page_title: Gauss's lemma (Riemannian geometry)
section_title: Proof
hierarchical_section_title: Gauss's lemma (Riemannian geometry) / Proof
caption_attribution_description: English: Scheme of the application chosen is the proof of the Gauss Lemma
context_page_description: 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.
context_section_description: Recall that 
  
  We proceed in three steps:
   : let us construct a curve
   such that  and . Since , we can put . 
  Therefore, 
  where  is the parallel transport operator and . The last equality is true because  is a geodesic, therefore  is parallel.
  Now let us calculate the scalar product . 
  We separate  into a component  parallel 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
  Note 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. Since the maps  are geodesics,
  Since the maps  are geodesics, 
  the function  is constant. Thus,
caption: The curve chosen to prove lemma
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