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Calculus and Beyond Homework Help
Solving Curvature Questions: Please Help!
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[QUOTE="JasonJo, post: 1960829, member: 3936"] [b]Curvature Questions, Please Help![/b] [h2]Homework Statement [/h2] 1) Prove that if M is locally symmetric (i.e. the Riemann tensor is constant), connected and 2 dimensional, then M has constant sectional curvature. 2) Prove that if M has constant (sectional) curvature, then M is a locally symmetric space. [h2]Homework Equations[/h2] R(X, Y) Z = constant along any geodesic, i.e. it is a parallel vector field. [h2]The Attempt at a Solution[/h2] For the first part: Since M is two dim, the sectional curvature coincides with the actual curvature. Why do we need that M is connected? Am I suppose to use that the sectional curvature (hence the Riemann curvature) does not change along geodesics? * What exactly does constant sectional curvature mean? Does it mean that the sectional curvature K does not depend on the 2-dimensional space and that it does not change along any curve? For the second part: Not quite sure about this part. I see the idea and the picture, but what is the first step? Thanks guys! [/QUOTE]
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Solving Curvature Questions: Please Help!
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