By M. Abate,F. Tovena
The e-book presents an advent to Differential Geometry of Curves and Surfaces. the speculation of curves begins with a dialogue of attainable definitions of the concept that of curve, proving specifically the category of 1-dimensional manifolds. We then current the classical neighborhood conception of parametrized aircraft and area curves (curves in n-dimensional area are mentioned within the complementary material): curvature, torsion, Frenet’s formulation and the basic theorem of the neighborhood idea of curves. Then, after a self-contained presentation of measure concept for non-stop self-maps of the circumference, we learn the worldwide idea of aircraft curves, introducing winding and rotation numbers, and proving the Jordan curve theorem for curves of sophistication C2, and Hopf theorem at the rotation variety of closed easy curves.
The neighborhood concept of surfaces starts with a comparability of the idea that of parametrized (i.e., immersed) floor with the idea that of normal (i.e., embedded) floor. We then increase the fundamental differential geometry of surfaces in R3: definitions, examples, differentiable maps and capabilities, tangent vectors (presented either as vectors tangent to curves within the floor and as derivations on germs of differentiable services; we will always use either techniques within the complete publication) and orientation. subsequent we research different notions of curvature on a floor, stressing either the geometrical that means of the gadgets brought and the algebraic/analytical equipment had to examine 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, uncomplicated houses, geodesic curvature, and, within the complementary fabric, a whole evidence of minimizing houses of geodesics and of the Hopf-Rinow theorem for surfaces). Then we will current an evidence of the prestigious Gauss-Bonnet theorem, either in its neighborhood and in its worldwide shape, utilizing easy houses (fully proved within the complementary fabric) of triangulations of surfaces. As an program, we will end up the Poincaré-Hopf theorem on zeroes of vector fields. eventually, the final bankruptcy could be dedicated to a number of very important effects at the international idea of surfaces, like for example the characterization of surfaces with consistent Gaussian curvature, and the orientability of compact surfaces in R3.