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.     

The Ricci flow approach to homogeneous Einstein metrics on flag manifolds

We give a global picture of the normalized Ricci flow on generalized flag manifolds with two or three isotropy summands. The normalized Ricci flow for these spaces reduces to a parameter-dependent system of two or three ordinary differential equations, respectively. Here, we present a qualitative study of these systems’ global phase portrait, which uses techniques of dynamical systems theory. This study allows us to draw conclusions about the existence and the analytical form of invariant Einstein metrics on such manifolds and seems to offer a better insight to the classification problem of invariant Einstein metrics on compact homogeneous spaces.     

Twistor forms on Riemannian products

We study twistor forms on products of compact Riemannian manifolds and show that they are defined by Killing forms on the factors. The main result of this note is a necessary step in the classification of compact Riemannian manifolds with non-generic holonomy carrying twistor forms.     

Berezin integration on non-compact supermanifolds

We investigate the Berezin integral of non-compactly supported quantities. In the framework of supermanifolds with corners, we give a general, explicit and coordinate-free representation of the boundary terms introduced by an arbitrary change of variables. As a corollary, a general Stokes’s theorem is derived—here, the boundary integral contains transversal derivatives of arbitrarily high order.     

Closed geodesics on Riemannian manifolds via the heat flow

The main topic discussed in this paper is the following question: Given a Riemannian manifold M and a closed C¹ curve f: S¹ → M does there exist a (unique) solution of the heat equation ?_tf_t = τ(f_t) defined for all t ≧ 0 which is continuous at t = 0 along with its first S¹-derivative and which coincides with f at t = 0.     

On the structure of superfields in a field theory on a supermanifold

In view of applications to the formulation of gauge field theories on supermanifolds, we study the relation between the sheaves of functions on a supermanifold M and its body manifold M ₀, respectively. The nonuniqueness of the local injections t: M ₀M is analysed in consideration of its role in supersymmetric field theories. A Banach space structure is given to the set of bounded, supersmooth, C ^k fields on M in order to get a rigorous formulation of variational principles for the class of theories under consideration.Research work partly supported by the National Group for Mathematical Physics (GNFM) of the Italian Research Council (CNR) and by the Italian Ministry of Public Education through the research project Geometria e Fisica.     

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.     

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.     

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.     

Existence of connections on superbundles

We show that the existence of a connection on a super vector bundle or on a principal super fibre bundle is equivalent to the vanishing of a cohomological invariant of the superbundle. This invariant is proved to vanish in the case of a De Witt base supermanifold. Finally, some examples are discussed.     

Einstein four-manifolds with skew torsion

We develop a notion of Einstein manifolds with skew torsion on compact, orientable Riemannian manifolds of dimension four. We prove an analogue of the Hitchin–Thorpe inequality and study the case of equality. We use the link with self-duality to study the moduli space of 1-instantons on S⁴$S^4$ for a family of metrics defined by Bonneau.     

Biminimal properly immersed submanifolds in the Euclidean spaces

We consider a complete nonnegative biminimal   submanifold M$M$ (that is, a complete biminimal submanifold with λ≥0$\lambda \ge 0$) in a Euclidean space E^N${\mathbb{E}}^N$. Assume that the immersion is proper  , that is, the preimage of every compact set in E^N${\mathbb{E}}^N$ is also compact in M$M$. Then, we prove that M$M$ is minimal. From this result, we give an affirmative partial answer to Chen’s conjecture. For the case of λ<0$\lambda <0$, we construct examples of biminimal submanifolds and curves.     

On a generalized 1-harmonic equation and the inverse mean curvature flow

We introduce and study generalized 1-harmonic equations (1.1). Using some ideas and techniques in studying 1-harmonic functions from Wei (2007)  [1], and in studying nonhomogeneous 1-harmonic functions on a cocompact set from Wei (2008)  [2, (9.1)], we find an analytic   quantity w$w$ in the generalized 1-harmonic equations  (1.1) on a domain in a Riemannian n$n$-manifold that affects the behavior of weak solutions of  (1.1), and establish its link with the geometry   of the domain. We obtain, as applications, some gradient bounds and nonexistence results for the inverse mean curvature flow, Liouville theorems for p$p$-subharmonic functions of constant p$p$-tension field, p≥n$p\ge n$, and nonexistence results for solutions of the initial value problem of inverse mean curvature flow.     

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.     

A generalization of Lancret’s theorem

General helices in a three dimensional Lie group with a bi-invariant metric are defined and a generalization of Lancret’s theorem is obtained. We conclude that the so-called spherical images of general helices are plane curves, and we obtain the so-called spherical general helices. We also give a relation between the geodesics of the so-called cylinders and general helices.     

Spectral geometry of eta-Einstein Sasakian manifolds

We extend a result of Patodi for closed Riemannian manifolds to the context of closed contact manifolds by showing the condition that a manifold is an η$\eta$-Einstein Sasakian manifold is spectrally determined. We also prove that the condition that a Sasakian space form has constant ?$?$-sectional curvature c$c$ is spectrally determined.     

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.