On positive quaternionic Kähler manifolds with certain symmetry rank

Let M be a positive quaternionic Kähler manifold of real dimension 4m. In this paper we show that if the symmetry rank of M is greater than or equal to [m/2] + 3, then M is isometric to HP ^m or Gr2(C ^m+2). This is sharp and optimal, and will complete the classification result of positive quaternionic Kähler manifolds equipped with symmetry. The main idea is to use the connectedness theorem for quaternionic Kähler manifolds with a group action and the induction arguments on the dimension of the manifold.     

A note on the {\rho}-Nitsche conjecture

Let (M, g) be a complete, simply connected Riemannian manifold of dimension 3 without conjugate points. We show that if M is asymptotically harmonic of constant h = 0, then M is a flat manifold. This theorem shows that any asymptotically harmonic manifold in dimension 3 is a symmetric space, thus completing the classification of asymptotically harmonic manifolds in dimension 3.     

Equivariant cohomology distinguishes the geometric structures of toric hyperkähler manifolds

Toric hyperkähler manifolds are the hyperkähler analogue of symplectic toric manifolds. The theory of Bielawski and Dancer tells us that, while a symplectic toric manifold is determined by a Delzant polytope, a toric hyperkähler manifold is determined by a smooth hyperplane arrangement. The purpose of this paper is to show that a toric hyperkähler manifold up to weak hyperhamiltonian T -isometry is determined not only by a smooth hyperplane arrangement up to weak linear equivalence but also by its equivariant cohomology H* _T (M; ?) with a point â in H ²(M;?) \ {0} up to weak H*(BT; ?)-algebra isomorphism preserving â.     

A class of metrics and foliations on tangent bundle of Finsler manifolds

Let (M,F) be a Finsler manifold, and let TM ₀ be the slit tangent bundle of M with a generalized Riemannian metric G, which is induced by F. In this paper, we extract many natural foliations of (TM ₀,G) and study their geometric properties. Next, we use this approach to obtain new characterizations of Finsler manifolds with positive constant flag curvature. We also investigate the relations between Levi-Civita connection, Cartan connection, Vaisman connection, vertical foliation, and Reinhart spaces.     

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.     

Metric projection onto subsets of compact connected two-dimensional Riemannian manifolds

The paper is focused on combinatorial properties of the metric projection P _E of a compact connected Riemannian two-dimensional manifold M ² onto its subset E consisting of k closed connected sets E _j . A point x ∈ M ² is called singular if P _E (x) contains points from at least three distinct E _j . An exact estimate of the number of singular points is obtained in terms of k and the type of the manifold M ². A similar estimate is proved for subsets E of a normed plane consisting of a finite number of connected components.     

Factorization properties of subdiagonal algebras

Let M be a von Neumann algebra equipped with a normal finite faithful normalized trace τ, and let A be a tracial subalgebra of M. Let E be a symmetric quasi-Banach space on [0, 1]. We show that A has an L_E(M)-factorization if and only if A is a subdiagonal algebra.     

On the hyperbolicity of flows

Let X be a C~1 vector field on a compact boundaryless Riemannian manifold M(dim M≥2),and A a compact invariant set of X.Suppose that A has a hyperbolic splitting,i.e.,T∧M = E~sX E~u with E~s uniformly contracting and E~u uniformly expanding.We prove that if,in addition,A is chain transitive,then the hyperbolic splitting is continuous,i.e.,A is a hyperbolic set.In general,when A is not necessarily chain transitive,the chain recurrent part is a hyperbolic set.Furthermore,we show that if the whole manifold M admits a hyperbolic splitting,then X has no singularity,and the flow is Anosov.     

The manifold of isospectral symmetric tridiagonal matrices and realization of cycles by aspherical manifolds

We consider the classical N. Steenrod’s problem of realization of cycles by continuous images of manifolds. Our goal is to find a class \(\mathcal{M}_n \) of oriented n-dimensional closed smooth manifolds such that each integral homology class can be realized with some multiplicity by an image of a manifold from the class \(\mathcal{M}_n \). We prove that as the class \(\mathcal{M}_n \) one can take a set of finite-fold coverings of the manifold M ⁿ of isospectral symmetric tridiagonal real (n + 1) × (n + 1) matrices. It is well known that the manifold M ⁿ is aspherical, its fundamental group is torsion-free, and its universal covering is diffeomorphic to ? ⁿ . Thus, every integral homology class of an arcwise connected space can be realized with some multiplicity by an image of an aspherical manifold with a torsion-free fundamental group. In particular, for any closed oriented manifold Q ⁿ , there exists an aspherical manifold that has torsion-free fundamental group and can be mapped onto Q ⁿ with nonzero degree.     

On the Compactness of the Hypercomplex Commutator in Hölder Continuous Functions Spaces

This paper is devoted to the compactness of the hypercomplex commutator S _γ M _a ? M _a S _γ, where S _γ is the Cauchy singular integral operator (in the Douglis sense), a is a Hölder continuous hypercomplex function and M _a is the multiplication operator given by M _a f = a f. We extend a known compactness sufficient condition for the commutator of the Cauchy singular integral operator to the frame of the hypercomplex analysis, where γ is merely required to be an arbitrary regular closed Jordan curve.     

Weakly complex Einstein-Finsler vector bundle

Let (E, F) be a complex Finsler vector bundle over a compact Kähler manifold (M, g) with Kähler form Φ. We prove that if (E, F) is a weakly complex Einstein-Finsler vector bundle in the sense of Aikou (1997), then it is modeled on a complex Minkowski space. Consequently, a complex Einstein-Finsler vector bundle (E, F) over a compact Kähler manifold (M, g) is necessarily Φ-semistable and (E, F) = (E₁, F₁) ? · · · ? (Ek; Fk); where F _j := F |E _j , and each (E _j , F _j ) is modeled on a complex Minkowski space whose associated Hermitian vector bundle is a Φ-stable Einstein-Hermitian vector bundle with the same factor c as (E, F).     

Holomorphic vector bundles on open subsets of Stein manifolds

LetM be a connected two-dimensional Stein manifold withH ²(M,Z)=0 andS∈M a discrete subset withS≠ Ø. SetX:=M/S. Fix an integerr≥2. Then there exists a rankr vector bundleE onX such that there is no line bundleL onX with a non-zero mapL →E.     

Cohomogeneity one Kähler and Kähler–Einstein manifolds with one singular orbit I

Let M be a cohomogeneity one manifold of a compact semisimple Lie group G with one singular orbit \(S_0 = G/H\). Then M is G-diffeomorphic to the total space \(G \times _H V\) of the homogeneous vector bundle over \(S_0\) defined by a sphere transitive representation of G in a vector space V. We describe all such manifolds M which admit an invariant Kähler structure of standard type. This means that the restriction \(\mu : S = Gx = G/L \rightarrow F = G/K \) of the moment map of M to a regular orbit \(S=G/L\) is a holomorphic map of S with the induced CR structure onto a flag manifold \(F = G/K\), where \(K = N_G(L)\), endowed with an invariant complex structure \(J^F\). We describe all such standard Kähler cohomogeneity one manifolds in terms of the painted Dynkin diagram associated with \((F = G/K,J^F)\) and a parameterized interval in some T-Weyl chamber. We determine which of these manifolds admit invariant Kähler–Einstein metrics.     

Proper Modifications of Generalized <Emphasis Type="Italic">p</Emphasis>-Kähler Manifolds

In this paper, we consider a proper modification \(f : \tilde{M} \rightarrow M\) between complex manifolds, and study when a generalized p-Kähler property goes back from M to \(\tilde{M}\). When f is the blow-up at a point, every generalized p-Kähler property is conserved, while when f is the blow-up along a submanifold, the same is true for \(p=1\). For \(p=n-1\), we prove that the class of compact generalized balanced manifolds is closed with respect to modifications, and we show that the fundamental forms can be chosen in the expected cohomology class. We also get some partial results in the non-compact case; finally, we end the paper with some examples of generalized p-Kähler manifolds.     

Second variation for L-minimal Legendrian submanifolds in pseudo-Sasakian manifolds

In this paper we provide the second variation formula for L-minimal Lagrangian submanifolds in a pseudo-Sasakian manifold. We apply it to the case of Lorentzian–Sasakian manifolds and relate the L-stability of L-minimal Legendrian submanifolds in a Sasakian manifold M to their L-stability in an associated Lorentzian–Sasakian structure on M.     

On holomorphic immersions into kähler manifolds of constant holomorphic sectional curvature

We study holomorphic immersions f:X →M from a complex manifoldX into a Kähler manifold of constant holomorphic sectional curvatureM, i.e. a complex hyperbolic space form, a complex Euclidean space form, or the complex projective space equipped with the Fubini-Study metric. ForX compact we show that the tangent sequence splits holomorphically if and only iff is a totally geodesic immersion. ForX not necessarily compact we relate an intrinsic cohomological invariantp(X) onX, viz. the invariant defined by Gunning measuring the obstruction to the existence of holomorphic projective connections, to an extrinsic cohomological invariant(f) measuring the obstruction to the holomorphic splitting of the tangent sequence. The two invariantsp(X) and?(f) are related by a linear map on cohomology groups induced by the second fundamental form. In some cases, especially whenX is a complex surface andM is of complex dimension 4, under the assumption thatX admits a holomorphic projective connection we obtain a sufficient condition for the holomorphic splitting of the tangent sequence in terms of the second fundamental form.     

Equivariant cohomology and localization for Lie algebroids

Let M be a manifold carrying the action of a Lie group G, and let A be a Lie algebroid on M equipped with a compatible infinitesimal G-action. Using these data, we construct an equivariant cohomology of A and prove a related localization formula for the case of compact G. By way of application, we prove an analog of the Bott formula.     

Quasitoric Totally Normally Split Manifolds

A smooth stably complex manifold is called a totally tangentially/normally split manifold (TTS/TNS manifold for short) if the respective complex tangential/normal vector bundle is stably isomorphic to a Whitney sum of complex line bundles, respectively. In this paper we construct manifolds M such that any complex vector bundle over M is stably equivalent to a Whitney sum of complex line bundles. A quasitoric manifold shares this property if and only if it is a TNS manifold. We establish a new criterion for a quasitoric manifold M to be TNS via non-semidefiniteness of certain higher degree forms in the respective cohomology ring of M. In the family of quasitoric manifolds, this generalizes the theorem of J. Lannes about the signature of a simply connected stably complex TNS 4-manifold. We apply our criterion to show the flag property of the moment polytope for a nonsingular toric projective TNS manifold of complex dimension 3.     

On the structure of the ambient manifold for Morse–Smale systems without heteroclinic intersections

It is shown that if a closed smooth orientable manifold M ⁿ , n ≥ 3, admits a Morse–Smale system without heteroclinic intersections (the absence of periodic trajectories is additionally required in the case of a Morse–Smale flow), then this manifold is homeomorphic to the connected sum of manifolds whose structure is interconnected with the type and number of points that belong to the non-wandering set of the Morse–Smale system.