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Classical theorems

Major theorems like the Hopf-Rinow theorem and the Cartan-Hadamard theorem.
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The statement of the theorem

Let (M,g)(M, g) be a complete Riemannian manifold, where MM is a smooth manifold and gg is a Riemannian metric tensor on MM. Let p be a fixed point in Mp \text{ be a fixed point in } M, and TpMT_p M be the tangent space at pp. The exponential map expp:TpM→M\text{exp}_p: T_p M \to M is defined by mapping a tangent vector v to the point reached by the unique geodesic expp(v)=geod(t)∣t=1 starting at p with initial velocity vv \text{ to the point reached by the unique geodesic } \text{exp}_p(v) = \text{geod}(t)|_{t=1} \text{ starting at } p \text{ with initial velocity } v. Classical theorems often establish conditions under which expp\text{exp}_p is a local diffeomorphism, or when the manifold is globally isometric to a simpler space. Specifically, the Cartan-Hadamard theorem states that if (M,g)(M, g) is a complete, simply connected manifold with non-positive sectional curvature K (i.e., K is negative or zero everywhere),K \text{ (i.e., } K \text{ is negative or zero everywhere)}, then the exponential map expp:TpM→M\text{exp}_p: T_p M \to M is a global diffeomorphism, implying that (M,g)(M, g) is globally isometric to a Hadamard manifold, which is a simply connected, complete manifold with non-positive sectional curvature.
Source: Wikipedia