Blow-up Theory for Elliptic PDEs in Riemannian Geometry (MN-45)
Chapter One
Background Material
We recall in this chapter basic facts concerning Riemannian geometry and nonlinear
analysis on manifolds. For reasons of length, we are obliged to be succinct and partial.
Possible references are Chavel, do Carmo, Gallot-Hulin-Lafontaine,
Hebey, Jost, Kobayashi-Nomizu, Sakai, and Spivak.
As a general remark, we mention that Einstein's summation convention is adopted:
an index occurring twice in a product is to be summed. This also holds for the rest
of this book.
1.1 RIEMANNIAN GEOMETRY
We start with a few notions in differential geometry. Let M be a Hausdorff topological
space. We say that M is a topological manifold of dimension n if each point of
M possesses an open neighborhood that is homeomorphic to some open subset of
the Euclidean space [[??].sup.n]. A chart of M is then a couple ([OMEGA], [phi]) where [OMEGA] is an
open subset of M, and [phi] is a homeomorphism of [OMEGA] onto some open ... read full excerpt from Blow-up Theory for Elliptic PDEs in Riemannian Geometry (MN-45) ebook
