The Spectral Geometry of Einstein Manifolds with Boundary

Let (M,g) be a compact Einstein manifold with smooth boundary. We consider the spectrum of the p form valued Laplacian with respect to a suitable boundary condition. We show that certain geometric properties of the boundary may be spectrally characterized in terms of this data where we fix the Einstein constant.Comment: 6Page

Moduli spaces of oriented Type A manifolds of dimension at least 3

We examine the moduli space of oriented locally homogeneous manifolds of Type A which have non-degenerate symmetric Ricci tensor both in the setting of manifolds with torsion and also in the torsion free setting where the dimension is at least 3. These exhibit phenomena that is very different than in the case of surfaces. In dimension 3, we determine all the possible symmetry groups in the torsion free setting.Comment: 22 page

Berry phase in the simple harmonic oscillator

Berry phase of simple harmonic oscillator is considered in a general representation. It is shown that, Berry phase which depends on the choice of representation can be defined under evolution of the half of period of the classical motions, as well as under evolution of the period. The Berry phases do {\em not} depend on the mass or angular frequency of the oscillator. The driven harmonic oscillator is also considered, and the Berry phase is given in terms of Fourier coefficients of the external force and parameters which determine the representation.Comment: LaTex, 1 figur

Totally geodesic submanifolds of Damek-Ricci spaces and Einstein hypersurfaces of the Cayley projective plane

We classify totally geodesic submanifolds of Damek-Ricci spaces and show that they are either homogeneous (such submanifolds are known to be "smaller" Damek-Ricci spaces) or isometric to rank-one symmetric spaces of negative curvature. As a by-product, we obtain that a totally geodesic submanifold of any known harmonic manifold is by itself harmonic. We prove that the Cayley hyperbolic plane admits no Einstein hypersurfaces and that the only Einstein hypersurfaces in the Cayley projective plane are geodesic spheres of a particular radius; this completes the classification of Einstein hypersurfaces in rank-one symmetric spaces. We also show that if a $2$-stein space admits a $2$-stein hypersurface, then both are of constant curvature, under some additional conditions.Comment: 17 page

A one-parameter family of totally umbilical hyperspheres in the nearly Kaehler 6-sphere

We discuss two kinds of almost contact metric structures on a one-parameter family of totally umbilical hyperspheres in the nearly Kaehler unit 6-sphere.Comment: 13 page

Heat content asymptotics for spectral boundary conditions

We study the short time heat content asymptotics for spectral boundary conditions. The heat content coefficients are shown to be non-local and some preliminary results concerning the structure of the first few terms are given.Comment: 10 pages, LaTe

Characterizing the harmonic manifolds by the eigenfunctions of the Laplacian

The space forms, the complex hyperbolic spaces and the quaternionic hyperbolic spaces are characterized as the harmonic manifolds with specific radial eigenfunctions of the Laplacian.Comment: 8 page

A remark concerning universal curvature identities on 4-dimensional Riemannian manifolds

We shall prove the universality of the curvature identity for the 4-dimensional Riemannian manifold using a different method than that used by Gilkey, Park, and Sekigawa \cite{GPS}.Comment: 8 page

Curvature identities on contact metric manifolds and their applications

We study curvature identities on contact metric manifolds on the geometry of the corresponding almost K\"aehler cones, and we provide applications of the derived curvature identities.Comment: 14 page

Hermitian structures on the product of Sasakian manifolds

We investigate the curvature properties of a two-parameter family of Hermitian structures on the product of two Sasakian manifolds, as well as intermediate relations. We give a necessary and sufficient condition for a Hermitian structure belonging to the family to be Einstein and provide concrete examples.Comment: 12 page