The Gauss Lemma and the Hopf-Rinow Theorem

Fix x ∈ M. In T _x M, we define the tangent spheres(6.1.1)$${S_x}(r): = \{ y \in {T_x}M:F(x,y) = r\} $$ and open tangent balls(6.1.2)$${B_x}(r): = \{ y \in {T_x}M:F(x,y) = r\} $$ 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.