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Manifolds with a non-degenerate symmetric bilinear form in each tangent space varying differentiably but with constant index and signature.

5 votes

Is it possible to do calculus and differential geometry the old school way, without any orth...

It's possible to do differential geometry in a purely intrinsic way, at least once you've gotten past the initial hurdle of defining what a manifold is. The standard definition of a manifold is a seco …
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