Linear connection - Revision history

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

2025-05-26T16:03:00Z

Definition part (global form)

← Older revision		Revision as of 16:03, 26 May 2025
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	* It satisfies the following relation called the Leibniz rule:		* It satisfies the following relation called the Leibniz rule:

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

Vipul at 17:29, 6 January 2012

2012-01-06T17:29:59Z

← Older revision		Revision as of 17:29, 6 January 2012
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	===Definition part (pointwise form)===		===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(E)</math>.		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>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>E</math>).		* 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:		* It satisfies the following relation called the Leibniz rule:

Vipul: /* Curvature of a linear connection */

2009-07-23T23:34:33Z

Curvature of a linear connection

← Older revision		Revision as of 23:34, 23 July 2009
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	===Curvature of a linear connection===		===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:		The curvature of a linear connection <math>\nabla</math> is denoted as <math>R_\nabla</math>. It is defined as:
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	<math>R_\nabla(X,Y) = \nabla_X\nabla_Y - \nabla_Y\nabla_X - \nabla_{[X,Y]}</math>		<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~~. In the case of a linear connection, it is tensorial, with lots of nice properties. It is in fact a (1,3)-tensor called the [[Riemann curvature tensor]]~~.		The notion of curvature actually makes sense for any connection.

Vipul: 8 revisions

2008-05-18T19:48:26Z

8 revisions

← Older revision	Revision as of 19:48, 18 May 2008
(No difference)

Vipul: /* Generalizations */

2008-04-04T23:51:17Z

Generalizations

← Older revision		Revision as of 23:51, 4 April 2008
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	===Generalizations===		===Generalizations===

	The notion of linear connection can be generalized to the more general notion of a [[connection]].		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==		==Operations on a linear connection==

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

2008-04-04T23:50:48Z

Definition part (pointwise form)

← Older revision		Revision as of 23:50, 4 April 2008
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	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(E)</math>.		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(E)</math>.

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

Vipul: /* Operations on a linear connection */

2007-12-11T20:55:00Z

Operations on a linear connection

← Older revision		Revision as of 20:55, 11 December 2007
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	===Torsion of 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:		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:
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	A connection whose torsion is zero is termed a [[torsion-free linear connection]].		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===		===Curvature of a linear connection===

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

2007-12-11T20:53:57Z

Definition part (global form)

← Older revision		Revision as of 20:53, 11 December 2007
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	===Definition part (global form)===		===Definition part (global form)===

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

Vipul: /* Torsion of a linear connection */

2007-04-05T11:21:34Z

Torsion of a linear connection

← Older revision		Revision as of 11:21, 5 April 2007
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	===Torsion of a linear connection===		===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:		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>.		<math>\tau(\nabla)(X,Y)= \nabla_XY - \nabla_YX - [X,Y]</math>.

Vipul: /* Torsion of a linear connection */

2007-03-07T17:00:22Z

Torsion of a linear connection

← Older revision		Revision as of 17:00, 7 March 2007
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	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:		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>.		<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]].		A connection whose torsion is zero is termed a [[torsion-free linear connection]].