Pullback of connection on a vector bundle: Difference between revisions

Latest revision as of 19:51, 18 May 2008

Definition

Suppose $f:N\to M$ is a smooth map between differential manifolds $N$ and $M$ . Let $E$ be a vector bundle over $M$ , and $f^{*}E$ denote the pullback of $E$ via $f$ (hence, $f^{*}E$ is a vector bundle over $N$ ).

Given a connection $\nabla$ for the vector bundle $E$ , we can define a connection $f^{*}(\nabla )$ for the vector bundle $f^{*}(E)$ , called the pullback of $\nabla$ , as the unique connection satisfying the following:

$(f^{*}\nabla )_{X}(f^{*}s)=f^{*}(\nabla _{(Df)(X)}(s))$

This is to be understood as follows. Start with a section $s\in \Gamma (E)$ . Take the pullback of $s$ to get a section $f^{*}s\in \Gamma (f^{*}E)$ . Then, given a vector field $X$ on $N$ , $(f^{*}\nabla )_{X}$ should send $f^{*}s$ to the pullback via $f$ of $\nabla _{Df(x)}(s)$ .

Related facts

Induced connection on submanifold

Further information: induced connection on submanifold

if $M$ is a Riemannian manifold and $N$ is a submanifold, then we can use a linear connection on $M$ to induce a linear connection on $N$ . This involves two steps:

Pull back the connection on $TM$ , to the connection on the pullback bundle on $N$ namely $TM|_{N}$
Project this to the connection on $TN$ , using the inner product structure on $TM|_{N}$

Connection along a curve

Further information: 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 $(0,1)$ .

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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>).
-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>
+===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>.