0

I want to know two things:

1) If the riemann is zero the manifold is necessarily ${R^n}$ and if is true, how can I prove it?

2)Can we have 2 manifolds with the same Riemann tensor?

What I really want to know with these questions is if we can know our manifold only knowing the geodesic deviation.

Victor Alencar
  • 3
  • What would it mean for two tensors defined on two different manifolds to be equal? I think you need to formulate your question a bit more carefully. This question might be of interest. – Anthony Carapetis Feb 20 '18 at 05:55
  • "Equal" for me would be two tensors that in a system of coordinate have the same components. But now that you asked, I realized that my definition of "equal" is not very good. Thx for the coment! – Victor Alencar Feb 20 '18 at 06:37

1 Answers1

0

The Riemann tensor only captures intrinsic curvature. Different manifolds embedded in an ambient space can have the same intrinsic but different extrinsic curvatures. For example, a plane and a cylinder both have zero intrinsic curvature (though different topologies) so their Riemann tensors are both identically zero. But there is obviously a sense in which the cylinder is "curved" while the plane isn't. This is captured by their extrinsic curvature (e.g. their mean curvature) but not by their Riemann tensors.

tparker
  • 6,219