Relating the second and fourth order terms in the asymptotic expansion of the heat equation

Leta _n be the coefficients in the asymptotic expansion of the heat equation. In this paper, we study the relationship betweena ₂ ² anda ₄ in the context of both Riemannian and affine geometry.Research partially supported by the DFG project on Affine differential geometryResearch partially supported by the NSF (USA) and MSRI (USA)     

The spectral geometry of the canonical Riemannian submersion of a compact Lie group

Let G$G$ be a compact connected Lie group which is equipped with a bi-invariant Riemannian metric. Let m(x,y)=xy$m(x,y)=xy$ be the multiplication operator. We show the associated fibration m:G×G→G$m:G\times G\to G$ is a Riemannian submersion with totally geodesic fibers and we study the spectral geometry of this submersion. We show that the pull-backs of eigenforms on the base have finite Fourier series on the total space and we give examples where arbitrarily many Fourier coefficients can be non-zero. We give necessary and sufficient conditions for the pull-back of a form on the base to be harmonic on the total space.     

Manifolds whose higher odd order curvature operators have constant eigenvalues at the basepoint

LetR ^k be the higher-order curvature operator of a Riemannian manifoldM. IfR ^k has constant eigenvalues at the basepointP ofM for all oddk, then all the odd covariant derivatives of the curvature tensor vanish atP. If the metric is real analytic, the geodesic involution is a local isometry atP.     

Heat Flow from Polygons

Potential Analysis - We study the heat flow from an open, bounded set D in $\mathbb {R}^{2}$ with a polygonal boundary ?D. The initial condition is the indicator function of D. A Dirichlet 0...     

Pseudo-riemannian manifolds with commuting jacobi operators

We study the geometry of pseudo-Riemannian manifolds which are Jacobi-Tsankov, i.e. ℊ(x)ℊ(y)=ℊ(y)ℊ(x) for allx, y. We also study manifolds which are 2-step Jacobi nilpotent, i.e. ℊ(x)ℊ(y)=0 for allx, y.     

Heat Trace Asymptotics Defined by Transfer Boundary Conditions

We compute the first five terms in the short-time heat trace asymptotics expansion for an operator of the Laplace type with transfer boundary conditions using the functorial properties of these invariants.     

Geometric realizations of curvature models by manifolds with constant scalar curvature

We show any pseudo-Riemannian curvature model can be geometrically realized by a manifold with constant scalar curvature. We also show that any pseudo-Hermitian curvature model, para-Hermitian curvature model, hyper-pseudo-Hermitian curvature model, or hyper-para-Hermitian curvature model can be realized by a manifold with constant scalar and -scalar curvature.