There are some more useful facts about isometries that I want to gather together in this post.

**Isometries are determined by one tangent space **

Unlike smooth maps, isometries have to be much more rigid:

Proposition 1Let be isometries such that and . Then .

(Recall that we’re assuming all manifolds to be connected.)

The idea is that we can take local “exponential coordinates” around , respectively. Then as I showed yesterday, and are necessarily given by a linear map in these coordinates. Since on , this means the two linear maps coincide, so and are locally equal. So we can let be the set of with on . is evidently closed, and it is open by the argument just given, so by connectedness we get the result.

**Continuation of isometries **

Because of the previous result, it is possible to talk about “analytic continuation” of an isometry. So let be a curve, and let be an isometry of a neighborhood of into (i.e. onto a submanifold thereof). Suppose we have a family of isometries , , of isometries of neighborhoods into , with , satisfying the following condition: *Whenever are close enough, and .* This is exactly analogous to the treatment of analytic continuation of holomorphic functions, and I have to wonder if there is a general framework using fancy techniques in category theory or something like that.

The first basic result is that this is unique:

Proposition 2If are continuations of along the curve , then for all , in some neighborhood of .

Indeed, the proof is now a direct corollary of the previous result: take the set of where the statement of the proposition is true, and it follows that said set is open, closed, and nonempty.

**Existence of continuations **

As in complex analysis, it is possible to perform a continuation, but one needs the assumptions of completeness and analyticity:

Proposition 3If is a complete Riemannian manifold, then any (analytic) isometry can be continued along the curve .

The curve doesn’t have to be analytic.

The idea is that we only need to do this one small piece at a time. The small pieces rely on the following lemma, which allows us to extend on normal neighborhoods:

Lemma 4Let be a disk around with respect to the metric on , and let . Suppose is the diffeomorphic image of an open subset of under the exponential map (which is true for small enough). Then an (analytic) isometry of into can be extended to an isometry of into .

Such a is called a **normal neighborhood** of . The normal neighborhood theorem, which I have not yet proven, states that there is a basis consisting open sets which are normal neighborhoods of each of their points.

Since the map is an isometry, it sends geodesics to geodesics (cf. yesterday’s post). Now let be the linear transformation . The map can be described as follows: it maps to . This follows from the previous discussions. Moreover, by the description it offers a way of defining on since each element of can be written as for a *unique* . The fact that this is an isometry follows because everything here (especially if are the metrics on ) is real-analytic, and is an isometry on an open subset of .

Now let’s prove the existence proposition.

Consider the set of such that an analytic continuation of on exists. It is open and nonempty. I claim it is closed.

Let , and suppose . There is a curve defined on , well-defined by uniqueness. It tends to a limit as by completeness and since is an isometry.

Let be a neighborhood of which is a normal neighborhood of each of its points. Then there is very close to with and a normal neighborhood of with an isometry into via continuation. This extends to by the lemma, which gives the continuation to a neighborhood of . This is a contradiction since was assumed to be the supremum.

**Independence up to homotopy **

Proposition 5Let be homotopic paths (i.e. with , and the homotopy respecting endpoints). Then the continuations of along agree in a neighborhood of .

The proof is similar to the one for analytic continuation.

**The Myers-Rinow Theorem **

The following theorem—and its proof—are strikingly analogous to the monodromy theorem:

Theorem 6 (Myers-Rinow)Let be analytic, simply connected, complete manifolds and an (analytic) isometry between an open subset of and an open subset of . Then can be extended to an isometry of onto .

The idea is the same as the monodromy theorem: to define the extension on by starting with a with defined, drawing a curve from to , and continuing along ; this can be used to define in a neighborhood of . Simple connectivity shows that the choice of is irrelevant. It is easy to check that is smooth locally by comparing paths with catenated with a small arc between and, say, close to .

We can do the same for and get an isometry . Now on open subsets of ; thus this is true everywhere, and the extensions are diffeomorphisms. We already know they preserve the inner product.

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