arXiv research

We characterize Willmore tori in the 4-sphere with nontrivial normal bundle as Twistor projections of elliptic curves in complex projective space or as inverted minimal tori (with planar ends) in Euclidean 4-space.

Study shows non-existence of almost complex structures on certain sphere bundles over complex projective spaces.

Researchers found all invariant contact structures on tangent sphere bundles of compact symmetric spaces.

Geodesically equivalent Finsler metrics share invariant volume forms and first integrals.

Theorems prove upper bounds for foliations on closed Alexandrov spaces.

It is well-known that if a curve is a geodesic line of the tangent (sphere) bundle with Sasaki metric of a locally symmetric Riemannian manifold then the projected curve has all its geodesic curvatures constant. In this paper we consider the case of tangent (sphere) bundle over the real, complex and quaternionic space …

We propose a special deformation of the Sasaki metric on tangent and unit tangent bundle of a Hermitian locally symmetric manifold. Geodesics of this deformed metric have different projections on a base manifold for tangent or unit tangent bundle cases in contrast to usual Sasaki metric. Nevertheless, the projections o…

In dimensions congruent to 1 modulo 4, we prove that the cotangent bundle of an exotic sphere which does not bound a parallelisable manifold is not symplectomorphic to the cotangent bundle of the standard sphere. More precisely, we prove that such an exotic sphere cannot embed as a Lagrangian in the cotangent bundle of…

The paper calculates the mapping class group of specific complex projective plane bundles.

New Sasaki metrics with constant scalar curvature on sphere bundles are constructed.

In this note we show that the configuration spaces of the kinematic system constructed in [4] and [12] gives rise to a natural tower of sphere bundles. Moreover, we prove that, each tower of projective bundles associated to special multi- flags (cf [1], [13], [2], [3]), we can associate such a tower of sphere bundles w…

A differential operator introduced by A. Gray on the unit sphere bundle of a Kähler-Einstein manifold is studied. A lower bound for the first eigenvalue of the Laplacian for the Sasaki metric on the unit sphere bundle of a Kähler-Einstein manifold is derived. Some rigidity theorems classifying complex space forms among…

Let M be one of the projective spaces CP^n, HP^n for n>1 or the Cayley projective plane OP^2, and let LM denote the free loop space on M. Using Morse theory methods, we prove that the suspension spectrum of (LM)_+ is homotopy equivalent to the suspension spectrum of M_+ wedge a family of Thom spaces of explicit vector …

We estimate the dimensions of the spaces of holomorphic sections of certain line bundles to give improved lower bounds on the index of complex isotropic harmonic maps to complex projective space from the sphere and torus, and in some cases from higher genus surfaces.

Let $S^{n}$ be the $n$-sphere of constant positive curvature. For $n \geq 2$, we will show that a measure on the unit tangent bundle of $S^{2n}$, which is even and invariant under the geodesic flow, is not uniquely determined by its projection to $S^{2n}$.

Study on topological rigidity of ALE vector bundles with specific conditions.

In this paper we develop the theory of Willmore sequences for Willmore surfaces in the 4-sphere. We show that under appropriate conditions this sequence has to terminate. In this case the Willmore surface either is the twistor projection of a holomorphic curve into complex projective space or the inversion of a minimal…

Study on triviality of tangent and generalized tangent bundles of manifolds.

We study the space of conformal immersions of a 2-torus into the 4-sphere. The moduli space of generalized Darboux transforms of such an immersed torus has the structure of a Riemann surface, the spectral curve. This Riemann surface arises as the zero locus of the determinant of a holomorphic family of Dirac type opera…

The paper studies Sasakian geometry on sphere bundles, focusing on extremal metrics and cohomology.

We produce new non-Kähler complete steady gradient Ricci solitons whose asymptotics combine those of the Bryant solitons and the Hamilton cigar. We also obtain a family of complete Ricci-flat metrics with asymptotically locally conical asymptotics. Finally, we obtain numerical evidence for complete steady soliton struc…

Compact character varieties of punctured spheres are proven.

We give a sufficient condition ensuring that the mean curvature flow commutes with a Riemannian submersion and we use this result to create new examples of evolution by mean curvature flow. In particular we consider evolution of pinched submanifolds of the sphere, of the complex projective space, of the Heisenberg grou…

We produce new non-Kähler, non-Einstein, complete expanding gradient Ricci solitons with conical asymptotics and underlying manifold of the form $\R^2 \times M_2 \times \cdots \times M_r$, where $r \geq 2$ and $M_i$ are arbitrary closed Einstein spaces with positive scalar curvature. We also find numerical evidence for…

We present the non-trivial example how to generate non-Euclidean geometries from associative unital algebras. We consider bundles of the sphere of the degenerate non-Eucleadian space and its two models. The first (conformal) model is obtained by the mapping S onto a plane pass through the origin. It is analogous to the…

New minimal surfaces in spheres with complex topologies from capillarity.

The study preserves positive Ricci curvature on connected sums of fibre bundles.

In this paper, we investigate the behaviour of the Serre spectral sequence with respect to the algebraic structures of string topology in generalized homology theories, specificially with the Chas-Sullivan product and the corresponding coproduct and module structures. We prove compatibility for two kinds of fibre bundl…

The present article studies the class of Einstein-Hermitian harmonic maps of constant Kaehler angle from the projective line into quadrics. We provide a description of their moduli spaces up to image, and gauge-equivalence using the language of vector bundles and representation theory. It is shown that the dimension of…

One says that a smooth manifold M is a pseudo-Riemannian manifold of signature (p,q) if the tangent bundle TM is equipped with a smooth non-degenerate symmetric inner product g of signature (p,q). Similarly one says that M is an affine manifold if TM is equipped with a torsion free connection. One says g is Osserman if…

When can a map between manifolds be deformed away from itself? We describe a (normal bordism) obstruction which is often computable and in general much stronger than the classical primary obstruction in cohomology. In particular, it answers our question completely in a large dimension range. As an illustration we give …

We describe a construction of Riemannian metrics of nonnegative sectional curvature on a closed smooth nonorientable 4-manifold with fundamental group of order two that realizes a homotopy class that was not previously known to contain nonnegatively curved manifolds. The procedure yields new metrics of nonnegative sect…

A self-transverse immersion of the 2-sphere into 4-space with algebraic number of self intersection points equal to -n induces an immersion of the circle bundle over the 2-sphere of Euler class 2n into 4-space. Precomposing the circle bundle immersions with their universal covering maps, we get for n>0 immersions g_n o…

We show a generic finiteness result for least area planes in 3-dimensional hyperbolic space. Moreover, we prove that the space of minimal immersions of disk into hyperbolic space is a submanifold of a product bundle over a space of immersions of circle into sphere at infinity. The bundle projection map when restricted …

An explicit construction of surfaces with flat normal bundle in the Euclidean space (unit hypersphere) in terms of solutions of certain linear system is proposed. In the case of 3-space our formulae can be viewed as the direct Lie sphere analog of the generalized Weierstrass representation of surfaces in conformal geom…

A well known result on pseudodifferential operators states that the noncommutative residue (Wodzicki residue) of a pseudodifferential projection vanishes. This statement is non-local and implies the regularity of the eta invariant at zero of Dirac type operators. We prove that in a filtered algebra the value of a proje…

We construct Einstein metrics of non-positive scalar curvature on certain solid torus bundles over a Fano Kahler-Einstein manifold. We show, among other things, that the negative Einstein metrics are conformally compact, and the Ricci-flat metrics have slower-than-Euclidean volume growth and quadratic curvature decay. …

Study nondegenerate fibrations of Euclidean spaces and their relation to sphere fibrations.

The loop space of the Riemann sphere consisting of all $C^k$ or Sobolev $W^{k,p}$ maps from the circle $S^1$ to the sphere is an infinite dimensional complex manifold. We compute the Picard group of holomorphic line bundles on this loop space as an infinite dimensional complex Lie group with Lie algebra the first Dolbe…

We construct and study a natural homeomorphism between the moduli space of polynomial cubic differentials of degree d on the complex plane and the space of projective equivalence classes of oriented convex polygons with d+3 vertices. This map arises from the construction of a complete hyperbolic affine sphere with pres…

We show that on a surface locally every affine torsion-free connection is projectively equivalent to a Weyl connection. First, this is done using exterior differential system theory. Second, this is done by showing that the solutions of the relevant PDE are in one-to-one correspondence with the sections of the `twistor…

We compute the Szego kernel of the unit circle bundle of a negative line bundle dual to a regular quantum line bundle over a compact Kaehler manifold. As a corollary we provide an infinite family of smoothly bounded strictly pseudoconvex domains on complex manifolds (disk bundles over homogeneous Hodge manifolds) for w…

The paper classifies Willmore 2-spheres in $S^n$.

Spherical T-duality for iterated sphere bundles

Study spherical T-duality and Massey products in iterated sphere bundles.

For a given Hilbert space $\mathcal H$, consider the space of self-adjoint projections $\mathcal P(\mathcal H)$. In this paper we study the differentiable structure of a canonical sphere bundle over $\mathcal P(\mathcal H)$ given by $$ \mathcal R=\{\, (P,f)\in \mathcal P(\mathcal H)\times \mathcal H \, : \, Pf=f , \, \…

We prove that all smooth sphere bundles that admit fiberwise 1/4-pinched metrics are induced bundles of vector bundles, so their structure groups reduce from the diffeomorphism group of the sphere to the orthogonal group. This result implies the existence of many smooth n-sphere bundles over a k-sphere that do not supp…

Sphere bundles over 4-manifolds are trivial after looping, except for two cases.