Generalized Curvatures Part 6 ppsx

Chapter 10

Background on Riemannian Geometry

We do not use much intrinsic Riemannian geometry in this book, since our mani￾folds are essentially submanifolds of Euclidean spaces. However, the Steiner for￾mula for convex subsets with smooth boundaries and the tubes formula of Weyl (see

Chaps. 16 and 17) are Riemannian results. Moreover, the invariant forms described

in Chap. 19 have a typical Riemannian flavor. That is why we give a brief survey of

Riemannian geometry. The reader interested in the subject can consult [10], [29],

[76], [57, Tome 1, Chaps. 4 and 5], or [67].

10.1 Riemannian Metric and Levi-Civita Connexion

Let Mn be an n-dimensional manifold. We denote by T Mn its tangent bundle and

by X (Mn) the module of C∞ vector fields on Mn. A Riemannian metric on Mn is

a positive definite symmetric tensor field g of type (0,2) defined on Mn. In other

words, at each point m ∈ Mn, gm is a scalar product on TmMn which varies differ￾entiably with m. This tensor will often be denoted by <,>. Every manifold admits

Riemannian metrics. To each Riemannian metric, <,> is associated a unique linear

connexion ∇, called the Levi-Civita connexion. Recall that a linear connexion ∇ on

Mn is an operator

∇ : X (Mn)× X (Mn) → X (Mn),

(X,Y) → ∇XY,

satisfying, for all X,Y,Z ∈ X (Mn) and for all f,g ∈ C∞(Mn):

• ∇f X+gY Z = f ∇X Z +g∇Y Z.

• ∇X (Y +Z) = ∇XY +∇X Z.

• ∇X f Z = f ∇X Z +X(f)Z.

The torsion and the curvature are two tensors associated to any linear connection:

• The torsion T is defined for all X,Y ∈ X (Mn) by

T(X,Y) = ∇XY −∇YX −[X,Y].

101

102 10 Background on Riemannian Geometry

• The curvature R is defined for all X,Y,Z ∈ X (Mn) by

R(X,Y)Z = ∇X∇Y Z −∇Y ∇X Z −∇[X,Y]Z.

The Levi-Civita connexion is completely determined by two supplementary con￾ditions:

1. It has no torsion: for all X,Y ∈ X (Mn),

∇XY −∇YX = [X,Y],

where [.,.] is the Lie bracket on Mn.

2. <,> is parallel with respect to ∇: for all X,Y,Z ∈ X (Mn),

Z < X,Y >=< ∇ZX,Y > + < X,∇ZY > .

10.2 Properties of the Curvature Tensor

The curvature tensor R of ∇ satisfies the following algebraic properties. For all

X,Y,Z,W ∈ χ(Mn):

• R(X,Y)Z +R(Y,X)Z = 0.

• R(X,Y)Z +R(Y,Z)X +R(Z,X)Y = 0, called the first Bianchi identity.

• < R(X,Y)Z,W > + < R(Y,X)W,Z >= 0.

• < R(X,Y)Z,W >=< R(Z,W)X,Y >= 0.

At each point m ∈ Mn and each two-dimensional plane π of TmMn, one defines

the sectional curvature K(π) of π by

K(π) =< R(X,Y)Y,X >,

where (X,Y) is an orthonormal frame of π.

If X and Y are two vectors in TmMn, one defines the Ricci tensor Ricc of Mn for

all X,Y ∈ χ(Mn) by

Ricc(X,Y) =

n

∑

n=1

< R(ei,X)Y,ei >,

where (e1,...,ei,...,en) is an orthonormal frame of TmMn.

The scalar curvature at m is the real number

τ = 1

n(n−1)

n

∑

i=1

Ricc(ei,ei).