Mathematical Proof: Does Euclidian Geometry Hold in Riemannian?

[Patrick Sossoumihen]

Some of proofs without words have been described using Euclidian Geometry, do those proofs still hold alike in Riemmanian Geometry?

 

[Buzz Bloom]

Some of proofs without words have been described using Euclidian Geometry, do those proofs still hold alike in Riemmanian Geometry?

Hi Patrick:

Riemannian space with zero curvature is the same as Euclidean space (at least for 2 and 3 dimensions). So in principle any Euclidean Geometry proof would also be valid in a Riemannian space with a zero curvature tensor. It would likely not be valid in a Riemannian space with non-zero curvature.

If you can post some examples of particular Euclidean Geometry proofs, I might be able to offer a more specific answer.

Hope this helps.

Regards,
Buzz