Kang-Hai Tan
In this note we address a notion of sublaplacians of sub-Riemannian manifolds. In particular for fat sub-Riemannian manifolds we answered the sublaplacian question proposed by R. Montgomery.
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We obtain a Bochner type formula and an estimate from below on the spectrum of the sublaplacian of a compact strictly pseudoconvex CR manifold.
In this paper, we study the heat kernel associated to the intrinsic sublaplacian on a quaternionic contact manifold considered as a subriemannian manifold. More precisely, we explicitly compute the first two coefficients $c_0$ and $c_1$ appearing in the small time asymptotics expansion of the heat kernel on the diagonal. We show that the second coefficient $c_1$ depends linearly on the qc scalar curvature $κ$. Finally we apply our results to compact qc-Einstein manifolds and prove the spectral invariance of geometric quantities in the subriemannian setting.
The authors first in this paper define a semi-symmetric metric non-holonomic connection (called in briefly a semi-sub-Riemannian connection) on sub-Riemannian manifolds, and study the relations between sub-Riemannian connections and semi-sub-Riemannian connections. An invariant under a connection transformation $\nabla\rightarrow D$ is obtained. The authors then further deduce a sufficient and necessary condition that a sub-Riemannian manifold associated with a semi-sub-Riemannian connection is flat, and derive that a sub-Riemannian manifold with vanishing curvature with respect to semi-sub-Ri…
In the present paper we show that the geodesic flows of a sub-Riemannian metric and that of a Riemannian extension commute if and only if the extended metric is parallel with respect to a certain connection. This helps us to describe the geodesic flow of sub-Riemannian metrics on totally geodesic Riemannian submersions. As a consequence we can characterize sub-Riemannian geodesics as the horizontal lifts of projections of Riemannian geodesics.
The authors define a SNS (semi-nearly-sub)-Riemannian connection on nearly sub-Riemannian manifolds and study the geometric properties of such a connection, and obtain the natures of horizontal curvature tensors between horizontal sub-Riemannian connection and SNS-Riemannian connection. The authors further investigate the geometric characteristics of the projective SNS-Riemannian connection, and obtain a necessary and sufficient condition for a nearly sub-Riemannian manifold being projectively flat.
Sub-Riemannian cubics are a generalisation of Riemannian cubics to a sub-Riemannian manifold. Cubics are curves which minimise the integral of the norm squared of the covariant acceleration. Sub-Riemannian cubics are cubics which are restricted to move in a horizontal subspace of the tangent space. When the sub-Riemannian manifold is also a Lie group, sub-Riemannian cubics correspond to what we call a sub-Riemannian Lie quadratic in the Lie algebra. The present article studies sub-Riemannian Lie quadratics in the case of $\mathfrak{su}(2)$, focusing on the long term dynamics.
We study some sub-Riemannian objects (such as horizontal connectivity, horizontal connection, horizontal tangent plane, horizontal mean curvature) in hypersurfaces of sub-Riemannian manifolds. We prove that if a connected hypersurface in a contact manifold of dimension more than three is noncharacteristic or with isolated characteristic points, then given two points, there exists at least one piecewise smooth horizontal curve in this hypersurface connecting them. In any sub-Riemannian manifold, we obtain the sub-Riemannian version of the fundamental theorem of Riemannian geometry. We use this …
In this paper, we derived biharmonic equations for pseudo-Riemannian submanifolds of pseudo-Riemannian manifolds which includes the biharmonic equations for submanifolds of Riemannian manifolds as a special case. As applications, we proved that a pseudo-umbilical biharmonic pseudo-Riemannian submanifold of a pseudo-Riemannian manifold has constant mean curvature, we completed the classifications of biharmonic pseudo-Riemannian hypersurfaces with at most two distinct principal curvatures, which were used to give four construction methods to produce proper biharmonic pseudo-Riemannian submanifol…
We prove that the image of a real analytic Riemannian manifold under a smooth Riemannian submersion is necessarily real analytic.
We study generic Riemannian submersions from nearly Kaehler manifolds onto Riemannian manifolds. We investigate conditions for the integrability of various distributions arising for generic Riemannian submersions and also obtain conditions for leaves to be totally geodesic foliations. We obtain conditions for a generic Riemannian submersion to be a totally geodesic map and also study generic Riemannian submersions with totally umbilical fibers. Finally, we derive conditions for generic Riemannian submersions to be harmonic map.
In those lecture notes, we review some applications of heat semigroups methods in Riemannian and sub-Riemannian geometry. The notes contain parts of courses taught at Purdue University, Institut Henri Poincaré, Levico Summer School and Tata Institute.
We give a generalized curvature-dimension inequality connecting the geometry of sub-Riemannian manifolds with the properties of its sub-Laplacian. This inequality is valid on a large class of sub-Riemannian manifolds obtained from Riemannian foliations. We give a geometric interpretation of the invariants involved in the inequality. Using this inequality, we obtain a lower bound for the eigenvalues of the sub-Laplacian. This inequality also lays the foundation for proving several powerful results in Part~II.
In this paper, we consider critical maps of a horizontal energy functional for maps from a sub-Riemannian manifold to a Riemannian manifold. These critical maps are referred to as subelliptic harmonic maps. In terms of the subelliptic harmonic map heat flow, we investigate the existence problem for subelliptic harmonic maps. Under the assumption that the target Riemannian manifold has non-positive sectional curvature, we prove some Eells-Sampson type existence results for this flow when the source manifold is either a step-2 sub-Riemannian manifold or a step-r sub-Riemannian manifold whose sub…
The present paper is devoted to the problem of (local) geodesic equivalence of Riemannian metrics and sub-Riemannian metrics on generic corank 1 distributions. Using Pontryagin Maximum Principle, we treat Riemannian and sub-Riemannian cases in an unified way and obtain some algebraic necessary conditions for the geodesic equivalence of (sub-)Riemannian metrics. In this way first we obtain a new elementary proof of classical Levi-Civita's Theorem about the classification of all Riemannian geodesically equivalent metrics in a neighborhood of so-called regular (stable) point w.r.t. these metrics.…
Generalized Chen's inequalities for Riemannian submersions and Riemannian maps with Applications
利用广义δ-不变量，建立了黎曼浸没和黎曼映射的最优广义Chen不等式，刻画了等号成立的代数条件，并应用于实空间形式和复空间形式。
B. Sahin [9] introduced the notion of anti-invariant Riemannian submersions from almost Hermitian manifolds onto Riemannian manifolds. In the present paper we extend the notion of anti-invariant and Lagrangian Riemannian submersions (a special anti-invariant Riemannian submersion) to the case of locally conformal Kaehler manifolds. We discuss the geometry of foliation and obtain some decomposition theorems for the total manifold of such submersions.
In this paper, we define and study CR-lightlike submanifolds of a golden semi-Riemannian manifold. We investigate some properties of geodesic CR-submanifolds of a golden semi-Riemannian manifold. Moreover, we obtain many interesting results for totally geodesic and totally umbilical CR-submanifolds on a golden Riemannian manifold. A non-trivial example of CR-lightlike submanifold of a golden semi-Riemannian manifold is also constructed.
There are many equivalent definitions of Riemannian geodesics. They are naturally generalised to sub-Riemannian manifold, but become non-equivalent. We give a review of different definitions of geodesics of a sub-Riemannian manifold and interrelation between them. We recall three variational definitions of geodesics as (locally) shortest curves (Euler-Lagrange, Pontyagin and Hamilton) and three definitions of geodesics as straightest curves (d'Alembert , Levi-Civita-Schouten and Cartan-Tanaka ), used in nonholonomic mechanics and discuss their interrelations. We consider a big class of sub-Rie…
We introduce anti-invariant Riemannian submersions from Sasakian manifolds onto Riemannian manifolds. We survey main results of anti-invariant Riemannian submersions defined on Sasakian manifolds. We investigate necessary and sufficient condition for an anti-invariant Riemannian submersion to be totally geodesic and harmonic. We give examples of anti-invariant submersions such that characteristic vector field ξ is vertical or horizontal.
We introduce anti-invariant Riemannian submersions from cosymplectic manifolds onto Riemannian manifolds. We survey main results of anti-invariant Riemannian submersions defined on cosymplectic manifolds. We investigate necessary and sufficient condition for an anti-invariant Riemannian submersion to be totally geodesic and harmonic. We give examples of anti-invariant submersions such that characteristic vector field ξ is vertical or horizontal. Moreover we give decomposition theorems by using the existence of anti-invariant Riemannian submersions.
We introduce bi-slant $ξ^{\perp}$-Riemannian submersions from Sasakian manifolds onto Riemannian manifolds as a generalization of slant and semi-slant $ξ^{\perp}$-Riemannian submersion. We give an example and investigate the geometry foliations. After we obtain necessary and sufficient conditions related to totally geodesicness of submersion. Finally we give decomposition theorems for total manifold of such submersions.
This paper is concerned with the study of the Monge optimal transport problem in sub-Riemannian manifolds where the cost is given by the square of the sub-Riemannian distance. Our aim is to extend previous results on existence and uniqueness of optimal transport maps to cases of sub-Riemannian structures which admit many singular minimizing geodesics. We treat here the case of sub-Riemannian structures of rank two in dimension four.
In this letter we exhibit the relation between the isometries of a Riemannian contraction of a sub-Riemannian manifold and those of the sub-Riemannian metric, for to use this relation with two goals: establishing a result about the existence of fixed points of isometries groups; and the other, defining a Multiresolution Analysis (MRA) on sub-Riemannian manifolds that it will permit to obtain Haar's bases on the manifolds before mentioned. Keywords: Sub-Riemannian geometry, minimizing geodesic, Haar functions, self-similarity.
In this paper, we study slant submanifolds of Riemannian manifolds with Golden structure. A Riemannian manifold $(\tilde{M},\tilde{g},φ)$ is called a Golden Riemannian manifold if the $(1,1)$ tensor field $φ$ on $\tilde{M}$ is a golden structure, that is $φ^{2}=φ+I$ and the metric $\tilde{g}$ is $φ-$ compatible. First, we get some new results for submanifolds of a Riemannian manifold with Golden structure. Later we characterize slant submanifolds of a Riemannian manifold with Golden structure and provide some non-trivial examples of slant submanifolds of Golden Riemannian manifolds.
In this paper we study geodesics of left-invariant sub-Riemannian metrics on SO(3) and almost-Riemannian metrics on $S^2$. These structures are connected with each other, and it is possible to use information about one of them to obtain results about another one. We give an explicit parameterization of sub-Riemannian geodesics on SO(3) and use it to get a parameterization of almost-Riemannian geodesics on $S^2$. We use symmetries of the exponential map to obtain some necessary optimality conditions. We present some upper bounds on the cut time in both cases and describe periodic geodesics on S…
Let $M$ be a surface with a Riemannian metric and $UM$ the unit tangent bundle over $M$ with the canonical contact sub-Riemannian structure $D$ on $UM$. In this paper, the complete local classification of singularities, under the Legendre fibration $UM$ over $M$, is given for sub-Riemannian geodesics of $(UM, D)$. Legendre singularities of sub-Riemannian geodesics are classified completely also for another Legendre fibration from $UM$ to the space of Riemannian geodesics on $M$. The duality on Legendre singularities is observed related to the pendulum motion.
We introduce slant Riemannian submersions from Sasakian manifolds onto Riemannian manifolds. We survey main results of slant Riemannian submersions defined on Sasakian manifolds. We also give an example of such slant submersions.
The aim of the present paper to define and study semi-slant $ξ^\perp-$Riemannian submersions from Sasakian manifolds onto Riemannian manifolds as a generalization of anti-invariant $ξ^\perp-$Riemannian submersions, semi-invariant $ξ^\perp-$Riemannian submersions and slant Riemannian submersions. We obtain characterizations, investigate the geometry of foliations which arise from the definition of this new submersion. After we investigate the geometry of foliations, we obtain necessary and sufficient condition for base manifold to be a locally product manifold and proving new conditions to be t…
We classify the semi-Riemannian submersions from a pseudo-hyperbolic space onto a Riemannian manifold under the assumption that the fibres are connected and totally geodesic. Also we obtain the classification of the semi-Riemannian submersions from a complex pseudo-hyperbolic space onto a Riemannian manifold under the assumption that the fibres are complex, connected and totally geodesic submanifolds.
Slant and semi-slant submanifolds in metallic Riemannian manifolds
该文研究金属黎曼流形中斜与半斜子流形的性质，给出相关子流形的刻画、分布可积性条件，并提供对应实例。
In this research article, we establish the geometrical bearing on Riemannian submersions in terms of $η$-Ricci-Yamabe Soliton with the potential field and giving the classification of any fiber of Riemannian submersion is an $η$-Ricci-Yamabe soliton, $η$-Ricci soliton and $η$-Yamabe soliton. We also discuss the various conditions for which the target manifold of Riemannian submersion is an $η$-Ricci-Yamabe soliton, $η$-Ricci soliton, $η$-Yamabe soliton and quasi-Yamabe soliton. In a particular case when the potential filed $V$ of the $η$-Ricci-Yamabe soliton is of gradient type, we derive a La…
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.
The purpose of this paper is to study anti-invariant Riemannian submersions from Kenmotsu manifolds onto Riemannian manifolds. Several fundamental results in this respect are proved. The integrability of the distributions and the geometry of foliations are investigated. We proved that there do not exist (anti-invariant) Riemannian submersions from Kenmotsu manifolds onto Riemannian manifolds such that characteristic vector field ξ is a vertical vector field. We gave a method to get horizontally conformal submersion examples from warped product manifolds onto Riemannian manifolds. Furthermore, …
Under a nondegeneracy condition, we show that an equiregular sub-Riemannian manifold of step size $r$ admits a canonical, $V$-rigid complement defined from the sub-Riemannian data that is preserved the by action of sub-Riemannian isometries. We explore how the existence of such a complement relates to results from the literature and study the step size 2 case in more detail.
Length spectra for Riemannian metrics are well studied, while sub-Riemannian length spectra have been largely unexplored. Here we give the length spectrum for a canonical sub-Riemannian structure attached to any compact Lie group by restricting its Killing form to the sum of the root spaces. Surprisingly, the shortest loops are the same in both the Riemannian and sub-Riemannian cases. We provide specific calculations for SU(2) and SU(3).
We study curvature invariants of a sub-Riemannian manifold (i.e., a manifold with a Riemannian metric on a non-holonomic distribution) related to mutual curvature of several pairwise orthogonal subspaces of the distribution, and prove geometrical inequalities for a sub-Riemannian submanifold. As applications, inequalities are proved for submanifolds with mutually orthogonal distributions that include scalar and mutual curvature. For compact submanifolds, inequalities are obtained that are supported by known integral formulas for almost-product manifolds.
Comparison theorems on H-type sub-Riemannian manifolds
本文在H型次黎曼流形上建立了一致的次Hessian和次Laplace比较定理，并证明了一个精确的次黎曼Bonnet-Myers定理，将先前结果推广至一般情形。
We study isometries in the contact sub-pseudo-Riemannian geometry. In particular we give an upper bound on the dimension of the isometry group of a general sub-pseudo-Riemannian manifold and prove that the maximal dimension is attained for the left invariant structures on the Heisenberg group.
In this paper we introduce slant Riemannian submersions from cosymplectic manifolds onto Riemannian manifolds. We obtain some results on slant Riemannian submersions of a cosymplectic manifolds. We also give examples and inequalities between the scalar curvature and squared mean curvature of fibres of such slant submersions according to characteristic vector field is vertical or horizontal.
We show that under certain conditions, a nontrivial Riemannian submersion from positively curved four manifolds does not exist. This gives a partial answer to a conjecture due to Fred Wilhelm. We also prove a rigidity theorem for Riemannian submersions with totally geodesic fibers from compact four-dimensional Einstein manifolds.
If $π:M\rightarrow B$ is a Riemannian Submersion and $M$ has positive sectional curvature, O'Neill's Horizontal Curvature Equation shows that $B$ must also have positive curvature. We show there are Riemannian submersions from compact manifolds with positive Ricci curvature to manifolds that have small neighborhoods of (arbitrarily) negative Ricci curvature, but that there are no Riemannian submersions from manifolds with positive Ricci curvature to manifolds with nonpositive Ricci curvature.
Classification of biharmonic Riemannian submersions from manifolds with constant sectional curvature
本文研究了从具有常截面曲率的n+1维黎曼流形到n维流形的双调和黎曼浸入，证明其当且仅当为调和浸入。
Using the curvature-dimension inequality proved in Part~I, we look at consequences of this inequality in terms of the interaction between the sub-Riemannian geometry and the heat semigroup $P_t$ corresponding to the sub-Laplacian. We give bounds for the gradient, entropy, a Poincaré inequality and a Li-Yau type inequality. These results require that the gradient of $P_t f$ remains uniformly bounded whenever the gradient of $f$ is bounded and we give several sufficient conditions for this to hold.
We classify the pseudo-Riemannian biharmonic submersion from a 3-dimensional space form into a surface.
In this paper, we study Riemannian submersions whose total manifolds admitting a Ricci soliton. Here, we characterize any fiber of such a submersion is Ricci soliton or almost Ricci soliton. Indeed, we obtain necessary conditions for which the target manifold of Riemannian submersion is a Ricci soliton. Moreover, we study the harmonicity of Riemannian submersion from Ricci soliton and give a characterization for such a submersion to be harmonic.
We establish lower bounds for the first non-zero eigenvalue for the natural geometric sub-elliptic Laplacian operator defined on sub-Riemannian manifolds of step 2 that satisfy a positive curvature condition. The methods are very general and can be applied even when the sub-Riemannian geometry has considerable torsion.
We construct a canonically defined affine connection in sub-Riemannian contact geometry. Our method mimics that of the Levi-Civita connection in Riemannian geometry. We compare it with the Tanaka-Webster connection in the three-dimensional case.
Submanifolds in metallic Riemannian manifolds
该文研究带金属黎曼结构的黎曼流形中子流形的性质，重点分析子流形上诱导的$Σ$-金属黎曼结构的正规性，并给出欧氏空间上的金属黎曼结构在余维1球面上诱导结构的实例。
Harmonic maps into sub-Riemannian Lie groups
该研究推广黎曼流形的相关定义，提出次黎曼流形间调和映射的定义，推导了映到左不变度量李群的水平映射为调和映射的条件，证明其可分为异常与正常两类，并以Heisenberg群的调和映射方程为例进行说明。
We solve the local equivalence problem for sub-Riemannian structures on (2n + 1)-dimensional manifolds. We show that two sub-Riemannian structures are locally equivalent if and only if? their corresponding canonical linear connections are equivalent. When n = 1, these connections coincide with the generalized Tanaka-Webster connection of the corresponding contact metric structure. We show that in dimension > 5, there may not be any contact metric manifolds associated with a given sub-Riemannian structure.
The Metallic Ratio is fascinating topic that continually generated news ideas. A Riemannian manifold endowed with a Metallic structure will be called a Metallic Riemannian manifold. The main purpose of the present paper is to study the geometry of transversal lightlike submanifolds and radical transversal lightlike submanifolds and of Metallic Semi-Riemannian manifolds. We investigate the geometry of distributions and obtain necessary and sufficient conditions for the induced connection on these manifolds to be metric connection. We also obtain characterization of transversal lightlike submani…
We prove that two-step analytic sub-Riemannian structures on a compact analytic manifold equipped with a smooth measure and Lipschitz Carnot groups satisfy measure contraction properties.
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.
The authors found geodesics, shortest arcs, cut loci, and conjugate sets for left-invariant sub-Riemannian matric on the Lie group $SL(2)$, which is right-invariant relative to the Lie subgroup $SO(2)\subset SL(2)$ (in other words, for invariant sub-Riemannian metric on weakly symmetric space $(SL(2)\times SO(2))/SO(2)$).
We give some fundamental properties of the induced structures on submanifolds immersed in almost product or locally product Riemannian manifolds. We study the induced structure by the composition of two isometric immersions on submanifolds in an almost product Riemannian manifold. We give an effective construction for some induced structures on submanifolds of codimension 1 or 2 in Euclidean space.
The objet of this paper is the study of variations of a functional whose integrant is the $σ_u$-curvature of closed submanifolds of arbitrary codimension in Riemannian manifolds.
Sub-Riemannian structures on odd-dimensional spheres respecting the Hopf fibration naturally appear in quantum mechanics. We study the curvature maps for such a sub-Riemannian structure and express them using the Riemannian curvature tensor of the Fubini-Study metric of the complex projective space and the curvature form of the Hopf fibration. We also estimate the number of conjugate points of a sub-Riemannian extremal in terms of the bounds of the sectional curvature and the curvature form. It presents a typical example for the study of curvature maps and comparison theorms for a general cora…
We bound the dimension of the fiber of a Riemannian submersion from a positively curved manifold in terms of the dimension of the base of the submersion and either its conjugate radius or the length of its shortest closed geodesic.
We build an analogue for the Levi-Civita connection on Riemannian manifolds for sub-Riemannian manfiolds modeled on the Heisenberg group. We demonstrate some geometric properties of this connection to justify our choice and show that this connection is unique in having these properties.
In this paper we introduce paraquaternionic CR-submanifolds of almost paraquaternionic hermitian manifolds and state some basic results on their differential geometry. We also study a class of semi-Riemannian submersions from paraquaternionic CR-submanifolds of paraquaternionic Kaehler manifolds.
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