Gauss's lemma (Riemannian geometry)
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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:
- Failed to parse (Cannot write to or create math output directory): \mathrm{exp} : T_pM \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.
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[edit] Introduction
We define on Failed to parse (Cannot write to or create math output directory): M\
the exponential map at Failed to parse (Cannot write to or create math output directory): p\in M by
- Failed to parse (Cannot write to or create math output directory): \exp_p:T_pM\supset B_{\epsilon}(0) \longrightarrow M,\qquad v\longmapsto \gamma(1, p, v),
where we have had to restrict the domain Failed to parse (Cannot write to or create math output directory): T_pM\
by definition of a ball Failed to parse (Cannot write to or create math output directory): B_\epsilon(0)\ of radius Failed to parse (Cannot write to or create math output directory): \epsilon>0\ and centre Failed to parse (Cannot write to or create math output directory): 0\ to ensure that Failed to parse (Cannot write to or create math output directory): \exp_p\ is well-defined, and where Failed to parse (Cannot write to or create math output directory): \gamma(1,p,v)\ is the point Failed to parse (Cannot write to or create math output directory): q\in M reached by following the unique geodesic Failed to parse (Cannot write to or create math output directory): \gamma\ passing through the point Failed to parse (Cannot write to or create math output directory): p\in M with tangent Failed to parse (Cannot write to or create math output directory): \frac{v}{\vert v\vert}\in T_pM for a distance Failed to parse (Cannot write to or create math output directory): \vert v\vert\
. It is easy to see that Failed to parse (Cannot write to or create math output directory): \exp_p\
is a local diffeomorphism around Failed to parse (Cannot write to or create math output directory): 0\in B_\epsilon(0)
. Let Failed to parse (Cannot write to or create math output directory): \alpha : I\rightarrow T_pM
be a curve differentiable in Failed to parse (Cannot write to or create math output directory): T_pM\ such that Failed to parse (Cannot write to or create math output directory): \alpha(0):=0\ and Failed to parse (Cannot write to or create math output directory): \alpha'(0):=v\
. Since Failed to parse (Cannot write to or create math output directory): T_pM\cong \mathbb R^n , it is clear that we can choose Failed to parse (Cannot write to or create math output directory): \alpha(t):=vt\ . In this case, by the definition of the differential of the exponential in Failed to parse (Cannot write to or create math output directory): 0\
applied over Failed to parse (Cannot write to or create math output directory): v\
, we obtain:
- Failed to parse (Cannot write to or create math output directory): T_0\exp_p(v) = \frac{\mathrm d}{\mathrm d t} \Bigl(\exp_p\circ\alpha(t)\Bigr)\Big\vert_{t=0} = \frac{\mathrm d}{\mathrm d t} \Bigl(\exp_p(vt)\Bigr)\Big\vert_{t=0}=\frac{\mathrm d}{\mathrm d t} \Bigl(\gamma(1,p,vt)\Bigr)\Big\vert_{t=0}= \gamma'(t,p,v)\Big\vert_{t=0}=v.
The fact that Failed to parse (Cannot write to or create math output directory): \exp_p\
is a local diffeomorphism and that Failed to parse (Cannot write to or create math output directory): T_0\exp_p(v)=v\ for all Failed to parse (Cannot write to or create math output directory): v\in B_\epsilon(0) allows us to state that Failed to parse (Cannot write to or create math output directory): \exp_p\ is a local isometry around Failed to parse (Cannot write to or create math output directory): 0\
, i.e.
- Failed to parse (Cannot write to or create math output directory): \langle T_0\exp_p(v), T_0\exp_p(w)\rangle_0 = \langle v, w\rangle_p\qquad\forall v,w\in B_\epsilon(0).
This means in particular that it is possible to identify the ball Failed to parse (Cannot write to or create math output directory): B_\epsilon(0)\subset T_pM
with a small neighbourhood around Failed to parse (Cannot write to or create math output directory): p\in M
. We can see that Failed to parse (Cannot write to or create math output directory): \exp_p\
is a local isometry, but we would like it to be rather more than that. We assert that it is in fact possible to show that this map is a radial isometry !
[edit] The exponential map is a radial isometry
Let Failed to parse (Cannot write to or create math output directory): p\in M . In what follows, we make the identification Failed to parse (Cannot write to or create math output directory): T_vT_pM\cong T_pM\cong \mathbb R^n . Gauss's Lemma states:
Let Failed to parse (Cannot write to or create math output directory): v,w\in B_\epsilon(0)\subset T_vT_pM\cong T_pM
and Failed to parse (Cannot write to or create math output directory): M\ni q:=\exp_p(v)
. Then,
- Failed to parse (Cannot write to or create math output directory): \langle T_v\exp_p(v), T_v\exp_p(w)\rangle_v = \langle v,w\rangle_q.
For Failed to parse (Cannot write to or create math output directory): p\in M , this lemma means that Failed to parse (Cannot write to or create math output directory): \exp_p\
is a radial isometry in the following sense: let Failed to parse (Cannot write to or create math output directory): v\in B_\epsilon(0)
, i.e. such that Failed to parse (Cannot write to or create math output directory): \exp_p\
is well defined. Moreover, let Failed to parse (Cannot write to or create math output directory): q:=\exp_p(v)\in M
. Then the exponential Failed to parse (Cannot write to or create math output directory): \exp_p\
remains an isometry in Failed to parse (Cannot write to or create math output directory): q\
, and, more generally, all along the geodesic Failed to parse (Cannot write to or create math output directory): \gamma\
(in so far as Failed to parse (Cannot write to or create math output directory): \gamma(1,p,v)=\exp_p(v)\ is well defined)! Then, radially, in all the directions permitted by the domain of definition of Failed to parse (Cannot write to or create math output directory): \exp_p\
, it remains an isometry.
[edit] Proof
Recall that
- Failed to parse (Cannot write to or create math output directory): T_v\exp_p \colon T_pM\cong T_vT_pM\supset T_vB_\epsilon(0)\longrightarrow T_{\exp_p(v)}M.
We proceed in three steps:
- Failed to parse (Cannot write to or create math output directory): T_v\exp_p(v)=v\
: let us construct a curve Failed to parse (Cannot write to or create math output directory): \alpha : \mathbb R \supset I \rightarrow T_pM
such that Failed to parse (Cannot write to or create math output directory): \alpha(0):=v\in T_pM and Failed to parse (Cannot write to or create math output directory): \alpha'(0):=v\in T_vT_pM\cong T_pM
. Since Failed to parse (Cannot write to or create math output directory): T_vT_pM\cong T_pM\cong \mathbb R^n , we can put Failed to parse (Cannot write to or create math output directory): \alpha(t):=v(t+1) . We find that, thanks to the identification we have made, and since we are only taking equivalence classes of curves, it is possible to choose Failed to parse (Cannot write to or create math output directory): \alpha(t) = vt\
(these are exactly the same curves, but shifted because of the domain of definition Failed to parse (Cannot write to or create math output directory): I
- however, the identification allows us to gather them around Failed to parse (Cannot write to or create math output directory): 0
. Hence,
- Failed to parse (Cannot write to or create math output directory): T_v\exp_p(v) = \frac{\mathrm d}{\mathrm d t}\Bigl(\exp_p\circ\alpha(t)\Bigr)\Big\vert_{t=0}=\frac{\mathrm d}{\mathrm d t}\gamma(t,p,v)\Big\vert_{t=0} = v.
Now let us calculate the scalar product Failed to parse (Cannot write to or create math output directory): \langle T_v\exp_p(v), T_v\exp_p(w)\rangle .
We separate Failed to parse (Cannot write to or create math output directory): w\
into a component Failed to parse (Cannot write to or create math output directory): w_T\ tangent to Failed to parse (Cannot write to or create math output directory): v\ and a component Failed to parse (Cannot write to or create math output directory): w_N\ normal to Failed to parse (Cannot write to or create math output directory): v\
. In particular, we put Failed to parse (Cannot write to or create math output directory): w_T:=\alpha v\ , Failed to parse (Cannot write to or create math output directory): \alpha\in \mathbb R .
The preceding step implies directly:
- Failed to parse (Cannot write to or create math output directory): \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
-
- Failed to parse (Cannot write to or create math output directory): =\alpha\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:
Failed to parse (Cannot write to or create math output directory): \langle T_v\exp_p(v), T_v\exp_p(w_N)\rangle = \langle v, w_N\rangle = 0.
- Failed to parse (Cannot write to or create math output directory): \langle T_v\exp_p(v), T_v\exp_p(w_N)\rangle = 0
:
Let us define the curve
- Failed to parse (Cannot write to or create math output directory): \alpha \colon \left]-\epsilon, \epsilon\right[\times [0,1] \longrightarrow T_pM,\qquad (s,t) \longmapsto t\cdot v(s),
with Failed to parse (Cannot write to or create math output directory): v(0) := v\
and Failed to parse (Cannot write to or create math output directory): v'(0):=w_N\
. We remark in passing that:
- Failed to parse (Cannot write to or create math output directory): \alpha(0,1) = v(0) = v,\qquad\frac{\partial \alpha}{\partial t}(0,t) = v(0) = v,\qquad\frac{\partial \alpha}{\partial s}(0,t) = tw_N.
Let us put:
- Failed to parse (Cannot write to or create math output directory): f \colon \left]-\epsilon, \epsilon \right[ \times [0,1] \longrightarrow M,\qquad (s,t)\longmapsto \exp_p(t\cdot v(s)),
and we calculate:
- Failed to parse (Cannot write to or create math output directory): T_v\exp_p(v)=T_{\alpha(0,1)}\exp_p\left(\frac{\partial \alpha}{\partial t}(0,1)\right)=\frac{\partial}{\partial t}\Bigl(\exp_p\circ\alpha(s,t)\Bigr)\Big\vert_{t=1, s=0}=\frac{\partial f}{\partial t}(0,1)
and
- Failed to parse (Cannot write to or create math output directory): T_v\exp_p(w_N)=T_{\alpha(0,1)}\exp_p\left(\frac{\partial \alpha}{\partial s}(0,1)\right)=\frac{\partial}{\partial s}\Bigl(\exp_p\circ\alpha(s,t)\Bigr)\Big\vert_{t=1,s=0}=\frac{\partial f}{\partial s}(0,1).
Hence
- Failed to parse (Cannot write to or create math output directory): \langle T_v\exp_p(v), T_v\exp_p(w_N)\rangle = \left\langle \frac{\partial f}{\partial t},\frac{\partial f}{\partial s}\right\rangle(0,1).
We can now verify that this scalar product is actually independent of the variable Failed to parse (Cannot write to or create math output directory): t\ , and therefore that, for example:
- Failed to parse (Cannot write to or create math output directory): \left\langle\frac{\partial f}{\partial t},\frac{\partial f}{\partial s}\right\rangle(0,1) = \left\langle\frac{\partial f}{\partial t},\frac{\partial f}{\partial s}\right\rangle(0,0) = 0,
because, according to what has been given above:
- Failed to parse (Cannot write to or create math output directory): \lim_{t\rightarrow 0}\frac{\partial f}{\partial s}(t,0) = \lim_{t\rightarrow 0}T_{tv}\exp_p(tw_N) = 0
being given that the differential is a linear map! This will therefore prove the lemma.
- We verify that Failed to parse (Cannot write to or create math output directory): \frac{\partial}{\partial t}\left\langle \frac{\partial f}{\partial t},\frac{\partial f}{\partial s}\right\rangle=0
: this is a direct calculation. We first take account of the fact that the maps Failed to parse (Cannot write to or create math output directory): t\mapsto f(s,t)
are geodesics, i.e. Failed to parse (Cannot write to or create math output directory): \frac{D}{\partial t}\frac{\partial f}{\partial t}=0
. Therefore,
- Failed to parse (Cannot write to or create math output directory): \frac{\partial}{\partial t}\left\langle \frac{\partial f}{\partial t},\frac{\partial f}{\partial s}\right\rangle=\left\langle\underbrace{\frac{D}{\partial t}\frac{\partial f}{\partial t}}_{=0}, \frac{\partial f}{\partial s}\right\rangle + \left\langle\frac{\partial f}{\partial t},\frac{D}{\partial t}\frac{\partial f}{\partial s}\right\rangle = \left\langle\frac{\partial f}{\partial t},\frac{D}{\partial s}\frac{\partial f}{\partial t}\right\rangle=\frac{\partial }{\partial s}\left\langle \frac{\partial f}{\partial t}, \frac{\partial f}{\partial t}\right\rangle - \left\langle\frac{\partial f}{\partial t},\frac{D}{\partial s}\frac{\partial f}{\partial t}\right\rangle.
Hence, in particular,
- Failed to parse (Cannot write to or create math output directory): 0 = \frac{1}{2} \frac{\partial }{\partial s} \left\langle \frac{\partial f}{\partial t}, \frac{\partial f}{\partial t}\right\rangle = \left\langle \frac{\partial f}{\partial t},\frac{D}{\partial s}\frac{\partial f}{\partial t} \right\rangle = \frac{\partial}{\partial t} \left\langle \frac{\partial f}{\partial t},\frac{\partial f}{\partial s}\right\rangle,
because, since the maps Failed to parse (Cannot write to or create math output directory): t\mapsto f(s,t)
are geodesics, the function Failed to parse (Cannot write to or create math output directory): \left\langle\frac{\partial f}{\partial t},\frac{\partial f}{\partial t}\right\rangle
is a constant function.
[edit] See also
[edit] References
- do Carmo, Manfredo (1992), Riemannian geometry, Basel, Boston, Berlin: Birkhäuser, ISBN 978-0-8176-3490-2 [1]
