Chapter 1

A survey of sphere

theorems in geometry

1.1. Riemannian geometry background

Let M be a smooth manifold of dimension n, and let g be a Riemannian

metric on M. The Levi-Civita connection is defined by

2 g(DXY, Z) = X(g(Y, Z)) + Y (g(X, Z)) − Z(g(X, Y ))

+ g([X, Y ],Z) − g([X, Z],Y ) − g([Y, Z],X)

for all vector fields X, Y, Z. The connection D is torsion-free and metric-

compatible; that is,

DXY − DY X = [X, Y ]

and

X(g(Y, Z)) = g(DXY, Z) + g(Y, DXZ)

for all vector fields X, Y, Z. The Riemann curvature tensor of (M, g) is

defined by

g

(

DXDY Z − DY DX Z − D[X,Y ]Z, W

)

= −R(X, Y, Z, W ).

Hence, if we write DX,Y

2

Z = DX DY Z − DDX

Y

Z, then we obtain

DX,Y

2

Z − DY,XZ

2

= DX DY Z − DY DXZ − D[X,Y ]Z

= −

n

k=1

R(X, Y, Z, ek) ek.

The Levi-Civita connection on (M, g) induces a connection on tensor

bundles. For example, if S is a (0, 4)-tensor, then the covariant derivative

1

http://dx.doi.org/10.1090/gsm/111/01