Appendix A. Basic Ricci Flow Theory
Heat, like gravity, penetrates every substance of the universe, its rays occupy
all parts of space. The object of our work is to set forth the mathematical
laws which this element obeys. The theory of heat will hereafter form one of
the most important branches of general physics. -Joseph FourierIn this appendix we recall some basic Ricci flow notation, formulas,
and results, mostly from Volume One. Unless otherwise indicated, all page
numbers, theorem references, chapter and section numbers, etc., refer to
Volume One. Some of the results below are slight modifications of those
stated therein. If an unnumbered formula appears on p. C?• of Volume One,
we refer to it as (Vl-p. C?•); if the equation is numbered <). .ft, then we refer
to it as (Vl-<).tft).
The reader who has read or is familiar with Volume One may essentially
skip this chapter, referring to it only when necessary.
- Riemannian geometry
1.1. Notation. Let ( M, g) be a Riemannian manifold. Throughout
this appendix we shall often sum over repeated indices and not bother to
raise (or lower) indices. For example, aijbij ~ gikgfe.aijbke·
- If a is a 1-form, then a~ denotes the dual vector field. Conversely,
if xis a vector field, then xb denotes the dual 1-form.
• TM, T M, A^2 T M, and S2T* M denote the tangent, cotangent,
2-form, and symmetric (2, 0)-tensor bundles, respectively.- r, 'V, and ~ denote the Christoffel symbols, covariant derivative,
and Laplacian, respectively. - R, Re, and Rm denote the scalar, Ricci, and Riemann curvature
tensors, respectively. - r often denotes the average scalar curvature (assuming M is com-
pact). - The upper index on the Riemann (3, 1)-tensor is lowered into the
4-th position: Rijk£ = Rijk9m£· - .A 2: μ 2: v denote the eigenvalues of the Riemann curvature oper-
ator of a 3-manifold, in decreasing order.
• d =dist, diam, and inj denote the Riemannian distance, diameter,
and injectivity radius, respectively.- L, A= Area, and V denote length, area, and volume, respectively.
445