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The Hopf-Rinow theorems and geodesic completeness
*November 14, 2009*

*Posted by Akhil Mathew in differential geometry, MaBloWriMo.*

Tags: completeness, geodesic completeness, geodesics, Hopf-Rinow theorem, Riemannian manifolds

10 comments

Tags: completeness, geodesic completeness, geodesics, Hopf-Rinow theorem, Riemannian manifolds

10 comments

Ok, yesterday I covered the basic fact that given a Riemannian manifold , the geodesics on (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 is a piecewise -path between and has the smallest length among piecewise paths, then is, up to reparametrization, a geodesic (in particular smooth). The way to see this is to pick very close to each other, so that is contained in a neighborhood of satisfying the conditions of yesterday’s theorem; then must be length-minimizing, so it is a geodesic. We thus see that is locally a geodesic, hence globally.

Say that is **geodesically complete** if can be defined on all of ; in other words, a geodesic can be continued to . The name is justified by the following theorem:

Theorem 1 (Hopf-Rinow)The following are equivalent:

- is geodesically complete.
- In the metric on induced by (see here), is a complete metric space (more…)

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Topologies determine the absolute value
*October 5, 2009*

*Posted by Akhil Mathew in algebra, algebraic number theory, number theory.*

Tags: absolute values, completeness, norms, topologies

1 comment so far

Tags: absolute values, completeness, norms, topologies

1 comment so far

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 1Let , be absolute values on inducing the same topology. Then is a power of .(more…)