Curvature and Topology

Curvature is essentially a local phenomenon, and topology a global one: it is a most remarkable fact of nature that these two are intricately connected to each other. This connection is best exemplified in the following theorem of Gauss: \begin{equation} \int_S K dA = 2\pi\chi(S) \end{equation} where $S$ is a surface, the term on the left is the integral of the Gauss curvature of the surface, and on the right we have the Euler characteristic $\chi(S)$, which is a topological invariant of $S$. Other connections between curvature and topology are found, for instance, in the Bonnet-Myers theorem (curvature and compactness) and Hadamard’s theorem (spaces of nonpositive sectional curvature).

to be continued

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