How Does Constant Sectional Curvature Relate to Locally Symmetric Spaces?

[JasonJo]

Curvature Questions, Please Help!

Homework Statement

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.

Homework Equations

R(X, Y) Z = constant along any geodesic, i.e. it is a parallel vector field.

The Attempt at a Solution

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!

 

[mighty2000]

Hello,

1) To prove that M has constant sectional curvature, we can use the fact that the Riemann tensor is constant. This means that Riem(X,Y)Z = c for all X,Y,Z at any point on M. Since M is 2-dimensional, the sectional curvature coincides with the Riemann curvature. Therefore, the sectional curvature is also constant.

To answer your question, M being connected is important because it ensures that any two points can be connected by a curve and therefore, the sectional curvature does not change along any curve.

2) To prove that M is a locally symmetric space, we can use the fact that M has constant sectional curvature. This means that the Riemann curvature is constant along any geodesic. Since M is 2-dimensional, the sectional curvature coincides with the Riemann curvature. Therefore, the Riemann curvature is constant along any geodesic. This implies that the Riemann tensor is symmetric. Hence, M is a locally symmetric space.

As for your question, constant sectional curvature means that the sectional curvature does not depend on the 2-dimensional space and it does not change along any curve.

I hope this helps. Let me know if you have any further questions.