Differential Geometry - Computer Graphics and Geometric Modeling - page 663

9.17.3. Theorem. (The Fundamental Lemma of Riemannian Geometry) A
Riemannian manifold M n possesses a unique symmetric connection that is com-
patible with its metric. This connection is called the Levi-Civita connection for the
metric.

Proof. See [Spiv75] or [MilP77]. The Christoffel symbols defined above are the same
as the Christoffel symbols defined in Section 9.9 for this connection.

Definition. Let X , Y , Z ΠVect( M ). The vector field R( X , Y ) Z defined by

( ) ( ) +— ( ) +— [ R XYZ , =-— Z Z Z

]

XY YX XY ,

is called the curvature tensor of the connection.

The key to Riemannian geometry is an analysis of the curvature tensor. (Tensors
were briefly alluded to in Section 4.9 and are a whole other large topic in differential
geometry and physics.) For example,

9.17.4. Theorem. If M n is a Riemannian manifold, then the curvature tensor R for
the Levi-Civita connection on M n is 0 if and only if M n is locally isometric to R n with
its usual Riemannian metric.

Proof. See [Spiv75].

It must be pointed out that Theorem 9.17.4 is a local theorem because there are
manifolds that are locally isometric to R n but are not homeomorphic to R n . See

[Spiv75].
Here is another way that one can see the geometry that is embedded in the concept
of the curvature tensor. The latter is really defined by lower-dimensional concepts that
are easier to understand. Let M n be a Riemannian manifold with curvature tensor R.
Let p ΠM and let V be a two-dimensional subspace of the tangent space T p ( M ). It turns
out that the geodesics in a neighborhood of p that pass through p and are tangent
to V at p actually define a piece of a surface S through p that is uniquely defined by V .
The surface S has a Riemannian structure induced by the structure on M .

Definition. The Gauss curvature of S at p is called the sectional curvature of M at
p along V and is denoted by K( p , V ).

9.17.5. Theorem. With the notation and hypotheses leading up to the definition of
sectional curvature, let X and Y be vectors in T p ( M ) that span V . Then

( )

< R XYYX , , >

( ) = K pV , ,

A

where A is the area of the parallelogram spanned by X and Y .

Proof. See [Spiv75].