512 C. GLOSSARYgeometric quantity. For example, given a variation of a metric, we have
variation formulas for the Christoffel symbols and the Riemann, Ricci, and
scalar curvature tensors.
volume form. On an oriented n-dimensional Riemannian manifold,
the n-form dμ = J det (9ij) dx^1 /\ · · · /\ dxn in a positively oriented local
coordinate system. The integral of the volume form is the volume of the
manifold.
volume ratio. For a ball B (p, r) , the quantity
VolB(p,r)
rn
W-functional. (See Perelman's entropy.)
warped product metric. Given Riemannian manifolds (Mn, g) and
(Nm, h), the metric on the product M x N of the form g (x) + f (x) h (y),
where f : M ---+ JR.. The natural space-time metrics are similar to warped
products.
Witten's black hole. (See cigar soliton.)
Yamabe flow. The geometric evolution equation of metrics: gt9ij =
-Rgij· When n = 2, this is the same as the Ricci flow. A technique applied
to the Yamabe flow described in this book is the Aleksandrov reflection
method. It is expected that the Yamabe flow evolves metrics on closed
manifolds to constant scalar curvature metrics.
Yamabe problem. The problem of showing that in any conformal
class there exists a metric with constant scalar curvature. This problem has
been solved through the works of Yamabe, Trudinger, Aubin, and Schoen.
Schoen's work uses the positive mass theorem of Schoen and Yau.