The book provides an introduction to Differential Geometry of Curves and
Surfaces. The theory of curves starts with a discussion of possible
definitions of the concept of curve, proving in particular the
classification of 1-dimensional manifolds. We then present the classical
local theory of parametrized plane and space curves (curves in
n-dimensional space are discussed in the complementary material):
curvature, torsion, Frenet's formulas and the fundamental theorem of the
local theory of curves. Then, after a self-contained presentation of
degree theory for continuous self-maps of the circumference, we study
the global theory of plane curves, introducing winding and rotation
numbers, and proving the Jordan curve theorem for curves of class C2,
and Hopf theorem on the rotation number of closed simple curves. The
local theory of surfaces begins with a comparison of the concept of
parametrized (i.e., immersed) surface with the concept of regular (i.e.,
embedded) surface. We then develop the basic differential geometry of
surfaces in R3: definitions, examples, differentiable maps and
functions, tangent vectors (presented both as vectors tangent to curves
in the surface and as derivations on germs of differentiable functions;
we shall consistently use both approaches in the whole book) and
orientation. Next we study the several notions of curvature on a
surface, stressing both the geometrical meaning of the objects
introduced and the algebraic/analytical methods needed to study them via
the Gauss map, up to the proof of Gauss' Teorema Egregium. Then we
introduce vector fields on a surface (flow, first integrals, integral
curves) and geodesics (definition, basic properties, geodesic curvature,
and, in the complementary material, a full proof of minimizing
properties of geodesics and of the Hopf-Rinow theorem for surfaces).
Then we shall present a proof of the celebrated Gauss-Bonnet theorem,
both in its local and in its global form, using basic properties (fully
proved in the complementary material) of triangulations of surfaces. As
an application, we shall prove the Poincaré-Hopf theorem on zeroes of
vector fields. Finally, the last chapter will be devoted to several
important results on the global theory of surfaces, like for instance
the characterization of surfaces with constant Gaussian curvature, and
the orientability of compact surfaces in R3.
