Induced connection on submanifold - Revision history

Vipul: 2 revisions

2008-05-18T19:47:13Z

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Vipul at 14:23, 13 April 2008

2008-04-13T14:23:16Z

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	Suppose <math>M</math> is a [[Riemannian manifold]], <math>N</math> is a submanifold. Then <math>TN</math> is a subbundle of the bundle <math>TM\|_N</math>. The Riemannian metric on <math>M</math> naturally gives a notion of "projection" from <math>TM\|_N</math> to <math>TN</math>.		Suppose <math>M</math> is a [[Riemannian manifold]], <math>N</math> is a submanifold. Then <math>TN</math> is a subbundle of the bundle <math>TM\|_N</math>. The Riemannian metric on <math>M</math> naturally gives a notion of "projection" from <math>TM\|_N</math> to <math>TN</math>.

	Then, given any linear connection <math>\nabla</math> on <math>M</math>, we obtain an induced connection <math>\nabla\|_N</math> on <math>N</math>. The induced connection is defined as follows. For <math>X,Y \in \Gamma(TM)</math>, we have:		Then, given any [[linear connection]] <math>\nabla</math> on <math>M</math>, we obtain an induced connection <math>\nabla\|_N</math> on <math>N</math>. The induced connection is defined as follows. For <math>X,Y \in \Gamma(TM)</math>, we have:

	<math>(\nabla\|_N)_X Y := (\nabla_XY)^{tan}</math>		<math>(\nabla\|_N)_X Y := (\nabla_XY)^{tan}</math>

	The definition ~~packs the following steps~~:		The definition has two key parts:

	* First, using the [[tubular neighborhood theorem]], consider a tubular neighborhood <math>U \supset N</math> in <math>M</math>		# We use a connection on <math>TM</math> to obtain a connection on <math>TM\|_N</math>. This can be done either by using the general notion of a [[pullback connection]], or by the following more concrete process: First, using the [[tubular neighborhood theorem]], consider a tubular neighborhood <math>U \supset N</math> in <math>M</math>. Extend <math>X,Y</math> to vector fields on <math>U</math> by {{fillin}}. Evaluate <math>\nabla_XY</math> on this open set <math>U</math>, to get a vector field on <math>U</math>. Restrict the vector field to <math>N</math>. This gives a section of the bundle <math>TM\|_N</math>
	* Extend <math>X,Y</math> to vector fields on <math>U</math> by {{fillin}}		# We project this from <math>TM\|_N</math> to <math>TN</math>, using the Riemannian metric
	* Evaluate <math>\nabla_XY</math> on this open set <math>U</math>, to get a vector field on <math>U</math>
	* Restrict the vector field to <math>N</math>. This gives a section of the bundle <math>TM\|_N</math>
	* Project this from <math>TM\|_N</math> to <math>TN</math>

	Crudely speaking, we use the connection on <math>M</math>, and project the output vector field we get, onto <math>TN</math>.		Crudely speaking, we use the connection on <math>M</math>, and project the output vector field we get, onto <math>TN</math>.

Vipul: New page: ==Definition== ===For a submanifold of a Riemannian manifold=== Suppose $M$ is a Riemannian manifold, $N$ is a submanifold. Then $TN$ is a subbundle ...

2008-04-10T23:06:49Z

New page: ==Definition== ===For a submanifold of a Riemannian manifold=== Suppose <math>M</math> is a Riemannian manifold, <math>N</math> is a submanifold. Then <math>TN</math> is a subbundle ...

New page

==Definition==

===For a submanifold of a Riemannian manifold===

Suppose <math>M</math> is a [[Riemannian manifold]], <math>N</math> is a submanifold. Then <math>TN</math> is a subbundle of the bundle <math>TM|_N</math>. The Riemannian metric on <math>M</math> naturally gives a notion of "projection" from <math>TM|_N</math> to <math>TN</math>.

Then, given any linear connection <math>\nabla</math> on <math>M</math>, we obtain an induced connection <math>\nabla|_N</math> on <math>N</math>. The induced connection is defined as follows. For <math>X,Y \in \Gamma(TM)</math>, we have:

<math>(\nabla|_N)_X Y := (\nabla_XY)^{tan}</math>

The definition packs the following steps:

* First, using the [[tubular neighborhood theorem]], consider a tubular neighborhood <math>U \supset N</math> in <math>M</math>
* Extend <math>X,Y</math> to vector fields on <math>U</math> by {{fillin}}
* Evaluate <math>\nabla_XY</math> on this open set <math>U</math>, to get a vector field on <math>U</math>
* Restrict the vector field to <math>N</math>. This gives a section of the bundle <math>TM|_N</math>
* Project this from <math>TM|_N</math> to <math>TN</math>

Crudely speaking, we use the connection on <math>M</math>, and project the output vector field we get, onto <math>TN</math>.

Induced connection on submanifold - Revision history

Vipul: 2 revisions

Vipul at 14:23, 13 April 2008

Vipul: New page: ==Definition== ===For a submanifold of a Riemannian manifold=== Suppose M is a Riemannian manifold, N is a submanifold. Then TN is a subbundle ...

Vipul: New page: ==Definition== ===For a submanifold of a Riemannian manifold=== Suppose $M$ is a Riemannian manifold, $N$ is a submanifold. Then $TN$ is a subbundle ...