Z

Vipul:

This privacy policy is common to subject wikis. For the original privacy policy, refer [[Ref:Ref:Privacy policy]].

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User:Vipul/Sandbox

2017-11-27T07:44:53Z

Vipul: Created page with "<math>e^{\pi^2 + \sqrt{2}}</math>"