Connection along a curve - Revision history

Vipul: 6 revisions

2008-05-18T19:35:15Z

6 revisions

← Older revision	Revision as of 19:35, 18 May 2008
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Vipul: /* Connection gives connection along a curve */

2008-04-13T14:29:01Z

Connection gives connection along a curve

← Older revision		Revision as of 14:29, 13 April 2008
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	<math>DV/dt = \nabla_{\gamma'(t)}V</math>		<math>DV/dt = \nabla_{\gamma'(t)}V</math>

	where <math>\gamma'(t)</math> is the tangent vector to <math>\gamma</math> at <math>\gamma(t)</math>.		where <math>\gamma'(t)</math> is the tangent vector to <math>\gamma</math> at <math>\gamma(t)</math>. This can also be viewed as the [[pullback connection]] for the map <math>\gamma</math> (which we might restrict to the open interval <math>(0,1)</math>, for convenience).

	However, not every connection along a curve arises from a connection. A particular case is self-intersecting curves. We can construct connections along self-intersecting curves that behave very differently for the same point at two different times, and hence cannot arise from a connection.		However, not every connection along a curve arises from a connection. A particular case is self-intersecting curves. We can construct connections along self-intersecting curves that behave very differently for the same point at two different times, and hence cannot arise from a connection.

Vipul: /* Connection gives connection along a curve */

2008-04-13T14:27:00Z

Connection gives connection along a curve

← Older revision		Revision as of 14:27, 13 April 2008
Line 19:		Line 19:
	<math>DV/dt = \nabla_{\gamma'(t)}V</math>		<math>DV/dt = \nabla_{\gamma'(t)}V</math>

	where <math>\gamma'(t)</math> is the tangent vector to <math>\gamma<math> at <math>\gamma(t)</math>.		where <math>\gamma'(t)</math> is the tangent vector to <math>\gamma</math> at <math>\gamma(t)</math>.

	However, not every connection along a curve arises from a connection. A particular case is self-intersecting curves. We can construct connections along self-intersecting curves that behave very differently for the same point at two different times, and hence cannot arise from a connection.		However, not every connection along a curve arises from a connection. A particular case is self-intersecting curves. We can construct connections along self-intersecting curves that behave very differently for the same point at two different times, and hence cannot arise from a connection.

Vipul at 00:33, 13 April 2008

2008-04-13T00:33:57Z

← Older revision		Revision as of 00:33, 13 April 2008
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	==Definition==		==Definition==

	Let <math>M</math> be a [[differential manifold]] and <math>E</math> be a [[vector bundle]] over <math>M</math>. Let <math>\gamma:[0,1] \to M</math> be a [[smooth curve]] in <math>M</math>. A [[connection]] along <math>\gamma<math>, of <math>E</math>, is defined as follows: it is a map <math>D/dt</math> from the space of [[section of a vector bundle along a curve\|section]]s of <math>E</math> along <math>\gamma</math>, to itself, such that:		Let <math>M</math> be a [[differential manifold]] and <math>E</math> be a [[vector bundle]] over <math>M</math>. Let <math>\gamma:[0,1] \to M</math> be a [[smooth curve]] in <math>M</math>. A [[connection]] along <math>\gamma</math>, of <math>E</math>, is defined as follows: it is a map <math>D/dt</math> from the space of [[section of a vector bundle along a curve\|section]]s of <math>E</math> along <math>\gamma</math>, to itself, such that:

	<math>DV/dt + DW/dt = D(V + W)/dt</math>		<math>DV/dt + DW/dt = D(V + W)/dt</math>

Vipul: /* Connection gives connection along a curve */

2007-09-01T13:19:51Z

Connection gives connection along a curve

← Older revision		Revision as of 13:19, 1 September 2007
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	===Connection gives connection along a curve===		===Connection gives connection along a curve===

	Given a [[connection]] on the whole vector bundle <math>E</math>, we can obtain a connection along the curve <math>\gamma<math>. Simply define:		Given a [[connection]] on the whole vector bundle <math>E</math>, we can obtain a connection along the curve <math>\gamma</math>. Simply define:

	<math>DV/dt = \nabla_{\gamma'(t)}V</math>		<math>DV/dt = \nabla_{\gamma'(t)}V</math>

Vipul: /* Definition */

2007-09-01T13:19:25Z

Definition

← Older revision		Revision as of 13:19, 1 September 2007
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	==Definition==		==Definition==

	Let <math>M</math> be a [[differential manifold]] and <math>E</math> be a [[vector bundle]] over <math>M</math>. Let <math>\gamma:[0,1] \to M</math> be a [[smooth curve]] in <math>M</math>. A [[connection]] along <math>\gamma<math>, of <math>E</math>, is defined as follows: it is a map <math>D/dt</math> from the space of [[section of a vector bundle along a curve\|section]]s of <math>E</math> along <math>\~~gammma~~</math>, to itself, such that:		Let <math>M</math> be a [[differential manifold]] and <math>E</math> be a [[vector bundle]] over <math>M</math>. Let <math>\gamma:[0,1] \to M</math> be a [[smooth curve]] in <math>M</math>. A [[connection]] along <math>\gamma<math>, of <math>E</math>, is defined as follows: it is a map <math>D/dt</math> from the space of [[section of a vector bundle along a curve\|section]]s of <math>E</math> along <math>\gamma</math>, to itself, such that:

	<math>DV/dt + DW/dt = D(V + W)/dt</math>		<math>DV/dt + DW/dt = D(V + W)/dt</math>

Vipul at 13:18, 1 September 2007

2007-09-01T13:18:40Z

New page

==Definition==

Let <math>M</math> be a [[differential manifold]] and <math>E</math> be a [[vector bundle]] over <math>M</math>. Let <math>\gamma:[0,1] \to M</math> be a [[smooth curve]] in <math>M</math>. A [[connection]] along <math>\gamma<math>, of <math>E</math>, is defined as follows: it is a map <math>D/dt</math> from the space of [[section of a vector bundle along a curve|section]]s of <math>E</math> along <math>\gammma</math>, to itself, such that:

<math>DV/dt + DW/dt = D(V + W)/dt</math>

and for <math>f:[0,1] \to \R</math> we have:

<math>D(fV)/dt = fDV/dt + V df/dt</math>

where <math>df/dt</math> is usual real differentiation.

==Facts==

===Connection gives connection along a curve===

Given a [[connection]] on the whole vector bundle <math>E</math>, we can obtain a connection along the curve <math>\gamma<math>. Simply define:

<math>DV/dt = \nabla_{\gamma'(t)}V</math>

where <math>\gamma'(t)</math> is the tangent vector to <math>\gamma<math> at <math>\gamma(t)</math>.

However, not every connection along a curve arises from a connection. A particular case is self-intersecting curves. We can construct connections along self-intersecting curves that behave very differently for the same point at two different times, and hence cannot arise from a connection.