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.

Introduction

We define the exponential map at p\in M by where is the unique geodesic with and tangent and \epsilon is chosen small enough so that for every the geodesic is defined. So, if M is complete, then, by the Hopf–Rinow theorem, \exp_p is defined on the whole tangent space. Let be a curve differentiable in T_pM such that and. Since, it is clear that we can choose. In this case, by the definition of the differential of the exponential in 0 applied over v, we obtain: So (with the right identification ) the differential of \exp_p is the identity. By the implicit function theorem, \exp_p is a diffeomorphism on a neighborhood of 0 \in T_pM. The Gauss Lemma now tells that \exp_p is also a radial isometry.

The exponential map is a radial isometry

Let p\in M. In what follows, we make the identification. Gauss's Lemma states: Let and. Then, For p\in M, this lemma means that \exp_p is a radial isometry in the following sense: let, i.e. such that \exp_p is well defined. And let. Then the exponential \exp_p remains an isometry in q, and, more generally, all along the geodesic \gamma (in so far as is well defined)! Then, radially, in all the directions permitted by the domain of definition of \exp_p, it remains an isometry.

Proof

Recall that We proceed in three steps: such that and. Since, we can put. Therefore, where \Gamma is the parallel transport operator and. The last equality is true because \gamma is a geodesic, therefore \gamma' is parallel. Now let us calculate the scalar product. We separate w into a component w_T parallel to v and a component w_N normal to v. In particular, we put w_T:=a v,. 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 t, and therefore that, for example: because, according to what has been given above: being given that the differential is a linear map. This will therefore prove the lemma. Since the maps are geodesics, the function is constant. Thus,