Lazzeri's Jacobian of oriented compact Riemannian manifolds

The subject of this paper is a Jacobian, introduced by F. Lazzeri (unpublished), associated with every compact oriented Riemannian manifold whose dimension is twice an odd number. We study the Torelli and Schottky problem for Lazzeri's Jacobian of flat tori and we compare Lazzeri's Jacobian of Kähler manifolds with other Jacobians.     

Finslerian foliations of compact manifolds are Riemannian

We prove that a Finslerian foliation of a compact manifold is Riemannian.     

Betti and Tachibana numbers of compact Riemannian manifolds

We present definitions and properties of conformal Killing forms on a Riemannian manifold and determine Tachibana numbers as analogs of the well known Betti numbers of a compact Riemannian manifold. We show some sets of conditions which characterize these numbers. Finally, we prove some results which establish relationships between Betti and Tachibana numbers.     

Equivariant connected sums of compact self-dual manifolds

Partially supported by NSF grant #DMS-9306950 and UC Riverside grant #5-510000-07427-5     

Negative exterior curvature foliations on compact Riemannian manifolds

The topological structure of compact Riemannian manifolds that admit hyperbolic foliations is studied. Translated fromMatematicheskie Zametki, Vol. 62, No. 5, pp. 673–676, November, 1997. Translated by S. S. Anisov     

Classes of submersions of Riemannian manifolds with compact fibers

In Riemannian geometry and its applications, the most popular is the class of Riemannian submersions (and foliations) [1–4] which are characterized by simplest mutual disposition of fibers. The purpose of the present article is to introduce other, more general, classes of submersions of Riemannian manifolds which, as well as the class of Riemannian submersions, are described by simple local properties of configuration tensors and to begin their study.Given a submersion :MM of differentiable manifolds with compact connected fibers and any metric onM, we define a metric on the base with the help of theL ²-norm of horizontal fields. In this caseT¯ M becomes a subbundle of some larger bundleM. The main class of totally geodesic submersions introduced in the article (Definition 1) corresponds to the metrics onM with simplest disposition ofT¯ M inM. In the article we obtain a criterion for such submersions (Corollary 1); existence is proved by means of the product with a metric varying along fibers (Example 2). To study totally geodesic submersions, we use ideas from the theory of Riemannian submersions and submanifolds with degenerate second form (Theorems 1 and 2 and Corollary 4).Foliations modeled by totally geodesic submersions (see equality (13)) are of interest too, but we leave them beyond the scope of the article.This work was supported by the Russian Foundation for Fundamental Research (Grant 94-01-00271).Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 35, No. 5, pp. 1154–1164, September–October, 1994.     

Tits topology and fundamental groups of compact Riemannian manifolds

Let M be a compact Riemannian manifold without conjugate points. We generalize the Tits topology on the ideal boundary of the universal covering space of M. Then we show that if π₁(M) is amenable and is compact with respect to the Tits topology, then M is flat. This work was supported by Grant No.R01-2006-000-10047-0(2006) from the Basic Research Program of the Korea Science & Engineering Foundation.     

Curvature of a Riemannian metric on compact homogeneous manifolds

Translated from Matematicheskie Zametki, Vol. 43, No. 2, pp. 169–177, February, 1988.     

Lie solutions of Riemannian transport equations on compact manifolds

Given a couple of smooth positive measures of same total mass on a compact Riemannian manifold, the associated optimal transport equation admits a symplectic Monge-Ampère structure, hence Lie solutions (in a restricted sense, though, still expressing measure-transport). Properties of such solutions are recorded; a structure result is obtained for regular ones (each consisting of a closed 1-form composed with a diffeomorphism) and a quadratic cost-functional proposed for them.     

Isospectral deformations on Riemannian manifolds which are diffeomorphic to compact Heisenberg manifolds

It is known that ifH _m is the classical (2m+1)-dimensional Heisenberg group, Γ a cocompact discrete subgroup ofH _m andg a left invariant metric, then (Γ/H _{m, g}) is infinitesimally spectrally rigid within the family of left invariant metrics. The purpose of this paper is to show that for everym≥2 and for a certain choice of Γ andg, there is a deformation (Γ/H _{m, g α}) withg=g ₁ such that for every α≠1, (Γ/H _{m, g α})does admit a nontrivial isospectral deformation. For α≠1 the metricsg _α will not beH _m-left invariant, and the (Γ/H _m, gx_α) will not be nilmanifolds, but still solvmanifolds.