Connection along a curve: Difference between revisions
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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> | ||
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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> | ||
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. | ||
Latest revision as of 19:35, 18 May 2008
Definition
Let be a differential manifold and be a vector bundle over . Let be a smooth curve in . A connection along , of , is defined as follows: it is a map from the space of sections of along , to itself, such that:
![{\displaystyle \gamma :[0,1]\to M}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0066953642fb00abb394327531cea098815cd1c8)
and for we have:
![{\displaystyle f:[0,1]\to \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/2de6d0d4c98d4ca7ad937c772dc3e3e914b062f5)
where is usual real differentiation.
Facts
Connection gives connection along a curve
Given a connection on the whole vector bundle , we can obtain a connection along the curve . Simply define:
where is the tangent vector to at . This can also be viewed as the pullback connection for the map (which we might restrict to the open interval , 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.