HOME

TheInfoList



OR:

In
Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as manifold, smooth manifolds with a ''Riemannian metric'' (an inner product on the tangent space at each point that varies smooth function, smo ...
, Gauss's lemma asserts that any sufficiently small
sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
centered at a point in a
Riemannian manifold In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
is perpendicular to every
geodesic In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a conn ...
through the point. More formally, let ''M'' be a
Riemannian manifold In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
, equipped with its
Levi-Civita connection In Riemannian or pseudo-Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold that preserves the ( pseudo-) Riemannian ...
, and ''p'' a point of ''M''. The exponential map is a mapping from the
tangent space In mathematics, the tangent space of a manifold is a generalization of to curves in two-dimensional space and to surfaces in three-dimensional space in higher dimensions. In the context of physics the tangent space to a manifold at a point can be ...
at ''p'' to ''M'': :\mathrm : T_pM \to M which is a
diffeomorphism In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are continuously differentiable. Definit ...
in a neighborhood of zero. Gauss' lemma asserts that the image of a
sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
of sufficiently small radius in ''T''p''M'' under the exponential map is perpendicular to all
geodesic In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a conn ...
s originating at ''p''. The lemma allows the exponential map to be understood as a radial
isometry In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' me ...
, and is of fundamental importance in the study of geodesic convexity and
normal coordinates In differential geometry, normal coordinates at a point ''p'' in a differentiable manifold equipped with a torsion tensor, symmetric affine connection are a local coordinate system in a neighborhood (mathematics), neighborhood of ''p'' obtained by ...
.


Introduction

We define the exponential map at p\in M by : \exp_p: T_pM\supset B_(0) \longrightarrow M,\quad vt \longmapsto \gamma_(t), where \gamma_ is the unique
geodesic In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a conn ...
with \gamma_(0)=p and tangent \gamma_'(0)=v \in T_pM and \epsilon is chosen small enough so that for every t \in
, 1 The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
vt \in B_(0) \subset T_pM the geodesic \gamma_(t) is defined. So, if M is complete, then, by the Hopf–Rinow theorem, \exp_p is defined on the whole tangent space. Let \alpha : I\rightarrow T_pM be a curve differentiable in T_pM such that \alpha(0):=0 and \alpha'(0):=v. Since T_pM\cong \mathbb R^n, it is clear that we can choose \alpha(t):=vt. In this case, by the definition of the differential of the exponential in 0 applied over v, we obtain: : T_0\exp_p(v) = \frac \Bigl(\exp_p\circ\alpha(t)\Bigr)\Big\vert_ = \frac \Bigl(\exp_p(vt)\Bigr)\Big\vert_=\frac \Bigl(\gamma_(t)\Bigr)\Big\vert_= \gamma_'(0)=v. So (with the right identification T_0 T_p M \cong T_pM) 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 T_vT_pM\cong T_pM\cong \mathbb R^n. Gauss's Lemma states: Let v,w\in B_\epsilon(0)\subset T_vT_pM\cong T_pM and M\ni q:=\exp_p(v). Then, \langle T_v\exp_p(v), T_v\exp_p(w)\rangle_q = \langle v,w\rangle_p. For p\in M, this lemma means that \exp_p is a radial isometry in the following sense: let v\in B_\epsilon(0), i.e. such that \exp_p is well defined. And let q:=\exp_p(v)\in M. Then the exponential \exp_p remains an isometry in q, and, more generally, all along the geodesic \gamma (in so far as \gamma_(1)=\exp_p(v) 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 : T_v\exp_p \colon T_pM\cong T_vT_pM\supset T_vB_\epsilon(0)\longrightarrow T_M. We proceed in three steps: * ''T_v\exp_p(v)=v'' : let us construct a curve \alpha : \mathbb R \supset I \rightarrow T_pM such that \alpha(0):=v\in T_pM and \alpha'(0):=v\in T_vT_pM\cong T_pM. Since T_vT_pM\cong T_pM\cong \mathbb R^n, we can put \alpha(t):=v(t+1). Therefore, T_v\exp_p(v) = \frac\Bigl(\exp_p\circ\alpha(t)\Bigr)\Big\vert_=\frac\Bigl(\exp_p(tv)\Bigr)\Big\vert_=\Gamma(\gamma)_p^v=v, where \Gamma is the parallel transport operator and \gamma(t)=\exp_p(tv). The last equality is true because \gamma is a geodesic, therefore \gamma' is parallel. Now let us calculate the scalar product \langle T_v\exp_p(v), T_v\exp_p(w)\rangle. 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, a \in \mathbb R. The preceding step implies directly: : \langle T_v\exp_p(v), T_v\exp_p(w)\rangle = \langle T_v\exp_p(v), T_v\exp_p(w_T)\rangle + \langle T_v\exp_p(v), T_v\exp_p(w_N)\rangle ::=a \langle T_v\exp_p(v), T_v\exp_p(v)\rangle + \langle T_v\exp_p(v), T_v\exp_p(w_N)\rangle=\langle v, w_T\rangle + \langle T_v\exp_p(v), T_v\exp_p(w_N)\rangle. We must therefore show that the second term is null, because, according to Gauss's Lemma, we must have: \langle T_v\exp_p(v), T_v\exp_p(w_N)\rangle = \langle v, w_N\rangle = 0. * ''\langle T_v\exp_p(v), T_v\exp_p(w_N)\rangle = 0'' : Let us define the curve : \alpha \colon \epsilon, \epsilontimes ,1\longrightarrow T_pM,\qquad (s,t) \longmapsto tv+tsw_N. Note that : \alpha(0,1) = v,\qquad \frac(s,t) = v+sw_N, \qquad\frac(0,t) = tw_N. Let us put: : f \colon \epsilon, \epsilon times ,1\longrightarrow M,\qquad (s,t)\longmapsto \exp_p(tv+tsw_N), and we calculate: : T_v\exp_p(v)=T_\exp_p\left(\frac(0,1)\right)=\frac\Bigl(\exp_p\circ\alpha(s,t)\Bigr)\Big\vert_=\frac(0,1) and : T_v\exp_p(w_N)=T_\exp_p\left(\frac(0,1)\right)=\frac\Bigl(\exp_p\circ\alpha(s,t)\Bigr)\Big\vert_=\frac(0,1). Hence : \langle T_v\exp_p(v), T_v\exp_p(w_N)\rangle = \left\langle \frac,\frac\right\rangle(0,1). We can now verify that this scalar product is actually independent of the variable t, and therefore that, for example: : \left\langle\frac,\frac\right\rangle(0,1) = \left\langle\frac,\frac\right\rangle(0,0) = 0, because, according to what has been given above: : \lim_\frac(0,t) = \lim_T_\exp_p(tw_N) = 0 being given that the differential is a linear map. This will therefore prove the lemma. * We verify that ''\frac\left\langle \frac,\frac\right\rangle=0'': this is a direct calculation. Since the maps t\mapsto f(s,t) are geodesics, : \frac\left\langle \frac,\frac\right\rangle=\left\langle\underbrace_, \frac\right\rangle + \left\langle\frac,\frac\frac\right\rangle = \left\langle\frac,\frac\frac\right\rangle=\frac12\frac\left\langle \frac, \frac\right\rangle. Since the maps t\mapsto f(s,t) are geodesics, the function t\mapsto\left\langle\frac,\frac\right\rangle is constant. Thus, : \frac\left\langle \frac, \frac\right\rangle =\frac\left\langle v+sw_N,v+sw_N\right\rangle =2\left\langle v,w_N\right\rangle=0.


See also

*
Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as manifold, smooth manifolds with a ''Riemannian metric'' (an inner product on the tangent space at each point that varies smooth function, smo ...
*
Metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...


References

* {{Manifolds Articles containing proofs Lemmas Riemannian geometry Riemannian manifolds Theorems in Riemannian geometry