Difference between revisions of "Curvature of direct sum of connections equals direct sum of curvatures"

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(Created page with '==Statement== Suppose <math>E,E'</math> are vector bundles over a differential manifold <math>M</math>. Suppose <math>\nabla,\nabla'</math> are connections on <math>…')
 
 
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Suppose <math>E,E'</math> are [[vector bundle]]s over a [[differential manifold]] <math>M</math>. Suppose <math>\nabla,\nabla'</math> are [[connection]]s on <math>E,E'</math> respectively. Let <math>R_\nabla</math> denote the [[fact about::Riemann curvature tensor]] of <math>\nabla</math> and <math>R_\nabla'</math> denote that Riemann curvature tensor of <math>\nabla'</math>. Then if <math>E \oplus E'</math> is the [[direct sum of vector bundles]], <math>\nabla \oplus \nabla'</math> is the [[fact about::direct sum of connections]], and <math>R_{\nabla \oplus \nabla'}</math> is its Riemann curvature tensor, we have:
 
Suppose <math>E,E'</math> are [[vector bundle]]s over a [[differential manifold]] <math>M</math>. Suppose <math>\nabla,\nabla'</math> are [[connection]]s on <math>E,E'</math> respectively. Let <math>R_\nabla</math> denote the [[fact about::Riemann curvature tensor]] of <math>\nabla</math> and <math>R_\nabla'</math> denote that Riemann curvature tensor of <math>\nabla'</math>. Then if <math>E \oplus E'</math> is the [[direct sum of vector bundles]], <math>\nabla \oplus \nabla'</math> is the [[fact about::direct sum of connections]], and <math>R_{\nabla \oplus \nabla'}</math> is its Riemann curvature tensor, we have:
  
<math>R_{\nabla \oplus \nabla'}(X,Y)(s ,s') = \left(R_\nabla(X,Y)(s),R_\nabla(X,Y)s'\right)</math>.
+
<math>R_{\nabla \oplus \nabla'}(X,Y)(s ,s') = \left(R_\nabla(X,Y)(s),R_{\nabla'}(X,Y)s'\right)</math>.

Latest revision as of 22:25, 24 July 2009

Statement

Suppose E,E' are vector bundles over a differential manifold M. Suppose \nabla,\nabla' are connections on E,E' respectively. Let R_\nabla denote the Riemann curvature tensor of \nabla and R_\nabla' denote that Riemann curvature tensor of \nabla'. Then if E \oplus E' is the direct sum of vector bundles, \nabla \oplus \nabla' is the direct sum of connections, and R_{\nabla \oplus \nabla'} is its Riemann curvature tensor, we have:

R_{\nabla \oplus \nabla'}(X,Y)(s ,s') = \left(R_\nabla(X,Y)(s),R_{\nabla'}(X,Y)s'\right).