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
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