By M. Abate,F. Tovena
The neighborhood concept of surfaces starts with a comparability of the idea that of parametrized (i.e., immersed) floor with the concept that of standard (i.e., embedded) floor. We then improve the elemental differential geometry of surfaces in R3: definitions, examples, differentiable maps and services, tangent vectors (presented either as vectors tangent to curves within the floor and as derivations on germs of differentiable services; we will continually use either ways within the entire ebook) and orientation. subsequent we examine different notions of curvature on a floor, stressing either the geometrical which means of the items brought and the algebraic/analytical tools had to learn them through the Gauss map, as much as the evidence of Gauss’ Teorema Egregium.
Then we introduce vector fields on a floor (flow, first integrals, necessary curves) and geodesics (definition, uncomplicated houses, geodesic curvature, and, within the complementary fabric, a whole evidence of minimizing homes of geodesics and of the Hopf-Rinow theorem for surfaces). Then we will current an explanation of the prestigious Gauss-Bonnet theorem, either in its neighborhood and in its international shape, utilizing easy 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. eventually, the final bankruptcy should be dedicated to numerous very important effects at the international conception 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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