- Gauss's lemma (Riemannian geometry)
Riemannian geometry, Gauss's lemma asserts that any sufficiently small spherecentered at a point in a Riemannian manifoldis perpendicular to every geodesicthrough 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 spaceat "p" to "M"::which is a diffeomorphismin a neighborhood of zero. Gauss' lemma asserts that the image of a sphereof sufficiently small radius in "T"p"M" 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 on the exponential map at by:where we have had to restrict the domain by definition of a ball of radius and centre to ensure that is well-defined, and where is the point reached by following the unique
geodesicpassing through the point with tangent for a distance . It is easy to see that is a local diffeomorphismaround . Let be a curve differentiable in such that and . Since , it is clear that we can choose . In this case, by the definition of the differential of the exponential in applied over , we obtain:
The fact that is a local diffeomorphism and that for all allows us to state that is a local isometry around , i.e.
This means in particular that it is possible to identify the ball with a small neighbourhood around . We can see that 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 !
The exponential map is a radial isometry
Let . In what follows, we make the identification .Gauss's Lemma states:
Let and . Then, :
For , this lemma means that is a radial isometry in the following sense: let , i.e. such that is well defined. Moreover, let . Then the exponential remains an isometry in , and, more generally, all along the geodesic (in so far as is well defined)! Then, radially, in all the directions permitted by the domain of definition of , it remains an isometry.
We proceed in three steps:
* "" : let us construct a curve such that and . Since , we can put . 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 (these are exactly the same curves, but shifted (###décalées###), because of the domain of definition ; however, the identification allows us to gather them (###ramener###) around !!!). Hence,
Now let us calculate the scalar product .
We separate into a component tangent to and a component normal to . In particular, we put , .
The preceding step implies directly:
We must therefore show that the second term is null, because, according to Gauss's Lemma, we must have:
* "" : Let us define the curve
:with and . We remark in passing that::
Let us put:
and we calculate:
:and:Hence:We can now verify that this scalar product is actually independent of the variable , and therefore that, for example:
:because, according to what has been given above::being given that the differential is a linear map! This will therefore prove the lemma.
* We verify that "" : this is a direct calculation. We first take account of the fact that the maps are geodesics, i.e. . Therefore,
:Hence, in particular,:because, since the maps are geodesics, we have .
Wikimedia Foundation. 2010.
См. также в других словарях:
Gauss's lemma — can mean any of several lemmas named after Carl Friedrich Gauss:* Gauss s lemma (polynomial) * Gauss s lemma (number theory) * Gauss s lemma (Riemannian geometry) See also * List of topics named after Carl Friedrich Gauss … Wikipedia
Differential geometry of surfaces — Carl Friedrich Gauss in 1828 In mathematics, the differential geometry of surfaces deals with smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives:… … Wikipedia
List of differential geometry topics — This is a list of differential geometry topics. See also glossary of differential and metric geometry and list of Lie group topics. Contents 1 Differential geometry of curves and surfaces 1.1 Differential geometry of curves 1.2 Differential… … Wikipedia
Lemme de Gauss (géométrie riemannienne) — En géométrie riemannienne, le lemme de Gauss permet de comprendre l application exponentielle comme une isométrie radiale. Dans ce qui suit, soit M une variété riemannienne dotée d une connexion de Levi Civita (i.e. en particulier, cette… … Wikipédia en Français
List of mathematics articles (G) — NOTOC G G₂ G delta space G networks Gδ set G structure G test G127 G2 manifold G2 structure Gabor atom Gabor filter Gabor transform Gabor Wigner transform Gabow s algorithm Gabriel graph Gabriel s Horn Gain graph Gain group Galerkin method… … Wikipedia
Bernhard Riemann — Infobox Scientist name =Bernhard Riemann box width =300px image width =225px caption =Bernhard Riemann, 1863 birth date =September 17, 1826 birth place =Breselenz, Germany death date =death date and age|1866|7|20|1826|9|17 death place =Selasca,… … Wikipedia
Darboux frame — In the differential geometry of surfaces, a Darboux frame is a natural moving frame constructed on a surface. It is the analog of the Frenet–Serret frame as applied to surface geometry. A Darboux frame exists at any non umbilic point of a surface … Wikipedia
List of theorems — This is a list of theorems, by Wikipedia page. See also *list of fundamental theorems *list of lemmas *list of conjectures *list of inequalities *list of mathematical proofs *list of misnamed theorems *Existence theorem *Classification of finite… … Wikipedia
Exponential map — In differential geometry, the exponential map is a generalization of the ordinary exponential function of mathematical analysis to all differentiable manifolds with an affine connection. Two important special cases of this are the exponential map … Wikipedia
Laplace operator — This article is about the mathematical operator. For the Laplace probability distribution, see Laplace distribution. For graph theoretical notion, see Laplacian matrix. Del Squared redirects here. For other uses, see Del Squared (disambiguation) … Wikipedia