Differential Geometry - Computer Graphics and Geometric Modeling - page 660

Proof. Equation (9.95) implies that the Euler characteristic of the surface is posi-
tive and the classification theorem for surface tells us that the sphere is only such
surface.

There are many other beautiful theorems showing that the topology of a surface
is influenced by its curvature.
It was Riemann who took differential geometry to the next plateau. His inaugu-
ral lecture at Göttingen in 1854 has been considered to be one of the most influential
in the field. See [Spiv70b] for an excellent discussion of what Riemann said. He moved
differential geometry from surfaces in R 3 to n-dimensional manifolds that are
endowed with an intrinsic metric (now called a Riemannian metric), which is defined
independent of approximations by straight lines (lengths of curves in calculus were
traditionally defined in terms approximating polygonal lines with the length of a
straight line segment as basis). Starting with an arbitrary metric

n

,

2

()

ds = g x dx dx (9.96)

ij i j

ij = 1

Riemann asked the question “when is this metric isometric to a given one?” For
example, perhaps a change of coordinates would turn the metric into

n

Â

2 2

ds = dx i ,

i = 1

that is, we have a “flat” space, one that is isometric to R n . Now, using the Taylor expan-
sion for g ij ( x ) we can rewrite equation (9.96) in the form

n n

Ê
Á

∂
∂∂

g
xx

ˆ
˜

 Â

ij
rt

2

() ds = d + 0 x x + ... dx dx . (9.97)

ij rt i j

ij , = 1 rt , = 1

By analyzing the dominating expression of second partials, Riemann was led to the
curvature tensor, which is an appropriate function of these partials. The modern way
to approach this subject is by means of what is called a connection. This is where
Section 9.16 was leading. Although it will take a string of definitions and we will give
no proofs, we feel that it is worthwhile to give the reader an inkling of what one has
to do.
Let M n be an arbitrary Riemannian manifold. To keep things concrete, the user
may assume that the manifold is a submanifold of some R m and a vector field simply
assigns to every point of M a vector in R m that is tangent to the manifold.

Notation. If h(t) is a curve in M , then Vect( M ,h) will denote the set of vector fields
X (t) defined along h. The zero vector field in Vect( M ) or Vect( M ,h) will be denoted
by 0 .

We want a general notion of a covariant derivative of one vector field with respect
to another one. It should be something that generalizes the covariant derivative of a