三维球面中的螺旋极小曲面：一种通过球面曲线的研究方法

Pseudo-spherical submanifolds with 1-type pseudo-spherical Gauss map

In this work, we study the pseudo-Riemannian submanifolds of a pseudo-sphere with 1-type pseudo-spherical Gauss map. First, we classify the Lorentzian surfaces in a 4-dimensional pseudo-sphere $\mathbb{S}^4_s(1)$ with index s, $s=1, 2$, and having harmonic pseudo-spherical Gauss map. Then we give a characterization theorem for pseudo-Riemannian submanifolds of a pseudo-sphere $\mathbb{S}^{m-1}_s(1)\subset\mathbb{E}^m_s$ with 1-type pseudo-spherical Gauss map, and we classify spacelike surfaces and Lorentzian surfaces in the de Sitter space $\mathbb{S}^4_1(1)\subset\mathbb{E}^5_1$ with 1-type p…

Characterization of curves that lie on a geodesic sphere or on a totally geodesic hypersurface in a hyperbolic space or in a sphere

The consideration of the so-called rotation minimizing frames allows for a simple and elegant characterization of plane and spherical curves in Euclidean space via a linear equation relating the coefficients that dictate the frame motion. In this work, we extend these investigations to characterize curves that lie on a geodesic sphere or totally geodesic hypersurface in a Riemannian manifold of constant curvature. Using that geodesic spherical curves are normal curves, i.e., they are the image of an Euclidean spherical curve under the exponential map, we are able to characterize geodesic spher…

The geodesic total curvature of spherical curves

The geodesic total curvature of rectifiable spherical curves is analyzed. We extend to the case of high dimension spheres the explicit formula that holds true for curves supported into the 2-sphere. For this purpose, we take advantage of some new integral-geometric formulas concerning both the Euclidean and geodesic total curvature of spherical curves.

Positively curved manifolds with large spherical rank

Rigidity results are obtained for Riemannian $d$-manifolds with $\sec \geqslant 1$ and spherical rank at least $d-2>0$. Conjecturally, all such manifolds are locally isometric to a round sphere or complex projective space with the (symmetric) Fubini--Study metric. This conjecture is verified in all odd dimensions, for metrics on $d$-spheres when $d \neq 6$, for Riemannian manifolds satisfying the Rakić duality principle, and for Kählerian manifolds.

Differential Geometry of Rotation Minimizing Frames, Spherical Curves, and Quantum Mechanics of a Constrained Particle

This thesis is devoted to the Differential Geometry of curves and surfaces along with applications in Quantum Mechanics. In the 1st part we introduce the well known Frenet frame. Later, we show that the curvature function is a lower bound for the scalar angular velocity of any other orthonormal moving frame, from which one defines Rotation Minimizing (RM) frames as the ones that achieve this minimum. Remarkably, RM frames are ideal to study spherical curves and allow us to characterize them through a simple linear equation. We also apply these ideas to curves that lie on level surfaces, by rei…

Constant Mean Curvature Hypersurfaces in the (n+1)-Sphere by Gluing Spherical Building Blocks

The techniques developed by Butscher in arXiv:math/0703469 for constructing constant mean curvature (CMC) hypersurfaces in the (n+1)-sphere by gluing together spherical building blocks are generalized to handle less symmetric initial configurations. The outcome is that the approximately CMC hypersurface obtained by gluing the initial configuration together can be perturbed into an exactly CMC hypersurface only when certain global geometric conditions are met. These `balancing conditions' are analogous to those that must be satisfied in the `classical' context of gluing constructions of CMC hyp…

Funk-Minkowski transform and spherical convolution of Hilbert type in reconstructing functions on the sphere

The Funk--Minkowski transform ${\mathcal F}$ associates a function $f$ on the sphere ${\mathbb S}^2$ with its mean values (integrals) along all great circles of the sphere. Thepresented analytical inversion formula reconstruct the unknown function $f$ completely if two Funk--Minkowski transforms, ${\mathcal F}f$ and ${\mathcal F} \nabla f$, are known. Another result of this article is related to the problem of Helmholtz--Hodge decomposition for tangent vector field on the sphere ${\mathbb S}^2$. We proposed solution for this problem which is used the Funk-Minkowski transform ${\mathcal F}$ and…

On the Burns-Epstein invariants of spherical CR 3-manifolds

In this paper we develop a method to compute the Burns-Epstein invariant of a spherical CR homology sphere, up to an integer, from its holonomy representation. As application, we give a formula for the Burns-Epstein invariant, modulo an integer, of a spherical CR structure on a Seifert fibered homology sphere in terms of its holonomy representation.

Spherical conical metrics and harmonic maps to spheres

A spherical conical metric $g$ on a surface $Σ$ is a metric of constant curvature $1$ with finitely many isolated conical singularities. The uniformization problem for such metrics remains largely open when at least one of the cone angles exceeds $2π$. The eigenfunctions of the Friedrichs Laplacian $Δ_g$ with eigenvalue $λ=2$ play a special role in this problem, as they represent local obstructions to deformations of the metric $g$ in the class of spherical conical metrics. In the present paper we apply the theory of multivalued harmonic maps to spheres to the question of existence of such eig…

关于三维球面Sasakian流形的分类

On the classificarion of 3-dimensional spherical Sasakian manifolds

本文通过建立三组参数之间的对应关系，对三维球面Sasakian流形中的刚性球面进行分类，描述其模空间与自同构群，并识别出其中的齐性Sasakian流形。

Platonic polyhedra tune the 3-sphere: II. Harmonic analysis on cubic spherical 3-manifolds

From the homotopy groups of two cubic spherical 3-manifolds we construct the isomorphic groups of deck transformations acting on the 3-sphere. These groups become the cyclic group of order eight and the quaternion group respectively. By reduction of representations from the orthogonal group to the identity representation of these subgroups we provide two subgroup-periodic bases for the harmonic analysis on the 3-manifolds. This harmonic analysis has applications to cosmic topology.

Explicit description of spherical rigid hypersurfaces in C^2

We provide an explicit description of all rigid hypersurfaces that are equivalent to a Heisenberg sphere. These hypersurfaces are determined by 4 real parameters. The defining equations of the rigid spheres can also be viewed as the complete solution of a non-linear PDE that expresses the vanishing Cartan curvature condition for rigid hypersurfaces.

Almost everywhere divergence of spherical harmonic expansions and equivalence of summation methods

We show that there exists an integrable function on the $n$-sphere $(n\ge 2)$, whose Cesàro (C,$\frac{n-1}{2}$) means with respect to the spherical harmonic expansion diverge unboundedly almost everywhere. By studying equivalence theorems, we also obtain the corresponding results for Riesz and Bochner-Riesz means. This extends results of Stein (1961) for flat tori and complements the work of Taibleson (1985) for spheres.

Platonic polyhedra tune the 3-sphere: Harmonic analysis on simplices

A spherical topological manifold of dimension n-1 forms a prototile on its cover, the (n-1)-sphere. The tiling is generated by the fixpoint-free action of the group of deck transformations. By a general theorem, this group is isomorphic to the first homotopy group. Multiplicity and selection rules appear in the form of reduction of group representations. A basis for the harmonic analysis on the (n-1)-sphere is given by the spherical harmonics which transform according to irreducible representations of the orthogonal group. The deck transformations form a subgroup, and so the representations of…

将Caffarelli收缩定理推广到近球面流形

A Generalization of Caffarelli's Contraction Theorem to Nearly Spherical Manifolds

本文将Caffarelli收缩定理推广到近球面流形，并通过最优传输映射证明了Milman猜想的扰动形式。