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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 the language of tensor calculus, making use of natural metrics and connections on tensor bundles, the Gauss equation can be written as H 2 − |h| 2 = R and the two Codazzi equations can be written as ∇ 1 h 12 = ∇ 2 h 11 and ∇ 1 h 22 = ∇ 2 h 12; the complicated expressions to do with Christoffel symbols and the first fundamental form ...
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. (1976), Differential Geometry of Curves and Surfaces, Prentice-Hall, ISBN 0-13-212589-7 do Carmo, Manfredo P. (1992), Riemannian geometry , Birkhäuser, ISBN 0-8176-3490-8 Driver, Bruce K. (1995), A primer on Riemannian geometry and stochastic analysis on path spaces (PDF) , Lectures given at the E.T.H., Zurich
[1] [2] [3] 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]
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} .
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 mathematics, specifically in differential geometry, isothermal coordinates on a Riemannian manifold are local coordinates where the metric is conformal to the Euclidean metric. This means that in isothermal coordinates, the Riemannian metric locally has the form = (+ +),