Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such ...

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Studying the family of curves $\beta(s,r) = \alpha(s) + r\,{\bf N}(s)$

I'm reviewing some stuff on plane curves, just because, and I would like to confirm some things. The whole exercise is: Let $\alpha(s) = (x(s),y(s))$ be a unit-speed parametrized curve, ${\bf ...
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Accounting for signs in divergence thm. on Lorentzian manifold

I am trying to learn about integration in Lorentzian manifolds (I will use signature -+++) and have some problems. Oft quoted (in books for GR) form of divergence theorem is: $\int _U div( X ...