The Gauss Lemma and the Hopf-Rinow Theorem

Abstract

Fix x ∈ M. In T _x M, we define the tangent spheres

$${S_x}(r): = \{ y \in {T_x}M:F(x,y) = r\} $$

((6.1.1))

and open tangent balls

$${B_x}(r): = \{ y \in {T_x}M:F(x,y) = r\} $$

((6.1.2))

of radii r. The exponential map exp _x is a local diffeomorphism at the origin of T _x M because its derivative there is the identity; see §5.3. Thus, for r small enough, not only does exp _x [S _x (r)] makes sense, it is also diffeomorphic to S _x (r). The image set

$${\exp _x}[{S_x}(r)]$$

is called a geodesic sphere in M centered at x. We later show why it can be said to have radius equal to r.