Summary: Math 241A
Instructor: Guofang Wei
Fall 2000
The basic idea of this course is that curvature bounds give information
about manifolds, which in turn gives topological results. A typical example
is the Bonnet-Myers Theorem. Intuitively,
Bigger curvature Y Smaller manifold1
.
1 Volume Comparison Theorem
1.1 Volume of Riemannian Manifold
Recall: For U Rn
,
vol(U) =
U
1 dv =
U
1 dx1 · · · dxn.
Note - dx1 · · · dxn is called the volume density element.
Change of variable formula: Suppose : V U is a diffeomorphism,
with U, V Rn

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

Collections: Mathematics