David Klapheck, Michael VanValkenburgh
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.
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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.
We prove that singular Riemannian foliations in Euclidean spheres can be defined by polynomial equations.
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. In this paper we show that Singular Riemannian Foliations in spheres, of dimension at most 3, are orbits of an isometric group action.
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.
In this paper, we study biharmonic isometric immersions of a surface into and biharmonic Riemannian submersion from 3-dimensional Berger spheres. We obtain a classification of proper biharmonic isometric immersions of a surface with constant mean curvature into Berger 3-spheres. We also give a complete classification of proper biharmonic Hopf tori in Berger 3-sphere. For Riemannian submersions, we prove that a Riemannian submersion from Berger 3-spheres into a surface is biharmonic if and only if it is harmonic.
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 geodesics on this sub-Riemannian manifold making use of the Hamiltonian formalism and solving the corresponding Hamiltonian system.
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.
In this paper we give bounds for the first eigenvalue of the conformal Laplacian and the Yamabe invariant of a compact Riemannian manifold, by using conditions on the Ricci curvature and the diameter and deduce certain conditions on the manifold to be conformal to a sphere.
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 sphere as well.
We prove the existence of immersed closed curves of constant geodesic curvature in an arbitrary Riemannian 2-sphere for almost every prescribed curvature. To achieve this, we develop a min-max scheme for a weighted length functional.
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.
We consider a $3$-dimensional differentiable manifold with two circulant structures -- a Riemannian metric and an additional structure, whose third power is the identity. The structure is compatible with the metric such that an isometry is induced in any tangent space of the manifold. Further, we consider an associated metric with the Riemannian metric, which is necessary indefinite. We find equations of a sphere and of a circle, which are given in terms of the associated metric, with respect to the Riemannian metric.
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 or an extrinsic sphere of N.
A contact metric manifold is said to be $H$-contact, if the characteristic vector field is harmonic. We prove that the unit tangent bundle of a Riemannian manifold $M$ equipped with the standard contact metric structure is $H$-contact if and only if $M$ is $2$-stein.
We consider the sub-Riemannian $3$-sphere $(\mathbb{S}^3,g_h)$ obtained by restriction of the Riemannian metric of constant curvature $1$ to the planar distribution orthogonal to the vertical Hopf vector field. It is known that $(\mathbb{S}^3,g_h)$ contains a family of spherical surfaces $\{\mathcal{S}_λ\}_{λ\geq 0}$ with constant mean curvature $λ$. In this work we first prove that the two closed half-spheres of $\mathcal{S}_0$ with boundary $C_0=\{0\}\times\mathbb{S}^1$ minimize the sub-Riemannian area among compact $C^1$ surfaces with the same boundary. We also see that the only $C^2$ solut…
We extend the Lyapunov-Schmidt analysis of outlying stable CMC spheres in the work of S. Brendle and the second-named author to the "far-off-center" regime and to include general Schwarzschild asymptotics. We obtain sharp existence and non-existence results for large stable CMC spheres that depend very delicately on the behavior of scalar curvature at infinity.
In this paper we study almost complex and almost para-complex Cayley structures on six-dimensional pseudo-Riemannian spheres in the space of purely imaginary octaves of the split Cayley algebra $\mathbf{Ca}'$. It is shown that the Cayley structures are non-integrable, their basic geometric characteristics are calculated. In contrast to the usual Riemann sphere $\mathbb{S}^6$, there exist (integrable) complex structures and para-complex structures on the pseudospheres under consideration.
Riemannian 3-spheres that are hard to sweep out by short curves
构造一族直径和体积小于1的黎曼三维球面，其中满足拓扑条件的双参数闭曲线族必含长于任意给定L的曲线，并阻碍极小极大方法对最短闭测地线及正交测地弦长度的估计。
We construct a co-dimension $3$ completely non-holonomic sub-bundle on the Gromoll-Meyer exotic $7$ sphere based on its realization as a base space of a Sp(2)-principal bundle with the structure group Sp(1). The same method is valid for constructing a co-dimension 3 completely non-holonomic sub-bundle on the standard 7 sphere (or more general on a $4n+3$ dimensional standard sphere). In the latter case such a construction based on the Hopf bundle is well-known. Our method provides an alternated simple proof for the standard sphere $\mathbb{S}^7$.
The famous theorems of Cartan, related to the axiom of $r$-planes, and Leung-Nomizu about the axiom of $r$-spheres were extended to Kähler geometry by several authors. In this paper we replace the strong notions of totally geodesic submanifolds ($r$-planes) and extrinsic spheres ($r$-spheres) by a wider class of special isometric immersions such that theorems of type "axioms of $r$-special submanifolds" could hold. We verify also that there are plenty of special submanifolds in real and complex space forms and, in the codimension one case, in Einstein manifolds.
In this article we study the sub-Riemannian geometry of the spheres $S^{2n+1}$ and $S^{4n+3}$, arising from the principal $S^1-$bundle structure defined by the Hopf map and the principal $S^3-$bundle structure given by the quaternionic Hopf map respectively. The $S^1$ action leads to the classical contact geometry of $S^{2n+1}$, while the $S^3$ action gives another type of sub-Riemannian structure, with a distribution of corank 3. In both cases the metric is given as the restriction of the usual Riemannian metric on the respective horizontal distributions. For the contact $S^7$ case, we give a…
Inspired by the Lichnerowicz-Obata theorem for the first eigenvalue of the Laplacian, we define a new family of invariants $\{Ω_k(g)\}$ for closed Riemannian manifolds. The value of $Ω_k(g)$ delicately reflects the spherical part of the manifold. Indeed, $Ω_1(g)$ and $Ω_2(g)$ characterize the standard sphere.
We consider the conformal class of the Riemannian product $g_0 + g$, where $g_0$ is the constant curvature metric on $S^m$ and $g$ is a metric of constant scalar curvature on some closed manifold. We show that the number of metrics of constant scalar curvature in the conformal class grows at least linearly with respect to the square root of the scalar curvature of $g$. This is obtained by studying radial solutions of the equation $Δu -λu + λu^p =0$ on $S^m$, and the number of solutions in terms of $λ$.
Morse Index bound of simple closed geodesics on 2-spheres and strong Morse Inequalities
本文利用Morse理论刻画黎曼二维球面上的简单闭测地线，证明在通用度量下存在Morse指标为1、2、3的简单闭测地线，并对可定向黎曼曲面上的简单闭曲线空间建立了长度泛函的强Morse不等式。
Let $M$ be a Riemannian $2$-sphere. A classical theorem of Lyusternik and Shnirelman asserts the existence of three distinct simple non-trivial periodic geodesics on $M$. In this paper we prove that there exist three simple periodic geodesics with lengths that do not exceed $20d$, where $d$ is the diameter of $M$. We also present an upper bound that depends only on the area and diameter for the lengths of the three simple periodic geodesics with positive indices that appear as minimax critical values in the classical proofs of the Lyusternik-Shnirelman theorem. Finally, we present better bound…
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 group and the tangent sphere bundle equipped with the Sasaki metric.
In this paper we develop new methods of study of generalized normal homogeneous Riemannian manifolds. In particular, we obtain a complete classification of generalized normal homogeneous Riemannian metrics on spheres. We prove that for any connected (almost effective) transitive on $S^n$ compact Lie group $G$, the family of $G$-invariant Riemannian metrics on $S^n$ contains generalized normal homogeneous but not normal homogeneous metrics if and only if this family depends on more than one parameters. Any such family (that exists only for $n=2k+1$) contains a metric $g_{\can}$ of constant sect…
For a singular Riemannian foliation $\mathcal{F}$ on a Riemannian manifold, a curve is called horizontal if it meets the leaves of $\mathcal{F}$ perpendicularly. For a singular Riemannian foliation $\mathcal{F}$ on a unit sphere $\mathbb{S}^{n}$, we show that if $\mathcal{F}$ is a polar foliation or if $\mathcal{F}$ is given by the orbits of an infinitesimally polar action, then the horizontal diameter of $\mathbb{S}^{n}$ is $π$, i.e., any two points in $\mathbb{S}^{n}$ can be connected by a horizontal curve of length $\leqπ$.
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 $(M, g)$.
Transverse sphere theorems for Riemannian foliations
本文将经典球定理理论扩展到黎曼流形的横截几何，证明了横截版本的Grove-Shiohama直径球定理和Berger-Klingenberg四分之一压缩球定理，探讨了流形的叶空间结构和Gromov-Hausdorff极限。
In this paper we construct smooth Riemannian metrics on the sphere which admit smooth Zoll families of minimal hypersurfaces. This generalizes a theorem of Guillemin for the case of geodesics. The proof uses the Nash-Moser Inverse Function Theorem in the tame maps setting of Hamilton. This answers a question of Yau on perturbations of minimal hypersurfaces in positive Ricci curvature. We also consider the case of the projective space and characterize those metrics on the sphere with minimal equators.
The key condition A3w of Ma, Trudinger and Wang for regularity of optimal transportation maps is implied by the nonnegativity of a pseudo-Riemannian curvature -- which we call cross-curvature -- induced by the transportation cost. For the Riemannian distance squared cost, it is shown that (1) cross-curvature nonnegativity is preserved for products of two manifolds; (2) both A3w and cross-curvature nonnegativity are inherited by Riemannian submersions; and (3) the $n$-dimensional round sphere satisfies cross-curvature nonnegativity. From these results, a large new class of Riemannian manifolds …
We prove the existence of branched immersed constant mean curvature 2-spheres in an arbitrary Riemannian 3-sphere for almost every prescribed mean curvature, and moreover for all prescribed mean curvatures when the 3-sphere is positively curved. To achieve this, we develop a min-max scheme for a weighted Dirichlet energy functional. There are three main ingredients in our approach: a bi-harmonic approximation procedure to obtain compactness of the new functional, a derivative estimate of the min-max values to gain energy upper bounds for min-max sequences for almost every choice of mean curvat…
Our main aim is to present a geometrically meaningful formula for the fundamental solutions to a second order sub-elliptic differential equation and to the heat equation associated with a sub-elliptic operator in the sub-Riemannian geometry on the unit sphere $\mathbb S^3$. Our method is based on the Hamiltonian approach, where the corresponding Hamitonian system is solved with mixed boundary conditions. A closed form of the modified action is given. It is a sub-Riemannian invariant and plays the role of a distance on $\mathbb S^3$.
We study stable surfaces, i.e., second order minima of the area for variations of fixed volume, in sub-Riemannian space forms of dimension $3$. We prove a stability inequality and provide sufficient conditions ensuring instability of volume-preserving area-stationary $C^2$ surfaces with a non-empty singular set of curves. Combined with previous results, this allows to describe any complete, orientable, embedded and stable $C^2$ surface $Σ$ in the Heisenberg group $\mathbb{H}^1$ and the sub-Riemannian sphere $\mathbb{S}^3$ of constant curvature $1$. In $\mathbb{H}^1$ we conclude that $Σ$ is a E…
In some other context, the question was raised how many nearly Kähler structures exist on the sphere $§^6$ equipped with the standard Riemannian metric. In this short note, we prove that, up to isometry, there exists only one. This is a consequence of the description of the eigenspace to the eigenvalue $λ= 12$ of the Laplacian acting on 2-forms. A similar result concerning nearly parallel $\G_2$-structures on the round sphere $§^7$ holds, too. An alternative proof by Riemannian Killing spinors is also indicated.
This article studies the harmonicity of vector fields on Riemannian manifolds, viewed as maps into the tangent bundle equipped with a family of Riemannian metrics. Geometric and topological rigidity conditions are obtained, especially for surfaces and vector fields of constant norm, and existence is proved on two-tori. Classifications are given for conformal, quadratic and Killing vector fields on spheres. Finally, the class of metric considered on the tangent bundle is enlarged, permitting new vector fields to become harmonic.
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.
Given a 3-dimensional Riemannian manifold $(M,g)$, we prove that if $(Φ_k)$ is a sequence of Willmore spheres (or more generally area-constrained Willmore spheres), having Willmore energy bounded above uniformly strictly by $8 π$, and Hausdorff converging to a point $\bar{p}\in M$, then $Scal(\bar{p})=0$ and $\nabla Scal(\bar{p})=0$ (resp. $\nabla Scal(\bar{p})=0$). Moreover, a suitably rescaled sequence smoothly converges, up to subsequences and reparametrizations, to a round sphere in the euclidean 3-dimensional space. This generalizes previous results of Lamm and Metzger contained in \cite{…
Small spheres with prescribed nonconstant mean curvature in Riemannian manifolds
本文证明了在无边界光滑黎曼流形中，函数非退化临界点附近存在由平均曲率与该函数成比例的球面构成的唯一叶状结构，并指出放宽非退化假设会丧失该叶状结构性质。
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 particular the description of Pansu spheres in the first Heisenberg group and the sub-Riemannian $3$-sphere. Then, we study variational properties of $\mathcal{S}_λ(p)$ related to the area functional. First, we obtain uniqueness results for the spheres $\mathcal{S}_λ(p)$ as critical…
Spheres with parallel mean curvature in $\mathbb{S}^2 \times \mathbb{H}^2$
本文研究球面与双曲平面乘积空间中平行平均曲率曲面，分类了因拓扑原因使全纯二次微分消失的球面。
Starting from $g$-natural pseudo-Riemannian metrics of suitable signature on the unit tangent sphere bundle $T_1 M$ of a Riemannian manifold $(M,\langle,\rangle)$, we construct a family of paracontact metric structures. We prove that this class of paracontact metric structures is invariant under $\mathcal D$-homothetic deformations, and classify paraSasakian and paracontact $(κ,μ)$-spaces inside this class. We also present a way to build paracontact $(κ,μ)$-spaces from corresponding contact metric structures on $T_1 M$.
Let $(M,g_M,\mathcal F)$ be a closed, connected Riemannian manifold with a Riemannian foliation $\mathcal F$ of nonzero constant transversal scalar curvature. When $M$ admits a transversal nonisometric conformal field, we find some generalized conditions that $\mathcal F$ is transversally isometric to the sphere.
We show that, the solutions of the isoperimetric problem for small volumes are $C^{2,α}$-close to small spheres. On the way, we define a class of submanifolds called pseudo balls, defined by an equation weaker than constancy of mean curvature. We show that in a neighborhood of each point of a compact riemannian manifold, there is a unique family concentric pseudo balls which contains all the pseudo balls $C^{2,α}$-close to small spheres. This allows us to reduce the isoperimetric problem for small volumes to a variational problem in finite dimension.
Spheres with minimal equators
研究球体上使所有赤道成为最小超曲面的黎曼度量分类及相关几何问题。
The first aim of the present paper is to compare various sub-Riemannian structures over the three dimensional sphere $S^3$ originating from different constructions. Namely, we describe the sub-Riemannian geometry of $S^3$ arising through its right Lie group action over itself, the one inherited from the natural complex structure of the open unit ball in $\comp^2$ and the geometry that appears when considering the Hopf map as a principal bundle. The main result of this comparison is that in fact those three structures coincide. In the second place, we present two bracket generating distribution…
Generalized Surgery on Riemannian Manifolds of Positive Ricci Curvature
本研究推广Wraith的正Ricci曲率手术定理，用带core metric的球面丛粘合替代球面与圆盘乘积的附加操作，在底空间容许core metric的2维球面丛上构造出core metric，得到新的6维正Ricci曲率流形例子。
Myers-Steenrod Theorems for Metric and Singular Riemannian Foliations
本文研究了度量与奇异黎曼叶的等距群结构，证明了其维数上界，并指出该上界由特定纤维丛诱导。
Geodesic graphs for geodesic orbit Finsler $(α,β)$ metrics on spheres
研究球面上由 Riemannian 测地轨道度量生成的不变 Finsler $(α,β)$ 度量，通过显式测地图构造刻画其轨道条件，并证明非实射影空间不存在纯 Finsler 情形。
We prove that the systolic ratio of a sphere of revolution $S$ does not exceed $π$ and equals $π$ if and only if $S$ is Zoll. More generally, we consider the rotationally symmetric Finsler metrics on a sphere of revolution which are defined by shifting the tangent unit circles by a Killing vector field. We prove that in this class of metrics the systolic ratio does not exceed $π$ and equals $π$ if and only if the metric is Riemannian and Zoll.
Given a sweepout of a Riemannian 2-sphere which is composed of curves of length less than L, we construct a second sweepout composed of curves of length less than L which are either constant curves or simple curves. This result, and the methods used to prove it, have several consequences; we answer a question of M. Freedman concerning the existence of min-max embedded geodesics, we partially answer a question due to N. Hingston and H.-B. Rademacher, and we also extend the results of [CL] concerning converting homotopies to isotopies in an effective way.
Let $G$ be the identity component of the isometry group for an arbitrary curved two-point homogeneous space $M$. We consider algebras of $G$-invariant differential operators on bundles of unit spheres over $M$. The generators of this algebra and the corresponding relations for them are found. The connection of these generators with two-body problem on two-point homogeneous spaces is discussed.
Natural metric structures on the tangent bundle and tangent sphere bundles $S_rM$ of a Riemannian manifold $M$ with radius function $r$ enclose many important unsolved problems. Admitting metric connections on $M$ with torsion, we deduce the equations of induced metric connections on those bundles. Then the equations of reducibility of $TM$ to the almost Hermitian category. Our purpose is the study of the natural contact structure on $S_rM$ and the $G_2$-twistor space of any oriented Riemannian 4-manifold.
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.
In a previous work, we studied isoparametric functions on Riemannian manifolds, especially on exotic spheres. One result there says that, in the family of isoparametric hypersurfaces of a closed Riemannian manifold, there exist at least one minimal isoparametric hypersurface. In this note, we show such minimal isoparametric hypersurface is also unique in the family if the ambient manifold has positive Ricci curvature. Moreover, we give a proof of the Theorem D claimed by Q.M.Wang (without proof) which asserts that the focal submanifolds of an isoparametric function on a complete Riemannian man…
A half-geodesic is a closed geodesic realizing the distance between any pair of its points. All geodesics in a round sphere are half-geodesics. Conversely, this note establishes that Riemannian spheres with all geodesics closed and sufficiently many half-geodesics are round.
Rigidity of riemannian manifolds containing an equator
本文利用古代平均曲率流方法，证明了包含特定极小2-球面的黎曼流形具有常截面曲率1的刚性，并推得Simon-Smith极小球面的刚性结果。
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 light rays from p are shaded away from p by a single point shade. We prove that Riemannian manifolds with cross and sphere blocking are isometric to round spheres.
We show that any effective isometric torus action of maximal rank on a compact Riemannian manifold with positive (sectional) curvature and maximal symmetry rank, that is, on a positively curved sphere, lens space, complex or real projective space, is equivariantaly diffeomorphic to a linear action. We show that a compact, simply connected Riemannian 4- or 5-manifold of quasipositive curvature and maximal symmetry rank must be diffeomorphic to the 4-sphere, complex projective plane or the 5-sphere.
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