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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.
Wire and plaster models illustrating the differential geometry and curvature of curves and surfaces, including surfaces of revolution, Dupin cyclides, helicoids, and minimal surfaces including the Enneper surface, with commentary by M. P. do Carmo, G. Fischer, U. Pinkall, H. and Reckziegel. [1] [3]
In differential geometry, Hilbert's theorem (1901) states that there exists no complete regular surface of constant negative gaussian curvature immersed in . This theorem answers the question for the negative case of which surfaces in R 3 {\displaystyle \mathbb {R} ^{3}} can be obtained by isometrically immersing complete manifolds with ...
Gauss' lemma asserts that the image of a sphere of 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.
In the mathematical field of differential geometry, the fundamental theorem of surface theory deals with the problem of prescribing the geometric data of a submanifold of Euclidean space. Originally proved by Pierre Ossian Bonnet in 1867, it has since been extended to higher dimensions and non-Euclidean contexts.
In vector calculus and differential geometry the generalized Stokes theorem (sometimes with apostrophe as Stokes' theorem or Stokes's theorem), also called the Stokes–Cartan theorem, [1] is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus.
Differential geometry of curves is the branch of geometry that deals with smooth curves in the plane and the Euclidean space by methods of differential and integral calculus. Many specific curves have been thoroughly investigated using the synthetic approach .