Pullback of connection on a vector bundle - Revision history

Vipul: 3 revisions

2008-05-18T19:51:15Z

3 revisions

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Vipul: /* Related facts */

2008-04-13T14:28:17Z

Related facts

← Older revision		Revision as of 14:28, 13 April 2008
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	* Pull back the connection on <math>TM</math>, to the connection on the pullback bundle on <math>N</math> namely <math>TM\|_N</math>		* Pull back the connection on <math>TM</math>, to the connection on the pullback bundle on <math>N</math> namely <math>TM\|_N</math>
	* Project this to the connection on <math>TN</math>, using the inner product structure on <math>TM\|_N</math>		* Project this to the connection on <math>TN</math>, using the inner product structure on <math>TM\|_N</math>

			===Connection along a curve===

			{{further\|[[connection along a curve]]}}

			A connection along a curve can be viewed as a special case of a pullback connection, where the pullback is to the interval <math>(0,1)</math>.

Vipul at 14:20, 13 April 2008

2008-04-13T14:20:46Z

← Older revision		Revision as of 14:20, 13 April 2008
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	Suppose <math>f:N \to M</math> is a [[smooth map]] between [[differential manifold]]s <math>N</math> and <math>M</math>. Let <math>E</math> be a [[vector bundle]] over <math>M</math>, and <math>f^E</math> denote the [[pullback of a vector bundle\|pullback]] of <math>E</math> via <math>f</math> (hence, <math>f^E</math> is a vector bundle over <math>N</math>).		Suppose <math>f:N \to M</math> is a [[smooth map]] between [[differential manifold]]s <math>N</math> and <math>M</math>. Let <math>E</math> be a [[vector bundle]] over <math>M</math>, and <math>f^E</math> denote the [[pullback of a vector bundle\|pullback]] of <math>E</math> via <math>f</math> (hence, <math>f^E</math> is a vector bundle over <math>N</math>).

	Given a [[connection]] <math>\nabla</math> for the vector bundle <math>E</math>, we can define a connection <math>f^(\nabla)</math> for the vector bundle <math>f^(E)</math>, called the '''pullback''' of <math>\nabla</math>, as ~~follows~~:		Given a [[connection]] <math>\nabla</math> for the vector bundle <math>E</math>, we can define a connection <math>f^(\nabla)</math> for the vector bundle <math>f^(E)</math>, called the '''pullback''' of <math>\nabla</math>, as the unique connection satisfying the following:

	<math>(f^*\nabla)_X(s) = ~~p \mapsto~~ f^~~{-1}~~(\nabla_{(Df)(X~~(p)~~)}(f(s)))</math>		<math>(f^\nabla)_X(f^s) = f^*(\nabla_{(Df)(X)}(s))</math>

	This is to be understood as follows. ~~Starting~~ with a ~~point~~ <math>p \in N</math>, a ~~tangent~~ vector <math>X~~(p)~~</math> at <math>p</math>, ~~and a section~~ <math>s</math> of <math>f^*(E)</math> ~~we proceed as follows:~~		This is to be understood as follows. Start with a section <math>s \in \Gamma(E)</math>. Take the pullback of <math>s</math> to get a section <math>f^s \in \Gamma(f^E)</math>. Then, given a vector field <math>X</math> on <math>N</math>, <math>(f^\nabla)_X</math> should send <math>f^s</math> to the pullback via <math>f</math> of <math>\nabla_{Df(x)}(s)</math>.

	* ~~Differentiate~~		==Related facts==

			===Induced connection on submanifold===

			{{further\|[[induced connection on submanifold]]}}

			if <math>M</math> is a [[Riemannian manifold]] and <math>N</math> is a submanifold, then we can use a linear connection on <math>M</math> to induce a linear connection on <math>N</math>. This involves two steps:

			* Pull back the connection on <math>TM</math>, to the connection on the pullback bundle on <math>N</math> namely <math>TM\|_N</math>
			* Project this to the connection on <math>TN</math>, using the inner product structure on <math>TM\|_N</math>

Vipul: New page: ==Definition== Suppose $f:N \to M$ is a smooth map between differential manifolds $N$ and $M$ . Let $E$ be a vector bundle over
2008-04-12T23:40:48Z

New page: ==Definition== Suppose <math>f:N \to M</math> is a smooth map between differential manifolds <math>N</math> and <math>M</math>. Let <math>E</math> be a vector bundle over <mat...

New page
==Definition==

Suppose <math>f:N \to M</math> is a [[smooth map]] between [[differential manifold]]s <math>N</math> and <math>M</math>. Let <math>E</math> be a [[vector bundle]] over <math>M</math>, and <math>f^E</math> denote the [[pullback of a vector bundle|pullback]] of <math>E</math> via <math>f</math> (hence, <math>f^E</math> is a vector bundle over <math>N</math>).

Given a [[connection]] <math>\nabla</math> for the vector bundle <math>E</math>, we can define a connection <math>f^(\nabla)</math> for the vector bundle <math>f^(E)</math>, called the '''pullback''' of <math>\nabla</math>, as follows:

<math>(f^\nabla)_X(s) = p \mapsto f^{-1}(\nabla_{(Df)(X(p))}(f(s)))</math>

This is to be understood as follows. Starting with a point <math>p \in N</math>, a tangent vector <math>X(p)</math> at <math>p</math>, and a section <math>s</math> of <math>f^(E)</math> we proceed as follows:

* Differentiate