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Do Carmo's main research interests were Riemannian geometry and the differential geometry of surfaces. [3]In particular, he worked on rigidity and convexity of isometric immersions, [26] [27] stability of hypersurfaces [28] [29] and of minimal surfaces, [30] [31] topology of manifolds, [32] isoperimetric problems, [33] minimal submanifolds of a sphere, [34] [35] and manifolds of constant mean ...
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.
do Carmo, Manfredo P. (2016). Differential geometry of curves & surfaces (Revised & updated second edition of 1976 original ed.). Mineola, NY: Dover Publications, Inc. ISBN 978-0-486-80699-0. MR 3837152. Zbl 1352.53002. Kobayashi, Shoshichi; Nomizu, Katsumi (1969). Foundations of differential geometry. Volume II. Interscience Tracts in Pure and ...
In the orientable case, the fundamental group Γ of M can be identified with a torsion-free uniform subgroup of G and M can then be identified with the double coset space Γ \ G / K. In the case of the sphere and the Euclidean plane, the only possible examples are the sphere itself and tori obtained as quotients of R 2 by discrete rank 2 subgroups.
This proves the uniqueness of a torsion-free and metric-compatible condition, since if g(W, Z) is equal to g(U, Z) for arbitrary Z, then W must equal U. This is a consequence of the non-degeneracy of the metric. In the local formulation above, this key property of the metric was implicitly used, in the same way, via the existence of g kl.
In 1841 Delaunay proved that the only surfaces of revolution with constant mean curvature were the surfaces obtained by rotating the roulettes of the conics. These are the plane, cylinder, sphere, the catenoid, the unduloid and nodoid.
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.
This proof is basically the same as in Hilbert's paper, although based in the books of Do Carmo and Spivak. Observations : In order to have a more manageable treatment, but without loss of generality , the curvature may be considered equal to minus one, K = − 1 {\displaystyle K=-1} .