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.     

Moduli and extremal metrics on nonorientable and twisted Riemannian manifolds

We present results on the moduli and extremal metrics of families of curves on certain nonorientable or twisted Riemannian manifolds. We also introduce and investigate weighted l-moduli of families of curves and the corresponding extremal metrics. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 10, pp. 1388–1398, October, 1998. This work was partially supported by the State Foundation for Fundamental Research at the Ministry of Science and Technology of Ukraine (project No. 1.4/263) and by INTAS (grant No. 94-1474).     

Asymptotic distribution and weak convergence on compact Riemannian manifolds

LetM be a compact Riemannian manifold. The discrepancy of two measures onM can be defined by means of geodesic balls onM. It is shown that a sequence of positive measures converges weakly to an absolutely continuous measure (w.r.t. the volume measure ofM) if and only if the discrepancy converges to 0, and the geodesic balls with vanishingv-measure of the boundary constitute a convergence determining class of sets for weak convergence of a sequence of positive measures tov. Estimates for the distance of probability measures with respect to the Kantorovich metric in terms of the discrepancy are obtained.     

On the first Betti number of certain compact nearly cosymplectic manifolds

We shall show that the first Betti number of some class of compact nearly cosymplectic manifolds is zero or even.     

On the non-vanishing of the first Betti number of hyperbolic three manifolds

We show the non-vanishing of cohomology groups of sufficiently small congruence lattices in SL(1,D), where D is a quaternion division algebra defined over a number field E contained inside a solvable extension of a totally real number field. As a corollary, we obtain new examples of compact, arithmetic, hyperbolic three manifolds, with non-torsion first homology group, confirming a conjecture of Waldhausen. The proof uses the characterisation of the image of solvable base change by the author, and the construction of cusp forms with non-zero cusp cohomology by Labesse and Schwermer.Mathematics Subject Classification (2000): 11F75, 22E40, 57M50Revised version: 18 February 2004     

Contact Moishezon threefolds with second Betti number one

We prove that the only contact Moishezon threefold having second Betti number equal to one is the projective space.     

Weak convergence of laws of stochastic processes on Riemannian manifolds

Weak convergence of the laws of discrete time re-metrized stochastic processes derived from Brownian motions on compact Riemannian manifolds with heat kernels uniformly bounded by a constant on each compact set of the time parameter and bounded volumes to a stochastic process is given. With a weak condition, we also give weak convergence of those of Brownian motions themselves on manifolds in the same class. Several examples are given, which cover the cases when the manifolds collapse, the cases when the original Brownian motions converge to a non-local Markov process, and the cases when the Gromov-Hausdorff limit and the spectral limit by Kasue and Kumura are different. Received: 22 February 2000?Published online: 9 March 2001     

The algebraic dimension of compact complex threefolds with vanishing second Betti number

This note investigates compact complex manifolds X of dimension 3 with second Betti number b₂(X) = 0. If X admits a non-constant meromorphic function, then we prove that either b₁(X) = 1 and b₃(X) = 0 or that b₁(X) = 0 and b₃(X) = 2. The main idea is to show that c₃(X) = 0 by means of a vanishing theorem for generic line bundles on X. As a consequence a compact complex threefold homeomorphic to the 6-sphere S⁶ cannot admit a non-constant meromorphic function. Furthermore we investigate the structure of threefolds with b₂(X) = 0 and algebraic dimension 1, in the case when the algebraic reduction X P₁ is holomorphic.     

Pointwise characterizations of curvature and second fundamental form on Riemannian manifolds

Let M be a complete Riemannian manifold possibly with a boundary?M.For any C~1-vector field Z,by using gradient/functional inequalities of the(reflecting)diffusion process generated by L:=?+Z,pointwise characterizations are presented for the Bakry-Emery curvature of L and the second fundamental form of?M if it exists.These characterizations extend and strengthen the recent results derived by Naber for the uniform norm‖RicZ‖∞on manifolds without boundaries.A key point of the present study is to apply the asymptotic formulas for these two tensors found by the first author,such that the proofs are significantly simplified.     

Collapsing to Riemannian manifolds with boundary and the convergence of the eigenvalues of the Laplacian

We prove that the eigenvalues of the Laplacian acting on functions converge to those of the limit manifold for a special collapsing family of closed Riemannian manifolds without curvature bounds. The proof uses L ²-analysis.Dedicated to Professor Hajime Urakawa on his sixtieth birthday.The author is partially supported by the Grant-in-Aid for Scientific Research No. 16740026 of the Japan Society for the Promotion of Science.     

Betti numbers of random manifolds

Betti numbers of configuration spaces of mechanical linkages (known also as polygon spaces) depend on a large number of parameters – the lengths of the bars of the linkage. Motivated by applications in topological robotics, statistical shape theory and molecular biology, we view these lengths as random variables and study asymptotic values of the average Betti numbers as the number of links n tends to infinity. We establish a surprising fact that for a reasonably ample class of sequences of probability measures the asymptotic values of the average Betti numbers are independent of the choice of the measure. The main results of the paper apply to planar linkages as well as for linkages in R ³. We also prove results about higher moments of Betti numbers. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Remark on the Helly number for strongly convex sets on Riemannian manifolds

Replacing convex by strongly convex we show that Helly's famous intersection theorem holds on every Riemannian n-manifold in the following form: The intersection of k relatively compact, strongly convex subsets of M (kn+i2) is nonvoid as soon as any n+i of these sets have a nonvoid intersection, where i=2 if M is homeomorphic to the standard n-sphere and i=1 otherwise.     

Viscosity solutions to second order partial differential equations on Riemannian manifolds

We prove comparison, uniqueness and existence results for viscosity solutions to a wide class of fully nonlinear second order partial differential equations F(x,u,du,d²u)=0 defined on a finite-dimensional Riemannian manifold M. Finest results (with hypothesis that require the function F to be degenerate elliptic, that is nonincreasing in the second order derivative variable, and uniformly continuous with respect to the variable x) are obtained under the assumption that M has nonnegative sectional curvature, while, if one additionally requires F to depend on d²u in a uniformly continuous manner, then comparison results are established with no restrictive assumptions on curvature.     

Betti numbers of 3-Sasakian manifolds

A vanishing theorem and constraints are given for the Betti numbers of compact 3-Sasakian manifolds.     

An improved unifying convergence analysis of Newton’s method in Riemannian manifolds

Using more precise majorizing sequences we provide a finer convergence analysis than before [1], [7] of Newton’s method in Riemannian manifolds with the following advantages: weaker hypotheses, finer error bounds on the distances involved and a more precise information on the location of the singularity of the vector field.     

On the second best constant in logarithmic Sobolev inequalities on complete Riemannian manifolds

We study the second best constant problem for logarithmic Sobolev inequalities on complete Riemannian manifolds and investigate its relationship with optimal heat kernel bounds and the existence of extremal functions.     

The manifolds with nonnegative Ricci curvature and collapsing volume

Let be a complete noncompact -manifold with collapsing volume and . The paper proves that is of finite topological type under some restrictions on volume growth.