There are a whole bunch of theorems in Riemannian geometry to the effect that “if the Riemannian manifold ${M}$ has property A of the curvature, then it has the topological property B.” Over the rest of MaBloWriMo and in the following weeks, I aim to talk about a few such results. The first one characterizes manifolds of negative curvature.

Negative curvature

Let ${M}$ be a Riemannian manifold with Riemannian metric ${g}$ Say that ${M}$ has negative curvature if for all ${p \in M, X,Y \in T_p(M)}$,

$\displaystyle g( R(X,Y)Y, X) \leq 0.$

(Later I will interpret this in terms of the sectional curvature, which I have not yet defined.)

Statements

Theorem 1 (Cartan-Hadamard) Let ${M}$ be a complete Riemannian manifold of negative curvature. Then for ${p \in M}$, the map ${\exp_p: T_p(M) \rightarrow M}$ is a covering map. In particular, if ${M}$ is simply connected, then it is diffeomorphic to ${\mathbb{R}^n}$.

Of course, the diffeomorphism doesn’t have to preserve the Riemannian metric.

The strategy of the proof is as follows. First, we will show that the map ${\exp_p}$ is an immersion (though in general not injective), using the discussion yesterday about how Jacobi fields determine the differential of the exponential map. Then we will invoke

Proposition 2 Let ${M}$ be a complete Riemannian manifold. Suppose ${p \in M}$ and the map ${\exp_p}$ is an immersion. Then ${\exp_p}$ is a covering map.

The condition of the result is often stated to the effect that “${M}$ has no conjugate points to ${p}$.”

To see this, we will appeal to yet another result:

Theorem 3 (Ambrose) Let ${f: M \rightarrow N}$ be a surjective morphisms of Riemannian manifolds with ${M}$ complete. Suppose ${f}$ preserves the metric on the tangent spaces. Then ${f}$ is a covering map.

I will work backwards to prove these three results. (more…)

Ok, yesterday I covered the basic fact that given a Riemannian manifold ${(M,g)}$, the geodesics on ${M}$ (with respect to the Levi-Civita connection) locally minimize length. Today I will talk about the phenomenon of “geodesic completeness.”

Henceforth, all manifolds are assumed connected.

The first basic remark to make is the following. If ${c: I \rightarrow M}$ is a piecewise ${C^1}$-path between ${p,q}$ and has the smallest length among piecewise ${C^1}$ paths, then ${c}$ is, up to reparametrization, a geodesic (in particular smooth). The way to see this is to pick ${a,b \in I}$ very close to each other, so that ${c([a,b])}$ is contained in a neighborhood of ${c\left( \frac{a+b}{2}\right)}$ satisfying the conditions of yesterday’s theorem; then ${c|_{[a,b]}}$ must be length-minimizing, so it is a geodesic. We thus see that ${c}$ is locally a geodesic, hence globally.

Say that ${M}$ is geodesically complete if ${\exp}$ can be defined on all of ${TM}$; in other words, a geodesic ${\gamma}$ can be continued to ${(-\infty,\infty)}$. The name is justified by the following theorem:

Theorem 1 (Hopf-Rinow)

The following are equivalent:

• ${M}$ is geodesically complete.
• In the metric ${d}$ on ${M}$ induced by ${g}$ (see here), ${M}$ is a complete metric space (more…)

Time to go back to basic algebraic number theory (which we’ll need for two of my future aims here: class field theory and modular representation theory), and to throw in a few more facts about absolute values and completions—as we’ll see, extensions in the complete case are always unique, so this simplifies dealing with things like ramification. Since ramification isn’t affected by completion, we can often reduce to the complete case.

Absolute Values

Henceforth, all absolute values are nontrivial—we don’t really care about the absolute value that takes the value one everywhere except at zero.

I mentioned a while back that absolute values on fields determine a topology. As it turns out, there is essentially a converse.

Theorem 1 Let ${\left|\cdot\right|_1}$, ${\left|\cdot\right|_2}$ be absolute values on ${K}$ inducing the same topology. Then ${\left|\cdot\right|_2}$ is a power of ${\left|\cdot\right|_1}$  (more…)