Gauss's lemma (Riemannian geometry)

In Riemannian geometry, Gauss's lemma asserts that any sufficiently small sphere centered at a point in a Riemannian manifold is perpendicular to every geodesic through the point.

More formally, let M be a Riemannian manifold, equipped with its Levi-Civita connection, and p a point of M. The exponential map is a mapping from the tangent space at p to M: which is a diffeomorphism in a neighborhood of zero.

Gauss' lemma asserts that the image of a sphere of sufficiently small radius in TpM under the exponential map is perpendicular to all geodesics originating at p. The lemma allows the exponential map to be understood as a radial isometry, and is of fundamental importance in the study of geodesic convexity and normal coordinates.

We define the exponential map at

is the unique geodesic with

is chosen small enough so that for every

is complete, then, by the Hopf–Rinow theorem,

is defined on the whole tangent space.

be a curve differentiable in

, it is clear that we can choose

In this case, by the definition of the differential of the exponential in

, we obtain: So (with the right identification

By the implicit function theorem,

is a diffeomorphism on a neighborhood of

The Gauss Lemma now tells that

is also a radial isometry.

In what follows, we make the identification

Gauss's Lemma states: Let

is a radial isometry in the following sense: let

remains an isometry in

Then, radially, in all the directions permitted by the domain of definition of

, it remains an isometry.

We proceed in three steps:

is the parallel transport operator and

The last equality is true because

Now let us calculate the scalar product

The preceding step implies directly: We must therefore show that the second term is null, because, according to Gauss's Lemma, we must have:

Let us define the curve Note that Let us put: and we calculate: and Hence We can now verify that this scalar product is actually independent of the variable

, and therefore that, for example: because, according to what has been given above: being given that the differential is a linear map.

are geodesics, the function