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: 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 p ∈ M {displaystyle pin M} by where γ p , v {displaystyle gamma _{p,v}} is the unique geodesic with γ p , v ( 0 ) = p {displaystyle gamma _{p,v}(0)=p} and tangent γ p , v ′ ( 0 ) = v ∈ T p M {displaystyle gamma _{p,v}'(0)=vin T_{p}M} and ϵ {displaystyle epsilon } is chosen small enough so that for every v ∈ B ϵ ( 0 ) ⊂ T p M {displaystyle vin B_{epsilon }(0)subset T_{p}M} the geodesic γ p , v {displaystyle gamma _{p,v}} is defined in 1. So, if M {displaystyle M} is complete, then, by the Hopf–Rinow theorem, exp p {displaystyle exp _{p}} is defined on the whole tangent space. Let α : I → T p M {displaystyle alpha :I ightarrow T_{p}M} be a curve differentiable in T p M {displaystyle T_{p}M} such that α ( 0 ) := 0 {displaystyle alpha (0):=0} and α ′ ( 0 ) := v {displaystyle alpha '(0):=v} . Since T p M ≅ R n {displaystyle T_{p}Mcong mathbb {R} ^{n}} , it is clear that we can choose α ( t ) := v t {displaystyle alpha (t):=vt} . In this case, by the definition of the differential of the exponential in 0 {displaystyle 0} applied over v {displaystyle v} , we obtain: So (with the right identification T 0 T p M ≅ T p M {displaystyle T_{0}T_{p}Mcong T_{p}M} ) the differential of exp p {displaystyle exp _{p}} is the identity. By the implicit function theorem, exp p {displaystyle exp _{p}} is a diffeomorphism on a neighborhood of 0 ∈ T p M {displaystyle 0in T_{p}M} . The Gauss Lemma now tells that exp p {displaystyle exp _{p}} is also a radial isometry. Let p ∈ M {displaystyle pin M} . In what follows, we make the identification T v T p M ≅ T p M ≅ R n {displaystyle T_{v}T_{p}Mcong T_{p}Mcong mathbb {R} ^{n}} . Gauss's Lemma states: Let v , w ∈ B ϵ ( 0 ) ⊂ T v T p M ≅ T p M {displaystyle v,win B_{epsilon }(0)subset T_{v}T_{p}Mcong T_{p}M} and M ∋ q := exp p ⁡ ( v ) {displaystyle M i q:=exp _{p}(v)} . Then, ⟨ T v exp p ⁡ ( v ) , T v exp p ⁡ ( w ) ⟩ q = ⟨ v , w ⟩ p . {displaystyle langle T_{v}exp _{p}(v),T_{v}exp _{p}(w) angle _{q}=langle v,w angle _{p}.} For p ∈ M {displaystyle pin M} , this lemma means that exp p {displaystyle exp _{p}} is a radial isometry in the following sense: let v ∈ B ϵ ( 0 ) {displaystyle vin B_{epsilon }(0)} , i.e. such that exp p {displaystyle exp _{p}} is well defined. And let q := exp p ⁡ ( v ) ∈ M {displaystyle q:=exp _{p}(v)in M} . Then the exponential exp p {displaystyle exp _{p}} remains an isometry in q {displaystyle q} , and, more generally, all along the geodesic γ {displaystyle gamma } (in so far as γ ( 1 , p , v ) = exp p ⁡ ( v ) {displaystyle gamma (1,p,v)=exp _{p}(v)} is well defined)! Then, radially, in all the directions permitted by the domain of definition of exp p {displaystyle exp _{p}} , it remains an isometry.