Kenro Furutani

The heat kernel and the spectrum of a class of nilmanifolds

Let L be a lattice in R{sup n}, then the Jacobi identity is written as (1.1) {summation}{sub {gamma}}{epsilon}{sub L} e{sup -4}{pi}{sup 2}{parallel}{sup 2}t = Vol(R{sup n}/L)/(4{pi}t){sup n/2} {summation}{sub {gamma}}{epsilon}{sub L} e{sup -{parallel}{sup 2}/4t}. As is well-known, the left side of (1.1) is the trace of the heat kernel on the flat torus R{sup n}/L and the right side reveals the lengths of closed geodesics on it corresponding to each element in L.

The Fourier Transform Method

The Fourier transform has been known as one of the most powerful and useful methods of finding fundamental solutions for operators with constant coefficients. Sometimes the application of a partial Fourier transform might be more useful than the full Fourier transform. In this chapter, by the application of the partial Fourier transform, we shall reduce the problem of finding the heat kernel of a complicated operator to a simpler problem involving an operator with fewer variables. After solving the problem for this simple operator, the inverse Fourier transform provides the heat kernel for the initial operator represented under an integral form. In general, this integral cannot be computed explicitly, but in certain particular cases it actually can be worked out. We shall also apply this method to some degenerate operators.

Sub-Riemannian structures in a principal bundle and their Popp measures

Abstract In this note, we explain a relation between the Popp measures of sub-Riemannian structures on the total space of a principal bundle and the base manifold. Then we determine several concrete cases explicitly.

The Eigenfunction Expansion Method

Finding the heat kernel of an elliptic operator on a compact manifold using the eigenvalues method is a well-known method in mathematical physics and quantum mechanics. Roughly speaking, the eigenvalues and eigenfunctions of an operator determine its heat kernel. The formula is an infinite series that involves products of eigenfunctions; see Theorem 6.1.1. It is interesting that in several cases this series can be written as an elementary function by using the associated bilinear generating function.

Pseudo-Differential Operator and Reproducing Kernels Arising in Geometric Quantization

We show that an operator defined on the quaternion projective space is a zeroth order selfadjoint pseudo-differential operator of Hormander class L 1,0 0 . This operator arises when we compare two quantization operators of the geodesic flow on the quaternion projective space. Such quantization operators are defined on a Hilbert space consisting of holomorphic functions and the Hilbert space has reproducing kernel. We describe the reproducing kernels in the cases of sphere and quaternion projective space in terms of hypergeometric functions, and discuss their relation through fiber integration with respect to the complexified Hopf fibration.

Pseudo-metric 2-step nilpotent Lie algebras

Abstract The metric approach to studying 2-step nilpotent Lie algebras by making use of non-degenerate scalar products is realised. We show that a 2-step nilpotent Lie algebra is isomorphic to its standard pseudo-metric form, that is a 2-step nilpotent Lie algebra endowed with some standard non-degenerate scalar product compatible with the Lie bracket. This choice of the standard pseudo-metric form allows us to study the isomorphism properties. If the elements of the centre of the standard pseudo-metric form constitute a Lie triple system of the pseudo-orthogonal Lie algebra, then the original 2-step nilpotent Lie algebra admits integer structure constants. Among particular applications we prove that pseudo H-type algebras have bases with rational structure constants, which implies that the corresponding pseudo H-type groups admit lattices.

Heat Kernel for the Kohn Laplacian on the Heisenberg Group

We shall deal next with the nonsymmetric form of the Heisenberg group. The Heisenberg group considered in this section will be the set ℍ n = ℝ n ×ℝ n ×ℝ with the following group law:

Heat Kernel for the Sub-Laplacian on the Sphere S 3

(x,y,t) {_\ast} ({x}^{{\prime}},{y}^{{\prime}},{t}^{{\prime}}) = (x + {x}^{{\prime}},y + {y}^{{\prime}},t + {t}^{{\prime}} + x \cdot {y}^{{\prime}}),