Partition of a unity on infinite-dimensional manifold of the Lipschiz class Lip<Superscript><Emphasis Type="Italic">k</Emphasis></Superscript>

In the present paper we prove a criterion of Lip ^k -paracompactness for infinitedimensional manifold M modeled in nonnormable topological vector Fréchet space F. We establish that a manifold is Lip ^k -paracompact if and only if the model space F is paracompact and Lip ^k -normal. We prove a sufficient condition for existence of Lip ^k -partition of a unity on a manifold of class Lip ^k .     

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).     

Torus Actions of Complexity 1 and Their Local Properties

We consider an effective action of a compact (n ? 1)-torus on a smooth 2n-manifold with isolated fixed points. We prove that under certain conditions the orbit space is a closed topological manifold. In particular, this holds for certain torus actions with disconnected stabilizers. There is a filtration of the orbit manifold by orbit dimensions. The subset of orbits of dimensions less than n ? 1 has a specific topology, which is axiomatized in the notion of a sponge. In many cases the original manifold can be recovered from its orbit manifold, the sponge, and the weights of tangent representations at fixed points. We elaborate on the introduced notions using specific examples: the Grassmann manifold G_4,2, the complete flag manifold F₃, and quasitoric manifolds with an induced action of a subtorus of complexity 1.     

Dimension scales of bicompacta

We introduce the notion of a (stable) dimension scale d-sc(X) of a space X, where d is a dimension invariant. A bicompactum X is called dimensionally unified if dim F = dim_G F for every closed F ? X and for an arbitrary abelian group G. We prove that there exist dimensionally unified bicompacta with every given stable scale dim-sc.     

Stable bundles on hypercomplex surfaces

A hypercomplex manifold is a manifold equipped with three complex structures I, J, K satisfying the quaternionic relations. Let M be a 4-dimensional compact smooth manifold equipped with a hypercomplex structure, and E be a vector bundle on M. We show that the moduli space of anti-self-dual connections on E is also hypercomplex, and admits a strong HKT metric. We also study manifolds with (4,4)-supersymmetry, that is, Riemannian manifolds equipped with a pair of strong HKT-structures that have opposite torsion. In the language of Hitchin’s and Gualtieri’s generalized complex geometry, (4,4)-manifolds are called “generalized hyperkähler manifolds”. We show that the moduli space of anti-self-dual connections on M is a (4,4)-manifold if M is equipped with a (4,4)-structure.     

Ideal Spaces of Measurable Operators Affiliated to A Semifinite Von Neumann Algebra

Suppose that M is a von Neumann algebra of operators on a Hilbert space H and τ is a faithful normal semifinite trace on M. Let E, F and G be ideal spaces on (M, τ). We find when a τ-measurable operator X belongs to E in terms of the idempotent P of M. The sets E+F and E·F are also ideal spaces on (M, τ); moreover, E·F = F·E and (E+F)·G = E·G+F·G. The structure of ideal spaces is modular. We establish some new properties of the L₁(M, τ) space of integrable operators affiliated to the algebra M. The results are new even for the *-algebra M = B(H) of all bounded linear operators on H which is endowed with the canonical trace τ = tr.     

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.     

On the structure of invariant attracting sets of systems of finite-difference equations generating Hénon-Lozi mappings

For Hénon-Lozi mappings F, we find sufficient conditions under which on the plane there exists a domain U such that its closure is mapped by F strictly inside U. This ensures the existence of a compact invariant set in U. We prove the existence of an open set of parameter values for which this invariant set contains a zero-dimensional locally maximal topologically transitive Markov set such that the restriction of the mapping to this set is topologically conjugate to the shift automorphism in the space of sequences of two symbols. We show that if this Markov set is hyperbolic, then the above-mentioned compact invariant set coincides with the closure of the unstable manifold of F at a fixed point lying in that set and is a topologically indecomposable one-dimensional continuum. We present the parameter values for which these results hold for the Hénon mapping. We thereby prove the existence of a parameter range in which the invariant set of the Hénon mapping is a one-dimensional topologically indecomposable Brauer-Janiszewski continuum that contains a zero-dimensional locally maximal set and lies in the attraction domain of itself.     

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.     

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.     

Hyperholomorphic connections on coherent sheaves and stability

Let M be a hyperkähler manifold, and F a reflexive sheaf on M. Assume that F (away from its singularities) admits a connection ? with a curvature Θ which is invariant under the standard SU(2)-action on 2-forms. If Θ is square-integrable, such sheaf is called hyperholomorphic. Hyperholomorphic sheaves were studied at great length in [21]. Such sheaves are stable and their singular sets are hyperkähler subvarieties in M. In the present paper, we study sheaves admitting a connection with SU(2)-invariant curvature which is not necessary L ²-integrable. We show that such sheaves are polystable.     

CR-structures of codimension 2 on tangent bundles in Riemann–Finsler geometry

We determine a 2-codimensional CR-structure on the slit tangent bundle \(T_0M\) of a Finsler manifold (M, F) by imposing a condition on the almost complex structure \(\Psi \) associated to F when restricted to the structural distribution of a framed f-structure. This condition is satisfied when (M, F) is of scalar flag curvature (particularly flat). In the Riemannian case (M, g) this last condition means that g is of constant curvature. This CR-structure is finally generalized by using one positive parameter but under more difficult conditions.     

The existence of <Emphasis Type="Italic">J</Emphasis>-holomorphic curves in almost Hermitian manifolds

In this paper, we investigate the existence of J-holomorphic curves on almost Hermitian manifolds. Let (M, g, J, F) be an almost Hermitian manifold and \(f:\Sigma \rightarrow M\) be an injective immersion. We prove that if the \(L_p\) functional has a critical point or a stable point in the same almost Hermitian class, then the immersion is J-holomorphic.     

Baum-Bott indices for curves of singularities

Let F be a holomorphic foliation on Pⁿ by curves such that the components of its singular locus are curves C_i and points p_j. We compute the Baum-Bott indices BB_φ(F, C_i) in terms of the main invariants of F and C_i. We also determine the sum of the BB_φ(F, p_i) in terms of the same invariants.When φ corresponds to the determinant, the latter result generalizes, from special to all holomorphic foliations, a formula for the number of isolated singularities of F, counted with multiplicities.     

Strongly convex weakly complex Berwald metrics and real Landsberg metrics

Under the assumption that' is a strongly convex weakly Khler Finsler metric on a complex manifold M, we prove that F is a weakly complex Berwald metric if and only if F is a real Landsberg metric.This result together with Zhong(2011) implies that among the strongly convex weakly Kahler Finsler metrics there does not exist unicorn metric in the sense of Bao(2007). We also give an explicit example of strongly convex Kahler Finsler metric which is simultaneously a complex Berwald metric, a complex Landsberg metric,a real Berwald metric, and a real Landsberg metric.     

Kähler geometry of hyperbolic type on the manifold of nondegenerate <Emphasis Type="Italic">m</Emphasis>-pairs

A nondegenerate m-pair (A, Ξ) in an n-dimensional projective space ?P _n consists of an m-plane A and an (n ? m ? 1)-plane Ξ in ?P _n , which do not intersect. The set \(\mathfrak{N}_m^n \) of all nondegenerate m-pairs ?P _n is a 2(n ? m)(n ? m ? 1)-dimensional, real-complex manifold. The manifold \(\mathfrak{N}_m^n \) is the homogeneous space \(\mathfrak{N}_m^n = {{GL(n + 1,\mathbb{R})} \mathord{\left/ {\vphantom {{GL(n + 1,\mathbb{R})} {GL(m + 1,\mathbb{R})}}} \right. \kern-\nulldelimiterspace} {GL(m + 1,\mathbb{R})}} \times GL(n - m,\mathbb{R})\) equipped with an internal Kähler structure of hyperbolic type. Therefore, the manifold \(\mathfrak{N}_m^n \) is a hyperbolic analogue of the complex Grassmanian ?G _m,n = U(n+1)/U(m+1) × U(n?m). In particular, the manifold of 0-pairs \(\mathfrak{N}_m^n {{GL(n + 1,\mathbb{R})} \mathord{\left/ {\vphantom {{GL(n + 1,\mathbb{R})} {GL(1,\mathbb{R})}}} \right. \kern-\nulldelimiterspace} {GL(1,\mathbb{R})}} \times GL(n,\mathbb{R})\) is a hyperbolic analogue of the complex projective space ?P _n = U(n+1)/U(1) × U(n). Similarly to ?P _n , the manifold \(\mathfrak{N}_m^n \) is a Kähler manifold of constant nonzero holomorphic sectional curvature (relative to a hyperbolic metrics). In this sense, \(\mathfrak{N}_0^n \) is a hyperbolic spatial form. It was proved in [6] that the manifold of 0-pairs \(\mathfrak{N}_0^n \) is globally symplectomorphic to the total space T*?P _n of the cotangent bundle over the projective space ?P _n . A generalization of this result (see [7]) is as follows: the manifold of nondegenerate m-pairs \(\mathfrak{N}_m^n \) is globally symplectomorphic to the total space T*?G _m,n of the cotangent bundle over the Grassman manifold ?G _m,n of m-dimensional subspaces of the space ?P _n .In this paper, we study the canonical Kähler structure on \(\mathfrak{N}_m^n \). We describe two types of submanifolds in \(\mathfrak{N}_m^n \), which are natural hyperbolic spatial forms holomorphically isometric to manifolds of 0-pairs in ?P _m +1 and in ?P _n?m , respectively. We prove that for any point of the manifold \(\mathfrak{N}_m^n \), there exist a 2(n ? m)-parameter family of 2(m + 1)-dimensional hyperbolic spatial forms of first type and a 2(m + 1)-parameter family of 2(n ? m)-dimensional hyperbolic spatial forms of second type passing through this point. We also prove that natural hyperbolic spatial forms of first type on \(\mathfrak{N}_m^n \) are in bijective correspondence with points of the manifold \(\mathfrak{N}_{m + 1}^n \) and natural hyperbolic spatial forms of second type on \(\mathfrak{N}_m^n \) are in bijective correspondence with points of the manifolds \(\mathfrak{N}_{m + 1}^n \).     

Symbol algebras and cyclicity of algebras after a scalar extension

For a field F and a family of central simple F-algebras we prove that there exists a regular field extension E/F preserving indices of F-algebras such that all the algebras from the family are cyclic after scalar extension by E. Let \( \mathcal{A} \) be a central simple algebra over a field F of degree n with a primitive nth root of unity ρ _n . We construct a quasi-affine F-variety Symb(\( \mathcal{A} \)) such that for a field extension L/F Symb(\( \mathcal{A} \)) has an L-rational point if and only if \( \mathcal{A}{ \otimes_F}L \) is a symbol algebra. Let \( \mathcal{A} \) be a central simple algebra over a field F of degree n and K/F be a cyclic field extension of degree n. We construct a quasi-affine F-variety C(\( \mathcal{A} \) ,K) such that, for a field extension L/F with the property [KL : L] = [K : F], the variety C(\( \mathcal{A} \) ,K) has an L-rational point if and only if KL is a subfield of \( \mathcal{A}{ \otimes_F}L \).     

Sobolev spaces and capacities theory on path spaces over a compact Riemannian manifold

We introduce Sobolev spaces and capacities on the path space P _{m 0} (M) over a compact Riemannian manifold M. We prove the smoothness of the Itô map and the stochastic anti-development map in the sense of stochastic calculus of variation. We establish a Sobolev norm comparison theorem and a capacity comparison theorem between the Wiener space and the path space P _{m 0} (M). Moreover, we prove the tightness of (r, p)-capacities on P _{m 0} (M), \(\), which generalises a result due to Airault-Malliavin and Sugita on the Wiener space. Finally, we extend our results to the fractional Hölder continuous path space \(\).     

On the geometry of spaces of oriented geodesics

Let M be either a simply connected pseudo-Riemannian space of constant curvature or a rank one Riemannian symmetric space, and consider the space L(M) of oriented geodesics of M. The space L(M) is a smooth homogeneous manifold and in this paper we describe all invariant symplectic structures, (para)complex structures, pseudo-Riemannian metrics and (para)Kähler structure on L(M).     

Isotropic functions revisited

To a real n-dimensional vector space V and a smooth, symmetric function f defined on the n-dimensional Euclidean space we assign an associated operator function F defined on linear transformations of V. F shall have the property that, for each inner product g on V, its restriction \(F_{g}\) to the subspace of g-selfadjoint operators is the isotropic function associated to f. This means that it acts on these operators via f acting on their eigenvalues. We generalize some well-known relations between the derivatives of f and each \(F_{g}\) to relations between f and F, while also providing new elementary proofs of the known results. By means of an example we show that well-known regularity properties of \(F_{g}\) do not carry over to F.