arXiv research

We show that a Riemannian foliation on a topological $n$-sphere has leaf dimension 1 or 3 unless n=15 and the Riemannian foliation is given by the fibers of a Riemannian submersion to an 8-dimensional sphere. This allows us to classify Riemannian foliations on round spheres up to metric congruence.

The paper classifies biharmonic immersions and submersions in specific spheres.

Singular Riemannian Foliations are particular types of foliations on Riemannian manifolds, in which leaves locally stay at a constant distance from each other. Singular Riemannian Foliations in round spheres play a special role, since they provide "infinitesimal information" about general Singular Riemannian Foliations…

Max systoles on spheres with punctures are counted.

Study on spheres with minimal equators.

The paper studies Pansu spheres in a sub-Riemannian 3-sphere and their area-minimizing properties.

Proves existence of sphere foliations with prescribed mean curvature on Riemannian manifolds.

Proves existence of curves with constant curvature in a sphere.

We prove that singular Riemannian foliations in Euclidean spheres can be defined by polynomial equations.

Characterizes geodesics on spheres with Morse index bounds and inequalities.

The unit sphere $\mathbb S^3$ can be identified with the unitary group SU(2). Under this identification the unit sphere can be considered as a non-commutative Lie group. The commutation relations for the vector fields of the corresponding Lie algebra define a 2-step sub-Riemannian manifold. We study sub-Riemannian geod…

We determine the lengths of all closed sub-Riemannian geodesics on the three-sphere. Our methods are elementary and allow us to avoid using explicit formulas for the sub-Riemannian geodesics.

Sphere theorems for specific manifolds with curvature constraints.

3D spheres can't be swept by short curves, complicating geodesic length estimates.

Proves existence of at least two minimal spheres in any 3D space.

We discuss the integrability of orthogonal almost complex structures on Riemannian products of even-dimensional round spheres and give a partial answer to the question raised by E. Calabi concerning the existence of complex structures on a product manifold of a round 2-sphere and a round 4-sphere.

New geometric invariant from min-max width of spheres on Riemannian 2-spheres.

The study classifies parallel mean curvature spheres in a sphere-hyperbolic product space.

Conditions for Riemannian manifolds to be Euclidean spheres or spaces.

Researchers create metrics on spheres with Ricci curvature ≥1, limiting to Grushin hemisphere.

We study a family of spheres with constant mean curvature (CMC) in the Riemannian Heisenberg group $H^1$. These spheres are conjectured to be the isoperimetric sets of $H^1$. We prove several results supporting this conjecture. We also focus our attention on the sub-Riemannian limit.

A closed Riemannian manifold is said to have cross blocking if whenever distinct points p and q are at distance less than the diameter, all light rays from p can be shaded away from q with at most two point shades. Similarly, a closed Riemannian manifold is said to have sphere blocking if for each point p, all the ligh…

The paper constructs metrics on spheres with families of minimal hypersurfaces.

New inequalities for spectral zeta kernels on spheres and manifolds.

Geodesic graphs for special Finsler metrics on spheres are studied.

Proves existence of special 2-spheres in curved 3-spaces.

Study of harmonic Riemannian submersions from 3D geometries.

3D spheres with certain properties approach the round sphere.

The paper studies stability of domains for the first eigenvalue on Riemannian manifolds.

The 3-sphere has either 2 minimal 2-spheres or an optimal foliation by 2-spheres.

Sphere theorems extended to Riemannian foliations with new results on curvature and leaf spaces.

The simple length spectrum of a Riemannian manifold is the set of lengths of its simple closed geodesics. We prove a theorem claimed by Lusternik: in any Riemannian 2-sphere whose simple length spectrum consists of only one element L, any geodesic is simple closed with length L.

A sphere has at least two geodesics whose product length is bounded by a constant times the area.

We prove that any Riemannian two-sphere with area at most 1 can be continuously mapped onto a tree in a such a way that the topology of fibers is controlled and their length is less than 7.6. This result improves previous estimates and relies on a similar statement for Riemannian two-disks.

Study verifies Homogeneity Conjecture for three odd-dimensional spheres in positive curvature.

Using representations of Clifford algebras we construct indecomposable singular Riemannian foliations on round spheres, most of which are non-homogeneous. This generalizes the construction of non-homogeneous isoparametric hypersurfaces due to by Ferus, Karcher and Munzner.

The study proves conditions for convex hypersurfaces in Riemannian manifolds to be rational homology spheres.

The present paper shows that for a given integer k greater than 2 it is possible to construct an at least k-differentiable Riemannian metric on the sphere of a certain dimension such that the cut locus of a point of it becomes a fractal. Moreover, we show that this construction can be extended to the case of Finsler sp…

Let Sol be the three-dimensional solvable Lie group equipped with its standard left-invariant Riemannian metric. We give a precise description of the cut locus of the identity, and a maximal domain in the Lie algebra on which the Riemannian exponential map is a diffeomorphism. As a consequence, we prove that the metric…

Let $(M, g)$ be a complete Riemannian $3$-manifold that is asymptotic to Schwarzschild with positive mass and whose scalar curvature vanishes. We \textsl{unconditionally} characterize the large, embedded stable constant mean curvature spheres in $(M, g)$.

The invariant metric affine connections on Berger spheres which are Einstein with skew torsion are determined in both Riemannian and Lorentzian signature. Expressions of such connections are explicitly given. In particular, every Berger sphere with Lorentzian signature admits invariant metric affine connections Einstei…

New metric found for 4-manifolds with specific properties.

Study shows limits of metrics with positive scalar curvature on spheres.

The paper studies geometric structures on tangent and sphere bundles over statistical manifolds.

Let $M$ be a complete Sasakian sub-Riemannian $3$-manifold of constant Webster scalar curvature $κ$. For any point $p\in M$ and any number $λ\in\mathbb{R}$ with $λ^2+κ>0$, we show existence of a $C^2$ spherical surface $\mathcal{S}_λ(p)$ immersed in $M$ with constant mean curvature $λ$. Our construction recovers in par…

Compact gradient ρ-Einstein solitons are isometric to Euclidean spheres.

The following Theorem is proved: Let M be an n-dimensional (n>2) submanifold of a Riemannian manifold N. Suppose that through each point p of M there exist two (n-1)-dimensional extrinsic spheres of N, which are contained in M in a neighbourhood of p and are tangent to each other at p. Then M is totally geodesic in N o…

We show, as our main theorem, that if a Lipschitz map from a compact Riemannian manifold $M$ to a connected compact Riemannian manifold $N$, where $\dim M \geq \dim N$, has no singular points on $M$ in the sense of F.H. Clarke, then the map admits a smooth approximation via Ehresmann fibrations. We also show the Reeb s…