C. GLOSSARY 503Cartan structure equations. Given an orthonormal frame { ei} and
dual coframe { wj}, they are the identities satisfied by the connection 1-forms { w;} and the curvature 2-forms { Rml}:
The Cartan structure equations are useful for computing curvatures, espe-
cially for metrics with some sort of symmetry. They may also be used for
general calculations ·in geometric analysis.
Cheeger-Gromov convergence. (See compactness theorem.)
Christoffel symbols. The components of the Levi-Civita connection
with respect to a local coordinate system:. k k
\la;axi8/EJxJ = rij8/8x.
The variation formula for the Christoffel symbols is the first step in comput-
ing the variation formula for the curvatures. The evolution equation for the
Christoffel symbols is also used to derive evolution equations for quantities
involving covariant derivatives.
cigar soliton. The rotationally symmetric steady gradient Ricci soliton
on the plane defined by
dx^2 + dy^2
92', (t) = et (^4) +x (^2) +y 2.
The scalar curvature of the cigar is
4 2
RI',= i 2 2 = 4sech s,
+x +y
wheres is the distance to the origin. Note that 92', is asymptotic to a cylinder
and RI', decays exponentially fast. Perelman's no local collapsing theorem
implies the cigar soliton and its product cannot occur as a limit of a finite
time singularity on a closed manifold.
classical entropy. An integral of the form JM f log f dμ.
collapsible manifold. A manifold admitting a sequence of metrics
with uniformly bounded curvature and maximum injectivity radius tending
to zero.
compactness theorem (Cheeger-Gromov-type). If a pointed se-
quence of complete metrics or solutions of Ricci flow has uniformly bounded
curvature and injectivity radius-at the origins uniformly bounded from be-
low, then there exists a subsequence which converges to a complete metric or
solution. The convergence is after the pull-back by diffeomorphisms, which
we call Cheeger-Gromov convergence.
conjugate heat equation. (See adjoint heat equation.)