Heat Kernel on Lie Groups and Maximally Symmetric Spaces

Preface
Acknowledgments
Contents
Notation
Part I Manifolds
	1 Introduction
		1.1 Heat Equation
		1.2 Geometric Framework
		1.3 Spherical Coordinates
		1.4 Hypergeometric Operators
	2 Geometry of Simple Groups
		2.1 Right-invariant and Left-invariant Basis
		2.2 Metric and Connection
		2.3 Heat Kernel on Simple Groups
		2.4 Spin-tensor Bundles over Simple Groups
		2.5 Geometry of SO(n+1) and SO(1,n)
	3 Geometry of SU(2)
		3.1 Representations of SU(2)
		3.2 Right-Invariant and Left-Invariant Basis
		3.3 Metric and Laplacian
		3.4 Heat Kernel on SU(2)
	4 Maximally Symmetric Spaces
		4.1 Normal Geodesic Coordinates
		4.2 Maximally Symmetric Spaces
		4.3 Local Coordinates
		4.4 Geodesic Spherical Coordinates
		4.5 Isometries
		4.6 Lie Derivatives
		4.7 Laplacian
	5 Three-dimensional Maximally Symmetric Spaces
		5.1 Geometry of S3
		5.2 Geometry of H3
Part II Heat Kernel
	6 Scalar Heat Kernel
		6.1 Reduction Formulas
		6.2 Scalar Heat Kernel on S1 and R
		6.3 Scalar Heat Kernel on S2
		6.4 Scalar Heat Kernel on H2
		6.5 Scalar Heat Kernel on Sn, n≥3
		6.6 Scalar Heat Kernel on Hn, n≥3
	7 Spinor Heat Kernel
		7.1 Spinor Heat Trace
		7.2 Spinor Heat Kernel
	8 Heat Kernel in Two Dimensions
		8.1 Heat Kernel on S2
		8.2 Heat Kernel on H2
	9 Heat Kernel on S3 and H3
		9.1 Scalar Heat Kernel on S3 and H3
		9.2 Spinor Heat Kernel on S3 and H3
		9.3 Heat Trace
		9.4 Heat Kernel on S3
		9.5 Heat Kernel on H3
	10 Algebraic Method for the Heat Kernel
		10.1 Algebraic Method for Heat Kernel
		10.2 Algebraic Method for S2 and H2
		10.3 Heat Kernel Diagonal on Sn and Hn
A Integrals, Series, and Special Functions
	A.1 Integrals
	A.2 Poisson Summation Formula
	A.3 Bernoulli Polynomials
	A.4 Gamma Function
	A.5 Hypergeometric Function
	A.6 Legendre Functions
	A.7 Polynomials
	A.8 Dirac Matrices
References
Index