Matrix of connection forms - Revision history

Vipul: 2 revisions

2008-05-18T19:48:38Z

2 revisions

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Vipul at 22:50, 10 April 2008

2008-04-10T22:50:51Z

← Older revision		Revision as of 22:50, 10 April 2008
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	Suppose <math>U</math> is an open set at <math>p</math> such that the bundle <math>E</math>, restricted to <math>U</math>, is trivial. Suppose, further, that <math>e_1,e_2,\ldots,e_r</math> is the collection of constant vector fields for basis directions in <math>U</math>. Then the matrix of connection forms at <math>p</math> is given by a <math>r \times r</math> matrix where the entry in the <math>(ij)^{th}</math> position is the 1-form:		Suppose <math>U</math> is an open set at <math>p</math> such that the bundle <math>E</math>, restricted to <math>U</math>, is trivial. Suppose, further, that <math>e_1,e_2,\ldots,e_r</math> is the collection of constant vector fields for basis directions in <math>U</math>. Then the matrix of connection forms at <math>p</math> is given by a <math>r \times r</math> matrix where the entry in the <math>(ij)^{th}</math> position is the 1-form:

	<math>\omega_{ij} := X \mapsto \left \langle (\nabla_X e_i)(p), e_j \rangle</math>		<math>\omega_{ij} := X \mapsto \left \langle (\nabla_X e_i)(p), e_j \right \rangle</math>

	In other words, the 1-forms are chosen so that:		In other words, the 1-forms are chosen so that:

Vipul: New page: ==Definition== Suppose $M$ is a differential manifold and $\nabla$ is a connection on $M$ . The '''matrix of connection forms''' at a point $p...$

2008-04-10T22:50:09Z

New page: ==Definition== Suppose <math>M</math> is a differential manifold and <math>\nabla</math> is a connection on <math>M</math>. The '''matrix of connection forms''' at a point <math>p...

New page

==Definition==

Suppose <math>M</math> is a [[differential manifold]] and <math>\nabla</math> is a [[connection]] on <math>M</math>. The '''matrix of connection forms''' at a point <math>p \in M</math> is a way of describing the connection in terms of coordinate charts. The matrix of connection forms is dependent on the choice of coordinate chart. It is described as follows.

Suppose <math>U</math> is an open set at <math>p</math> such that the bundle <math>E</math>, restricted to <math>U</math>, is trivial. Suppose, further, that <math>e_1,e_2,\ldots,e_r</math> is the collection of constant vector fields for basis directions in <math>U</math>. Then the matrix of connection forms at <math>p</math> is given by a <math>r \times r</math> matrix where the entry in the <math>(ij)^{th}</math> position is the 1-form:

<math>\omega_{ij} := X \mapsto \left \langle (\nabla_X e_i)(p), e_j \rangle</math>

In other words, the 1-forms are chosen so that:

<math>\nabla_Xe_i = \sum_j \omega_{ij}(X)(p) e_j</math>

==Relation with Christoffel symbols==

The matrix of connection forms is closely related to the [[Christoffel symbols]]. To obtain the Christoffel symbols from the matrix of connection forms, we need to choose a basis for the tangent space at the point, and express the 1-form in terms of the coefficients of its dual basis.

The key difference is thus:

* The matrix of connection forms does ''not'' depend on a choice of basis for <math>TM</math>. It only depends on a local trivialization for the bundle <math>E</math>.
* The Christoffel symbols, on the other hand, depend on a choice of basis for <math>TM</math> as well.

Matrix of connection forms - Revision history

Vipul: 2 revisions

Vipul at 22:50, 10 April 2008

Vipul: New page: ==Definition== Suppose M is a differential manifold and \nabla is a connection on M. The '''matrix of connection forms''' at a point p...

Vipul: New page: ==Definition== Suppose $M$ is a differential manifold and $\nabla$ is a connection on $M$ . The '''matrix of connection forms''' at a point $p...$