On the first homology of automorphism groups of manifolds with geometric structures

Hermann and Thurston proved that the group of diffeomorphisms with compact support of a smooth manifold M which are isotopic to the identity is a perfect group. We consider the case where M has a geometric structure. In this paper we shall survey on the recent results of the first homology of the diffeomorphism groups which preserve a smooth G-action or a foliated structure on M. We also work in Lipschitz category. This research was partially supported by Grant-in-Aid for Scientific Research (No. 16540058), Japan Society for the Promotion of Science. This research was partially supported by Grant-in-Aid for Scientific Research (No. 14540093), Japan Society for the Promotion of Science.     

On geometry of flat complete strictly causal Lorentzian manifolds

A flat complete causal Lorentzian manifold is called strictly causal if the past and future of its every point are closed near this point. We consider the strictly causal manifolds with unipotent holonomy groups and assign to a manifold of this type four nonnegative integers (a signature) and a parabola in the cone of positive definite matrices. Two manifolds are equivalent if and only if their signatures coincide and the corresponding parabolas are equal (up to a suitable automorphism of the cone and an affine change of variable). Also, we give necessary and sufficient conditions distinguishing the parabolas of this type among all parabolas in the cone.     

On the bochner conformal curvature of Kähler-Norden manifolds

Using the one-to-one correspondence between Kähler-Norden and holomorphic Riemannian metrics, important relations between various Riemannian invariants of manifolds endowed with such metrics were established in my previous paper [19]. In the presented paper, we prove that there is a strict relation between the holomorphic Weyl and Bochner conformal curvature tensors and similarly their covariant derivatives are strictly related. Especially, we find necessary and sufficient conditions for the holomorphic Weyl conformal curvature tensor of a Kähler-Norden manifold to be holomorphically recurrent.     

Isotropy of non-nilpotent Riemannian solvable Lie groups

We describe a family of non-nilpotent Riemannian solvable Lie groups whose isotropy group has a prescribed compact Lie algebra.     

Spherical functions and invariant differential operators on complex Grassmann manifolds

Proofs are given of two theorems of Berezin and Karpelevič, which as far as we know never have been proved correctly. By using eigenfunctions of the Laplace-Beltrami operator it is shown that the spherical functions on a complex Grassmann manifold are given by a determinant of certain hypergeometric functions. By application of this result, it is proved that a certain system of operators, fow which explicit expressions are given, generates the algebra of radial parts of invariant differential operators.     

Flat Pseudo-Reimannian manifolds with a nilpotent transitive group of isometries

Flat pseudo-Riemannian manifolds with a nilpotent transitive group of isometries are shown to be complete. Also flat pseudo-Riemannian homogeneous manifolds with non-trivial holonomy are shown to contain a complete geodesic.     

On conformally flat almost Hermitian manifolds

We study some of 2n-dimensional conformally flat almost Hermitian manifolds with J-(anti)-invariant Ricci tensor. Received 13 May 2000; revised 15 February 2001.     

Standard pseudo-Hermitian structure on manifolds and seifert fibration

A strictly pseudoconvex pseudo-Hermitian manifoldM admits a canonical Lorentz metric as well as a canonical Riemannian metric. Using these metrics, we can define a curvaturelike function onM. AsM supports a contact form, there exists a characteristic vector field dual to the contact structure. If induces a local one-parameter group ofCR transformations, then a strictly pseudoconvex pseudo-Hermitian manifoldM is said to be a standard pseudo-Hermitian manifold. We study topological and geometric properties of standard pseudo-Hermitian manifolds of positive curvature or of nonpositive curvature . By the definition, standard pseudo-Hermitian manifolds are calledK-contact manifolds by Sasaki. In particular, standard pseudo-Hermitian manifolds of constant curvature turn out to be Sasakian space forms. It is well known that a conformally flat manifold contains a class of Riemannian manifolds of constant curvature. A sphericalCR manifold is aCR manifold whose Chern-Moser curvature form vanishes (equivalently, Weyl pseudo-conformal curvature tensor vanishes). In contrast, it is emphasized that a sphericalCR manifold contains a class of standard pseudo-Hermitian manifolds of constant curvature (i.e., Sasakian space forms). We shall classify those compact Sasakian space forms. When 0, standard pseudo-Hermitian closed aspherical manifolds are shown to be Seifert fiber spaces. We consider a deformation of standard pseudo-Hermitian structure preserving a sphericalCR structure.Dedicated to Professor Sasao Seiya for his sixtieth birthday     

A classification of locally homogeneous connections on 2-dimensional manifolds via group-theoretical approach

The aim of this paper is to classify (lócally) all torsion-less locally homogeneous affine connections on two-dimensional manifolds from a group-theoretical point of view. For this purpose, we are using the classification of all non-equivalent transitive Lie algebras of vector fields in ℝ² according to P.J. Olver [7].     

Hypercomplex structures on Stiefel manifolds

This paper describes a family of hypercomplex structures {% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf% gDOfdaryqr1ngBPrginfgDObYtUvgaiuaacqWFqessaaa!4076!\[\mathcal{I}\]a(p)}a=1,2,3 depending on n real non-zero parameters p = (p ₁,...,p n) on the Stiefel manifold of complex 2-planes in n for all n > 2. Generally, these hypercomplex structures are inhomogenous with the exception of the case when all the p i's are equal. We also determine the Lie algebra of infinitesimal hypercomplex automorphisms for each structure. Furthermore, we solve the equivalence problem for the hypercomplex structures in the case that the components of p are pairwise commensurable. Finally, some of these examples admit discrete hypercomplex quotients whose topology we also analyze.During the preparation of this work all three authors were supported by NSF grants.     

Lie algebraic characterization of manifolds

Results on characterization of manifolds in terms of certain Lie algebras growing on them, especially Lie algebras of differential operators, are reviewed and extended. In particular, we prove that a smooth (real-analytic, Stein) manifold is characterized by the corresponding Lie algebra of linear differential operators, i.e. isomorphisms of such Lie algebras are induced by the appropriate class of diffeomorphisms of the underlying manifolds. The research of Janusz Grabowski supported by the Polish Ministry of Scientific Research and Information Technology under the grant No. 2 P03A 020 24, that of Norbert Poncin by grant C.U.L./02/010.     

Partial decay on simple manifolds

Existence, uniqueness and weighted regularity of solutions of linear and nonlinear second-order uniformly elliptic differential equations on complete punctured compact N-manifolds, N > 2. Application to prescribed curvature problems: scalar curvature in a quasi-isometry class (including a contribution to the Lichnérowicz-York equation of General Relativity); Ricci curvature in a weighted Kähler class (with a related result in equiaffine geometry). A new asymptonic behaviour is allowed throughout, called partial decay, which requires its own maximum principle.Current support: CNRS; partial support; CEE contract GADGET # SC1-0105-C.     

The spectrum of twisted Dirac operators on compact flat manifolds

Let be an orientable compact flat Riemannian manifold endowed with a spin structure. In this paper we determine the spectrum of Dirac operators acting on smooth sections of twisted spinor bundles of , and we derive a formula for the corresponding eta series. In the case of manifolds with holonomy group , we give a very simple expression for the multiplicities of eigenvalues that allows us to compute explicitly the -series, in terms of values of Hurwitz zeta functions, and the -invariant. We give the dimension of the space of harmonic spinors and characterize all -manifolds having asymmetric Dirac spectrum.

Furthermore, we exhibit many examples of Dirac isospectral pairs of -manifolds which do not satisfy other types of isospectrality. In one of the main examples, we construct a large family of Dirac isospectral compact flat -manifolds, pairwise nonhomeomorphic to each other of the order of .

     

Form preserving diffeomorphisms on open manifolds

We prove that on open manifolds of bounded geometry satisfying a certain spectral condition the component of the identity D _infw,0 ^supr of form preserving diffeomorphisms is a submanifold of the identity component of all bounded Sobolev diffeomorphisms. D _infw,0 ^supr inherits a natural Riemannian geometry and we can solve Euler equations in this context.Research supported by NSF grant # DMS-9303215 and Emory-Greifswald Exchange Program     

Yamabe 流下完备Riemannian 流形在无穷远的性质

本文研究完备的局部共形平坦的Riemannian 流形Mⁿ. 证明了在Yamabe 流下, 流形在无穷远处曲率趋向于零的性质是随时间保持的. 作为应用, 可以得到这个流形的渐近体积比是一个常数.     

A note on 4-dimensional locally conformally flat Walker manifolds

A 4-dimensional Walker metric on a semi Riemanian manifold M, for the canonical metric with c = 0, have been investigated by M. Chaichi, E. García—Río and Y. Matsushita. The paper generalizes these notions to the case of constant c ≠ 0. Specially the form of defining functions of this metric in locally conformally flat 4-dimensional Walker manifolds is found.     

Extended affine Weyl groups and Frobenius manifolds

We define certain extensions of affine Weyl groups (distinct from these considered by K. Saito [S1] in the theory of extended affine root systems), prove an analogue of Chevalley Theorem for their invariants, and construct a Frobenius structure on their orbit spaces. This produces solutions F(t₁, ..., t_n) of WDVV equations of associativity polynomial in t₁, ..., t_n-1, exp t_n.     

Topological,affine and isometric actions on flat riemannian manifolds II

Explicit examples of finite subgroups of the group of homotopy classes of self-homotopy equivalences of some flat Riemannian manifolds which cannot be lifted to effective actions are given. It is also shown that no finite subgroups of the kernel of π₀(Homeo(M))→Out π₁(M) can be lifted back to Homeo(M), for a large class of flat manifolds M. Some results of an earlier paper by the authors are refined and related to recent work of R. Schoen and S.T. Yau.     

Submanifolds of constant sectional curvature in Pseudo-Riemannian manifolds

The generalized equation and the intrinsic generalized equation are considered. The solutions of the first one are shown to correspond to Riemannian submanifolds M ⁿ (K) of constant sectional curvature of psedo-Riemannian manifolds % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca% WGnbaaamaaDaaaleaaieGacaWFZbaabaGaaGOmaiaad6gacqGHsisl% caaIXaaaaOGaaiikamaanaaabaGaam4saaaacaGGPaaaaa!3D97!\[\overline M _s^{2n - 1} (\overline K )\] of index s, with % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saiabgc% Mi5oaanaaabaGaam4saaaaaaa!3965!\[K \ne \overline K \], flat normal bundle and such that the normal principal curvatures are different from % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saiabgk% HiTmaanaaabaGaam4saaaaaaa!388B!\[K - \overline K \]. The solutions of the intrinsic generalized equation correspond to Riemannian metrics defined on open subsets of R ⁿ which have constant sectional curvature. The relation between solutions of those equations is given. Moreover, it is proven that the submanifolds M under consideration are determined, up to a rigid motion of % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca% WGnbaaaaaa!36D0!\[\overline M \], by their first fundamental forms, as solutions of the intrinsic generalized equation. The geometric properties of the submanifolds M associated to the solutions of the intrinsic generalized equation, which are invariant under an (n – 1)-dimensional group of translations, are given. Among other results, it is shown that such submanifolds are foliated by (n – 1)-dimensional flat submanifolds which have constant mean curvature in M. Moreover, each leaf of the foliation is itself foliated by curves of % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca% WGnbaaaaaa!36D0!\[\overline M \] which have constant curvatures.