On Ricci tensors of Randers metrics

In this paper, we study Randers metrics and find a condition on the Ricci tensors of these metrics for being Berwaldian. This generalizes Shen’s Theorem which says that every R-flat complete Randers metric is locally Minkowskian. Then we find a necessary and sufficient condition on the Ricci tensors under which a Randers metric of scalar flag curvature is of zero flag curvature.     

Brownian motion on manifolds with time-dependent metrics and stochastic completeness

The Brownian motion on a Riemannian manifold is a stochastic process such that the heat kernel is the density of the transition probability. If the total probability of the particle being found in the state space is constantly 1, then the Brownian motion is called stochastically complete. For manifolds with time-dependent metrics, the heat equation should be modified. With the modified heat equation, we study the Brownian motion on manifolds with time-dependent metrics and find conditions on metrics and the volume growth for stochastic completeness.     

On Randers spaces of constant flag curvature

This paper concerns a ubiquitous class of Finsler metrics on smooth manifolds of dimension n. These are the Randers metrics. They figure prominently in both the theory and the applications of Finsler geometry. For n ≥ 3, we consider only those with constant flag curvature. For n = 2, we focus on those whose flag curvature is a (possibly constant) function of position only. We characterize such metrics by three efficient conditions. With the help of examples in 2 and 3 dimensions, we deduce that the Yasuda-Shimada classification of Randers space forms actually addresses only a special case. The corrected classification for that special case is sharp, holds for n ≥ 2, and follows readily from our three necessary and sufficient conditions.     

Deformations of 2k-Einstein structures

It is shown that the space of infinitesimal deformations of 2k-Einstein structures is finite dimensional on compact non-flat space forms. Moreover, spherical space forms are shown to be rigid in the sense that they are isolated in the corresponding moduli space.     

Some rigidity theorems for locally symmetrical Finsler manifolds

In this paper we study some rigidity properties for locally symmetrical Finsler manifolds and obtain some results. We obtain the local equivalent characterization for a Finsler manifold to be locally symmetrical and prove that any locally symmetrical Finsler manifold with nonzero flag curvature must be Riemannian. We also generalize a rigidity result due to Akbar-Zadeh.     

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.     

On the curvature of warped product Finsler spaces and the Laplacian of the Sasaki–Finsler metrics

As an opening, we prove that a warped product Finsler space F=_F1×fF2$F=F_1{\times }_f{F}_2$ is of constant curvature c$c$ if and only if the base space _F1$F_1$ is also of constant curvature c$c$, the fiber space _F2$F_2$ is of some constant curvature α$\alpha$, and five other partial differential equations are satisfied. A rather similar result is proved for the case of warped product Finsler spaces of scalar curvature. Close relationships between the geometry of the warped product Finsler spaces of constant curvature and the spectral theory of the Laplacian (Laplace–Beltrami operator) of the well-known Sasaki–Finsler metrics of the base space _F1$F_1$ is established by detailed investigation of the above mentioned PDEs. We also define a new tensor for warped product Finsler spaces, which we call a warped-Cartan tensor. Using the tensor we define a new class of warped product Finsler spaces, calling them C-Warped spaces, which contain Landsberg, Berwald, locally Minkowski and Riemannian spaces, but not necessarily all of the constant curvature Finsler spaces of warped product type. Several results are obtained and special cases, for example the case of Riemannian, C-Warped and projectively flat spaces are also considered.     

Gradient estimate for Schrödinger operators on manifolds

In this paper, we prove a new localized version of a gradient estimate for Schrödinger operators on the complete manifolds without boundary and with Ricci curvature bounded below by a negative constant. As its application, we derive the Liouville type theorem, the Harnack inequality and the Gaussian lower bound of the heat kernel of Schrödinger operators.     

One-Loop β Functions of a Translation-Invariant Renormalizable Noncommutative Scalar Model

In this paper we explain how to define “lower dimensional” volumes of any compact Riemannian manifold as the integrals of local Riemannian invariants. For instance we give sense to the area and the length of such a manifold in any dimension. Our reasoning is motivated by an idea of Connes and involves in an essential way noncommutative geometry and the analysis of Dirac operators on spin manifolds. However, the ultimate definitions of the lower dimensional volumes do not involve noncommutative geometry or spin structures at all.      

Renormalised Chern–Weil forms associated with families of Dirac operators

We provide local expressions for Chern–Weil type forms built from superconnections associated with families of Dirac operators previously investigated in [S. Scott, Zeta–Chern forms and the local family index theorem, Trans. Amer. Math. Soc. (in press). arXiv: math.DG/0406294] and later in [S. Paycha, S. Scott, Chern–Weil forms associated with superconnections, in: B. Booss-Bavnbeck, S. Klimek, M. Lesch, W. Zhang (Eds.), Analysis, Geometry and Topology of Elliptic Operators, World Scientific, 2006].     

Gravitational interpretation of the Hitchin equations

By referring to theorems of Donaldson and Hitchin, we exhibit a rigorous AdS/CFT-type correspondence between classical 2+1-dimensional vacuum general relativity theory on Σ×R$\Sigma \times \mathbb{R}$ and SO(3) Hitchin theory (regarded as a classical conformal field theory) on the spacelike past boundary Σ$\Sigma$, a compact, oriented Riemann surface of genus greater than 1. Within this framework we can interpret the 2+1-dimensional vacuum Einstein equation as a decoupled “dual” version of the two-dimensional SO(3) Hitchin equations.     

Goldman flows on the Jacobian

We show that the Goldman flows preserve the holomorphic structure on the moduli space of homomorphisms of the fundamental group of a Riemann surface into U(1)$U\left(1\right)$, which identifies with the Jacobian.     

Real hypersurfaces in complex two-plane Grassmannians with commuting Ricci tensor

In this paper, first we introduce the full expression for the Ricci tensor of a real hypersurface M$M$ in complex two-plane Grassmannians _G2(C^m+2)$G_2\left({\mathbb{C}}^{m+2}\right)$ from the equation of Gauss. Next we prove that a Hopf hypersurface in complex two-plane Grassmannians _G2(C^m+2)$G_2\left({\mathbb{C}}^{m+2}\right)$ with commuting Ricci tensor is locally congruent to a tube of radius r$r$ over a totally geodesic _G2(C^m+1)$G_2\left({\mathbb{C}}^{m+1}\right)$. Finally it can be verified that there do not exist any Hopf Einstein hypersurfaces in _G2(C^m+2)$G_2\left({\mathbb{C}}^{m+2}\right)$.     

Uniqueness of balanced metrics on holomorphic vector bundles

Let E→M$E\to M$ be a holomorphic vector bundle over a compact Kähler manifold (M,ω)$(M,\omega )$. We prove that if E$E$ admits a ω$\omega$-balanced metric (in X. Wang’s terminology (Wang, 2005 [3])) then it is unique. This result together with Biliotti and Ghigi (2008) [14] implies the existence and uniqueness of ω$\omega$-balanced metrics of certain direct sums of irreducible homogeneous vector bundles over rational homogeneous varieties. We finally apply our result to show the rigidity of ω$\omega$-balanced Kähler maps into Grassmannians.     

Some calibrated surfaces in manifolds with density

Hyperplanes, hyperspheres and hypercylinders in Rⁿ${\mathbb{R}}^n$ with suitable densities are proved to be weighted area-minimizing by a calibration argument.     

Kähler metrics generated by functions of the time-like distance in the flat Kähler–Lorentz space

We prove that every Kähler metric, whose potential is a function of the time-like distance in the flat Kähler–Lorentz space, is of quasi-constant holomorphic sectional curvatures, satisfying certain conditions. This gives a local classification of the Kähler manifolds with the above-mentioned metrics. New examples of Sasakian space forms are obtained as real hypersurfaces of a Kähler space form with special invariant distribution. We introduce three types of even dimensional rotational hypersurfaces in flat spaces and endow them with locally conformal Kähler structures. We prove that these rotational hypersurfaces carry Kähler metrics of quasi-constant holomorphic sectional curvatures satisfying some conditions, corresponding to the type of the hypersurfaces. The meridians of those rotational hypersurfaces, whose Kähler metrics are Bochner–Kähler (especially of constant holomorphic sectional curvatures), are also described.     

On the geometrical representation of the path integral reduction Jacobian: The case of dependent variables in a description of reduced motion

The geometrical representation of the path integral reduction Jacobian obtained in the problem of the path integral quantization of a scalar particle motion on a smooth compact Riemannian manifold with the given free isometric action of the compact semisimple Lie group has been found for the case when the local reduced motion is described by means of dependent coordinates. The result is based on the scalar curvature formula for the original manifold which is viewed as a total space of the principal fiber bundle.     

The classification of 4-dimensional non-degenerate affine hypersurfaces with parallel cubic form

In this paper, we complete the classification of 4-dimensional non-degenerate affine hypersurfaces with parallel cubic form with respect to the Levi-Civita connection of the affine Berwald–Blaschke metric.     

A two-component geodesic equation on a space of constant positive curvature

We propose a new two-component geodesic equation with the unusual property that the underlying space has constant positive curvature. In the special case of one space dimension, the equation reduces to the two-component Hunter–Saxton equation.