A note on K<Emphasis Type="Italic">ä</Emphasis>hler manifolds with almost nonnegative bisectional curvature

In this note, we prove the following result. There is a positive constant ε(n, Λ) such that if M ⁿ is a simply connected compact Kähler manifold with sectional curvature bounded from above by Λ, diameter bounded from above by 1, and with holomorphic bisectional curvature H ≥ ?ε(n, Λ), then M ⁿ is diffeomorphic to the product M ₁ × ? × M _k , where each M _i is either a complex projective space or an irreducible Kähler–Hermitian symmetric space of rank ≥ 2. This resolves a conjecture of Fang under the additional upper bound restrictions on sectional curvature and diameter.     

Volume Growth and the Topology of Manifolds with Nonnegative Ricci Curvature

Let M ⁿ be a complete, open Riemannian manifold with Ric≥0. In 1994, Grigori Perelman showed that there exists a constant δ _n >0, depending only on the dimension of the manifold, such that if the volume growth satisfies \(\alpha_{M}:=\lim_{r\rightarrow \infty}\frac{\operatorname{Vol}(B_{p}(r))}{\omega_{n}r^{n}}\geq 1-\delta_{n}\), then M ⁿ is contractible. Here we employ the techniques of Perelman to find specific lower bounds for the volume growth, α(k,n), depending only on k and n, which guarantee the individual k-homotopy group of M ⁿ is trivial.     

On generalized Heawood inequalities for manifolds: a van Kampen–Flores-type nonembeddability result

The fact that the complete graph K₅ does not embed in the plane has been generalized in two independent directions. On the one hand, the solution of the classical Heawood problem for graphs on surfaces established that the complete graph K_n embeds in a closed surface M (other than the Klein bottle) if and only if (n?3)(n?4) ≤ 6b₁(M), where b₁(M) is the first Z₂-Betti number of M. On the other hand, van Kampen and Flores proved that the k-skeleton of the n-dimensional simplex (the higher-dimensional analogue of K_n+1) embeds in R^2k if and only if n ≤ 2k + 1.Two decades ago, Kühnel conjectured that the k-skeleton of the n-simplex embeds in a compact, (k ? 1)-connected 2k-manifold with kth Z₂-Betti number b_k only if the following generalized Heawood inequality holds: ( _k+1 ^n?k?1 ) ≤ ( _k+1 ^2k+1 )b_k. This is a common generalization of the case of graphs on surfaces as well as the van Kampen–Flores theorem.In the spirit of Kühnel’s conjecture, we prove that if the k-skeleton of the n-simplex embeds in a compact 2k-manifold with kth Z₂-Betti number bk, then n ≤ 2b_k( _k ^2k+2 )+2k+4. This bound is weaker than the generalized Heawood inequality, but does not require the assumption that M is (k?1)-connected. Our results generalize to maps without q-covered points, in the spirit of Tverberg’s theorem, for q a prime power. Our proof uses a result of Volovikov about maps that satisfy a certain homological triviality condition.     

A characterization of Moebius isoparametric hypersurfaces of the sphere

We consider a proper, umbilic-free immersion of an n-dimensional manifold M in the sphere S ⁿ⁺¹. We show that M is a Moebius isoparametric hypersurface if, and only if, it is a cyclide of Dupin or a Dupin hypersurface with constant Moebius curvature.     

Spectral (isotropic) manifolds and their dimension

A set of n × n symmetric matrices whose ordered vector of eigenvalues belongs to a fixed set in ?ⁿ is called spectral or isotropic. In this paper, we establish that every locally symmetric C^k submanifoldMof ?ⁿ gives rise to a C^k spectral manifold for k ∈ {2, 3, …,∞,ω}. An explicit formula for the dimension of the spectral manifold in terms of the dimension and the intrinsic properties of M is derived. This work builds upon the results of Sylvester and ?ilhavý and uses characteristic properties of locally symmetric submanifolds established in recent works by the authors.     

Teichmller space of negatively curved metrics on complex hyperbolic manifolds is not contractible

We prove that for all n = 4k- 2 and k 2 there exists a closed smooth complex hyperbolic manifold M with real dimension n having non-trivial π1(T0(M)). T0(M) denotes the Teichm¨uller space of all negatively curved Riemannian metrics on M, which is the topological quotient of the space of all negatively curved metrics modulo the space of self-diffeomorphisms of M that are homotopic to the identity.     

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 .     

Rigidity of closed submanifolds in a locally symmetric Riemannian manifold

Let M~n(n ≥ 4) be an oriented closed submanifold with parallel mean curvature in an(n + p)-dimensional locally symmetric Riemannian manifold N~(n+p). We prove that if the sectional curvature of N is positively pinched in [δ, 1], and the Ricci curvature of M satisfies a pinching condition, then M is either a totally umbilical submanifold, or δ = 1, and N is of constant curvature. This result generalizes the geometric rigidity theorem due to Xu and Gu[15].     

The Webster Scalar Curvature and Sharp Upper and Lower Bounds for the First Positive Eigenvalue of the Kohn-Laplacian on Real Hypersurfaces

Let(M, θ) be a compact strictly pseudoconvex pseudohermitian manifold which is CR embedded into a complex space. In an earlier paper, Lin and the authors gave several sharp upper bounds for the first positive eigenvalue λ_1 of the Kohn-Laplacian □_b on(M, θ). In the present paper, we give a sharp upper bound for λ_1, generalizing and extending some previous results. As a corollary, we obtain a Reilly-type estimate when M is embedded into the standard sphere. In another direction, using a Lichnerowicz-type estimate by Chanillo, Chiu, and Yang and an explicit formula for the Webster scalar curvature, we give a lower bound for λ_1 when the pseudohermitian structure θ is volume-normalized.     

Estimates of sums of zero multiplicities for eigenfunctions of the Laplace-Beltrami operator

We obtain an upper estimate N?_χ(M) for the sum Q _N of singular zero multiplicities of the Nth eigenfunction of the Laplace-Beltrami operator on the two-dimensional, compact, connected Riemann manifold M, where _χ M is the Euler characteristic ofM. Stronger estimates, but equivalent asymptotically (N å ∞), are given for the cases of the sphere S ² and the projective plane ?². Asymptotically sharper estimates are shown for the case of a domain on the plane.     

Sagitta,Lenses, and Maximal Volume

We give a characterization of critical points that allows us to define a metric invariant on all Riemannian manifolds M with a lower sectional curvature bound and an upper radius bound. We show there is a uniform upper volume bound for all such manifolds with an upper bound on this invariant. We generalize results by Grove and Petersen by showing any such M that has volume sufficiently close to this upper bound is homeomorphic to the standard sphere \(S^{n}\) or a standard lens space \(S^n/{\mathbb {Z}}_m\) where \(m\in \{2,3,\ldots \}\) is no larger than an a priori constant.     

Fast Algorithm of Square Rooting in Some Finite Fields of Odd Characteristic

It was proved that the complexity of square root computation in the Galois field GF(3^s), s = 2^kr, is equal to O(M(2^k)M(r)k + M(r) log₂r) + 2^kkr^1+o(1), where M (n) is the complexity of multiplication of polynomials of degree n over fields of characteristics 3. The complexity of multiplication and division in the field GF(3^s) is equal to O(M(2^k)M(r)) and O(M(2^k)M(r)) + r^1+o(1), respectively. If the basis in the field GF(3^r) is determined by an irreducible binomial over GF(3) or is an optimal normal basis, then the summands 2^kkr^1+o(1) and r^1+o(1) can be omitted. For M(n) one may take n log₂nψ(n) where ψ(n) grows slower than any iteration of the logarithm. If k grow and r is fixed, than all the estimates presented here have the form O_r (M (s) log ₂s) = s (log ₂s)²ψ(s).     

Peixoto graph of Morse-Smale diffeomorphisms on manifolds of dimension greater than three

Let M ⁿ be a closed orientable manifold of dimension greater than three and G ₁(M ⁿ ) be the class of orientation-preserving Morse-Smale diffeomorphisms on M ⁿ such that the set of unstable separatrices of every f ∈ G ₁(M ⁿ ) is one-dimensional and does not contain heteroclinic orbits. We show that the Peixoto graph is a complete invariant of topological conjugacy in G ₁(M ⁿ ).     

On submanifolds whose tubular hypersurfaces have constant higher order mean curvatures

Motivated by the theory of isoparametric hypersurfaces,we study submanifolds whose tubular hypersurfaces have some constant higher order mean curvatures.Here a k-th order mean curvature Q_k~v(k ≥ 1) of a submanifold M~n-is defined as the k-th power sum of the principal curvatures,or equivalently,of the shape operator with respect to the unit normal vector v.We show that if all nearby tubular hypersurfaces of M have some constant higher order mean curvatures,then the submanifold M itself has some constant higher order mean curvatures Q_k~v independent of the choice of v.Many identities involving higher order mean curvatures and Jacobi operators on such submanifolds are also obtained.In particular,we generalize several classical results in isoparametric theory given by E.Cartan,K.Nomizu,H.F.Miinzner,Q.M.Wang,et al.As an application,we finally get a geometrical filtration for the focal submanifolds of isoparametric functions on a complete Riemannian manifold.     

On the Extremal Zagreb Indices of Graphs with Cut Edges

For a (molecular) graph, the first Zagreb index M ₁ is equal to the sum of squares of the vertex degrees, and the second Zagreb index M ₂ is equal to the sum of products of degrees of pairs of adjacent vertices. In this paper, we show that all connected graphs with n vertices and k cut edges, the maximum (resp. minimum) M ₁- and M ₂-value are obtained, respectively, and uniquely, at K _n ^k (resp. P _n ^k ), where K _n ^k is a graph obtained by joining k independent vertices to one vertex of K _n?k and P _n ^k is a graph obtained by connecting a pendent path P _k+1 to one vertex of C _n?k.     

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.     

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.     

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.     

A new characterization of Willmore submanifolds

Let M ⁿ be a compact Willmore submanifold in the unit sphere S ^n+p . In this note, we investigate the first eigenvalue of the Schrödinger operator L = ?Δ?q on M, where q is some potential function on M, and present a gap estimate for the first eigenvalue of L.     

Classification of hypersurfaces with parallel Möbius second fundamental form in <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis>+1</Superscript>

Let M ⁿ(n ≥ 2) be an immersed umbilic-free hypersurface in the (n+1)-dimensional unit sphere S ⁿ⁺¹. Then M ⁿ is associated with a so-called Möbius metric g, and a Möbius second fundamental form B which are invariants of M ⁿunder the Möbius transformation group of S ⁿ⁺¹. In this paper, we classify all umbilic-free hypersurfaces with parallel Möbius second fundamental form.