By M. Abate,F. Tovena
The neighborhood concept of surfaces starts off with a comparability of the concept that of parametrized (i.e., immersed) floor with the idea that of standard (i.e., embedded) floor. We then advance the elemental differential geometry of surfaces in R3: definitions, examples, differentiable maps and features, tangent vectors (presented either as vectors tangent to curves within the floor and as derivations on germs of differentiable features; we will continually use either techniques within the complete booklet) and orientation. subsequent we examine the different notions of curvature on a floor, stressing either the geometrical that means of the items brought and the algebraic/analytical tools had to learn them through the Gauss map, as much as the facts of Gauss’ Teorema Egregium.
Then we introduce vector fields on a floor (flow, first integrals, vital curves) and geodesics (definition, easy houses, geodesic curvature, and, within the complementary fabric, an entire evidence of minimizing homes of geodesics and of the Hopf-Rinow theorem for surfaces). Then we will current an explanation of the distinguished Gauss-Bonnet theorem, either in its neighborhood and in its international shape, utilizing uncomplicated homes (fully proved within the complementary fabric) of triangulations of surfaces. As an software, we will end up the Poincaré-Hopf theorem on zeroes of vector fields. ultimately, the final bankruptcy should be dedicated to a number of vital effects at the worldwide idea of surfaces, like for example the characterization of surfaces with consistent Gaussian curvature, and the orientability of compact surfaces in R3.
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Curves and Surfaces (UNITEXT) by M. Abate,F. Tovena