In this section there follows a treatment of differential geometry topics in which, after introductory material, consideration is given to manifolds and tensor bundles, operations on tensor fields, connections, interplay between local and global properties, the Gauss-Bonnet formula, elliptic operators, and, finally, modern development in surface theory.
The equations of a
surface S in a
Euclidean space with the coordinates x
, x
,
x
can be given in parametric form (see 268).
At the point (u
, u
)
on the surface the components of a normal vector of unit length
can also be given (see 269).
The geometrical properties of S are then completely described
by two quadratic differential forms (see 270),
referred to respectively as the
first and
second
fundamental forms of
S. Gauss emphasized the importance of the properties
of S that depend only on the first fundamental form,
such as the length of a curve, the area of a domain, the
geodesics (i.e., curves
of shortest length), and the
Gaussian curvature. Another
German mathematician, Berhard Riemann, founded in 1854 what
is now known as
Riemannian geometry; that
is, the geometry in a space of n dimensions with the
coordinates u
, 
= 1, . .
. , n, based on a quadratic differential form (see
271). This geometry is of general form and includes as special
cases the non-Euclidean geometries. It serves as a model of
the physical universe in Einstein's general theory of relativity
(see below Riemannian
geometry).
Modern differential geometry stems from the basis that the objects of study are a class of spaces called manifolds equipped with additional structures. The main problems deal with the global properties of manifolds; i.e., properties that arise only when the manifolds are looked on as a whole. In global differential geometry topology is a major tool.
