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Manfredo Perdigão do Carmo (15 August 1928, Maceió – 30 April 2018, Rio de Janeiro) was a Brazilian mathematician. He spent most of his career at IMPA and is seen as the doyen of differential geometry in Brazil.
In Riemannian geometry and pseudo-Riemannian geometry, the Gauss–Codazzi equations (also called the Gauss–Codazzi–Weingarten-Mainardi equations or Gauss–Peterson–Codazzi formulas [1]) are fundamental formulas that link together the induced metric and second fundamental form of a submanifold of (or immersion into) a Riemannian or pseudo-Riemannian manifold.
Lemma 2: For each ′ exists a parametrization : ′, (), such that the coordinate curves of are asymptotic curves of () = ′ and form a Tchebyshef net. Lemma 3 : Let V ′ ⊂ S ′ {\displaystyle V'\subset S'} be a coordinate neighborhood of S ′ {\displaystyle S'} such that the coordinate curves are asymptotic curves in V ...
An extension of the fundamental theorem states that given a pseudo-Riemannian manifold there is a unique connection preserving the metric tensor, with any given vector-valued 2-form as its torsion. The difference between an arbitrary connection (with torsion) and the corresponding Levi-Civita connection is the contorsion tensor .
Given an open region D in R 2, let g and h be symmetric 2-tensors on D, with g additionally required to be positive-definite. If these are smooth and satisfy the Gauss–Codazzi equations, then Bonnet's theorem says that D is covered by open sets which can be smoothly embedded into R 3 with first fundamental form g and second fundamental form ...
By contrast, most higher-dimensional manifolds do not admit isothermal coordinates anywhere; that is, they are not usually locally conformally flat. In dimension 3, a Riemannian metric is locally conformally flat if and only if its Cotton tensor vanishes.
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