Certain Curvature Conditions on P-Sasakian Manifolds Admitting a Quater-Symmetric Metric Connection

The authors consider a quarter-symmetric metric connection in a P-Sasakian manifold and study the second order parallel tensor in a P-Sasakian manifold with respect to the quarter-symmetric metric connection. Then Ricci semisymmetric P-Sasakian manifold with respect to the quarter-symmetric metric connection is considered. Next the authors study ξ-concircularly flat P-Sasakian manifolds and concircularly semisymmetric P-Sasakian manifolds with respect to the quarter-symmetric metric connection. Furthermore, the authors study P-Sasakian manifolds satisfying the condition ■(ξ,Y)·■,where ■,■ are the concircular curvature tensor and Ricci tensor respectively with respect to the quarter-symmetric metric connection. Finally, an example of a 5-dimensional P-Sasakian manifold admitting quarter-symmetric metric connection is constructed.     

The Chern Conjecture for Affinely Flat Manifolds Using Combinatorial Methods

An affine manifold is a manifold with a flat affine structure, i.e. a torsion-free flat affine connection. We slightly generalize the result of Hirsch and Thurston that if the holonomy of a closed affine manifold is isomorphic to amenable groups amalgamated or HNN-extended along finite groups, then the Euler characteristic of the manifold is zero confirming an old conjecture of Chern. The technique is from Kim and Lee's work using the combinatorial Gauss–Bonnet theorem and taking the means of the angles by amenability. We show that if an even-dimensional manifold is obtained from a connected sum operation from K(, 1)s with amenable fundamental groups, then the manifold does not admit an affine structure generalizing a result of Smillie.     

Curvatures of strongly convex Kähler-Finsler manifolds

In this article, we study curvatures on a strongly convex (weakly) Kähler-Finsler manifold. First, we prove that the holomorphic sectional curvature is just half of the flag curvature in a holomorphic plane section on a strongly convex weakly Kähler-Finsler manifold. Second, we compare curvatures associated to the Rund connection with curvatures associated to the Chern-Finsler connection or the complex Berward connection on a strongly convex Kähler-Finsler manifold. Finally, we discuss relationships between flag curvatures and holomorphic bisectional curvatures, and compare two kinds of S-curvatures on a strongly convex Kähler-Finsler manifold.     

THE MODULI SPACE OF COMPLEX LAGRANGIAN SUBMANIFOLDS IN THE HYPER-KAEHLER MANIFOLD

51.IntroductionSpecialLagrangiansubmanifoldsofaCalabi-Yaumanifoldareoneoftherecentattractivesubjectsinmathematics(see[5-81).In1996,R.C.Mclean[7]obtainedthedeformationtheoremofspeciaILagrangiansubmanifold,whichshowsthat,givenonecompactspecialLagrangiansubmanifoldL,thereisalocalmodulispaceMlwhichisamanifoldandwhosetangelltspaceatLiscanonicallyidentifiedwiththespaceofharmonic1-formsonL.TheLzinnerproductonharmonicformsthengivesthemodulispaceanaturalRiemannianmetric.Strominger,YauandZaslow[1…     

Complex affine distributions

Geometry of affine immersions is the study of hypersurfaces that are invariant under affine transformations. As with the hypersurface theory on the Euclidean space, an affine immersion can induce a torsion-free affine connection and a (pseudo)-Riemannian metric on the hypersurface. Moreover, an affine immersion can induce a statistical manifold, which plays a central role in information geometry. Recently, a statistical manifold with a complex structure is actively studied since it connects information geometry and Kähler geometry. However, a holomorphic complex affine immersion cannot induce such a statistical manifold with a Kähler structure. In this paper, we introduce complex affine distributions, which are non-integrable generalizations of complex affine immersions. We then present the fundamental theorem for a complex affine distribution, and show that a complex affine distribution can induce a statistical manifold with a Kähler structure.     

Complex Product Structures and Affine Foliations

A complex product structure on a manifold is an appropriate combination of a complex structure and a product structure. The existence of such a structure determines many interesting properties of the underlying manifold, notably that the manifold admits a pair of complementary foliations whose leaves carry affine structures. This is due to the existence of a unique torsion-free connection which preserves both the complex and the product structure; this connection is not necessarily flat. We study the existence of complex product structures on tangent bundles of smooth manifolds, and we investigate the structure of manifolds admitting a complex product structure and a compatible hypersymplectic metric, showing that the foliations mentioned earlier are either symplectic or Lagrangian, depending on the symplectic form under consideration.     

复Finsler流形上的Koppelman-Leray-Norguet公式

利用不变积分核(Berndtsson核),复Finsler度量和联系于Chern-Finsler联络的非线性联络,研究复Finsler流形上具有逐块光滑C~((1))边界的有界域上(p,q)型微分形式的积分表示,得到了(p,q)型微分形式的Koppelman-Leray-Norguet公式和■-方程的解．作为应用,利用复Finsler度量和联系于Chern-Finsler联络的非线性联络,给出了Stein流形上具有逐块光滑C~((1))边界的有界域上(p,q)型微分形式的Koppelman- Leray-Norguet公式以及■-方程的解,并且得到了Stein流形上实非退化强拟凸多面体上(p,q)型微分形式的积分表示式和■-方程的解．     

Vrănceanu connections and foliations with bundle-like metrics

We show that the Vrănceanu connection which was initially introduced on non-holonomic manifolds can be used to study the geometry of foliated manifolds. We prove that a foliation is totally geodesic with bundle-like metric if and only if this connection is a metric one. We introduce the notion of a foliated Riemannian manifold of constant transversal Vrănceanu curvature and the notion of a transversal Einstein foliated Riemannian manifold. The geometry of these two classes of manifolds is studied and the relationship between them is determined.     

Flat Partial Connections on a Three Manifold Equipped with a Codimension One Foliation

Given a compact connected oriented three manifold, equipped with a codimension one foliation, such that the Bott connection on the normal bundle is flat, a 2-form on the space parametrizing flat partial connections on it has been constructed. This form is closed. In the special case where the foliated three manifold is a surface bundle over the circle, this 2-form is identified with a certain 2-form on the parameter space for a class of paths in the representation space for the surface group. The 2-form, in question, on the parameter space for paths is constructed from the natural symplectic form on the representation space for a surface group.     

Metrics and Connections on the Bundle of Affinor Frames

In this paper the authors consider the bundle of affinor frames over a smooth manifold,define the Sasaki metric on this bundle,and investigate the Levi-Civita connection of Sasaki metric.Also the authors determine the horizontal lifts of symmetric linear connection from a manifold to the bundle of affinor frames and study the geodesic curves corresponding to the horizontal lift of the linear connection.     

The double exponential map on spaces of constant curvature

In the article, we derive an explicit formula for the double exponential map on spaces of constant curvature. In addition, we consider some applications of the resulting formula to computing the principal symbol of the product of two pseudodifferential operators on a manifold with connection.     

The double exponential map and covariant derivation

We study the double exponential map which is a composition of a special form of two exponential maps on a manifold with connection. We relate this map and the composition of covariant derivations as well as the composition of pseudodifferential operators on these manifolds.     

Cartan connection and its geodesics on a manifold of nondegenerate affinor fields

We consider the quotient manifold of the manifold of nondegenerate affinor fields on a compact manifold with respect to the action of the group of nowhere vanishing functions. On this manifold, we construct a Cartan connection and find its torsion tensor. We also find the geodesics of the Cartan connection.     

A Note on P—Sasakian Manifold

s51.IntroductionLetMbeadifferentiablemanifo1dofdimensionn,ifMadmitsa(l,l)-tensorfieldgl,aPoSitivedefiniteRiemannianmetricg,avectorfieldeandal-formVwhichsatisfythefollow-ingconditions.thensuchamanifoldMiscalledaPara-SasakianmanifoldorbrieflyaP-Sasakianmanifoldby7'.AdatiandK.Matsumoto[ljwhichareconsideredasspecialcaseofanalmostparacontactmanifoldintroducedbyI.Sato[2].WhereVdenotestheoperatorofcovariantdifferentiationwithrespecttothemetrictensorg'X(M)denotesthesetofdifferentiablevectorfiel…     

UNIFORM ESTIMATES OF SOLUTIONS OF (δ)-EQUATION ON STEIN MANIFOLDS

By means of the Hermitian metric and Chern connection, Qiu [4] obtained the Koppelman-Leray-Norguet formula for (p, q) differential forms on an open set with C1 piecewise smooth boundary on a Stein manifold, and under suitable conditions gave the solutions of (δ)-equation on a Stein manifold. In this article, using the method of Range and Siu [5], under suitable conditions, the authors complicatedly calculate to give the uniform estimates of solutions of (δ)-equation for (p, q) differential forms on a Stein manifold.     

BOCHNER TECHNIQUE IN REAL FINSLER MANIFOLDS

Using non-linear connection of Finsler manifold M, the existence of local coordinates which is normalized at a point x is proved, and the Laplace operator A on 1-form of M is defined by non-linear connection and its curvature tensor. After proving the maximum principle theorem of Hopf-Bochner on M, the Bochner type vanishing theorem of Killing vectors and harmonic 1-form are obtained.     

Compact Hyperhermitian–Weyl and Quaternion Hermitian–Weyl Manifolds

Let (M, g, H) be a quaternion Hermitian manifold. The additional datum of a torsion-free connection D preserving both the quaternionic structure H and the conformal class of g defines on M the structure of quaternion Hermitian–Weyl manifold. Under the compactness assumption of both M and the leaves of a canonical foliation, M is here shown to project on a locally 3-Sasakian orbifold P. Then M is proved to admit both a compatible global complex structure and a finite covering M carrying a hyperhermitian–Weyl structure. The uniqueness of the Weyl structure compatible with a given quaternion Hermitian metric and some restrictions on the Betti numbers are also obtained.     

Scalar Curvature Rigidity for Asymptotically Locally Hyperbolic Manifolds

Rigidity results for asymptotically locally hyperbolic manifolds with lower bounds on scalar curvature are proved using spinor methods related to the Witten proof of the positive mass theorem. The argument is based on a study of the Dirac operator defined with respect to the Killing connection. The existence of asymptotic Killing spinors is related to the spin structure on the end. The expression for the mass is calculated and proven to vanish for conformally compact Einstein manifolds with conformal boundary a spherical space form, giving rigidity. In the four dimensional case, the signature of the manifold is related to the spin structure on the end and explicit formulas for the relevant invariants are given.     

Sasakian structures on CR-manifolds

A contact manifold M can be defined as a quotient of a symplectic manifold X by a proper, free action of \(\mathbb{R}\), with the symplectic form homogeneous of degree 2. If X is also Kähler, and its metric is homogeneous of degree 2, M is called Sasakian. A Sasakian manifold is realized naturally as a level set of a Kähler potential on a complex manifold, hence it is equipped with a pseudoconvex CR-structure. We show that any Sasakian manifold M is CR-diffeomorphic to an S ¹-bundle of unit vectors in a positive line bundle on a projective Kähler orbifold. This induces an embedding of M into an algebraic cone C. We show that this embedding is uniquely defined by the CR-structure. Additionally, we classify the Sasakian metrics on an odd-dimensional sphere equipped with a standard CR-structure.     

On Flat Complete Causal Lorentzian Manifolds

We describe up to finite coverings causal flat affine complete Lorentzian manifolds such that the past and the future of any point are closed near this point. We say that these manifolds are strictly causal. In particular, we prove that their fundamental groups are virtually abelian. In dimension 4, there is only one, up to a scaling factor, strictly causal manifold which is not globally hyperbolic. For a generic point of this manifold, either the past or the future is not closed and contains a lightlike straight line