Kristopher Tapp
We prove that any base space of Riemannian submersion from a compact Lie group (with bi-invariant metric) must have a basic property previously known for normal biquotients; namely, any zero-curvature plane exponentiates to a flat.
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We give a general Lie-theoretic construction for anti-invariant almost Hermitian Riemannian submersions, anti-invariant quaternion Riemannian submersions, anti-invariant para-Hermitian Riemannian submersions, anti-invariant para-quaternion Riemannian submersions, and anti-invariant octonian Riemannian submersions. This yields many compact Einstein examples.
In this paper, we study biharmonic Riemannian submersions. We first derive bitension field of a general Riemannian submersion, we then use it to obtain biharmonic equations for Riemannian submersions with $1$-dimensional fibers and Riemannian submersions with basic mean curvature vector fields of fibers. These are used to construct examples of proper biharmonic Riemannian submersions with $1$-dimensional fibers and to characterize warped products whose projections onto the first factor are biharmonic Riemannian submersions.
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.
We prove that the image of a real analytic Riemannian manifold under a smooth Riemannian submersion is necessarily real analytic.
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.
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.
In this paper, we study harmonic Riemannian submersions from 3-dimensional geometries using the ( generalized) integrability data associated to an orthonormal frame natural to a Riemannian submersion. We give complete classifications of harmonic Riemannian submersions from Thurston's 3-dimensional geometries, 3-dimensional BCV spaces and Berger sphere into a surface. We also give some explicit constructions of these harmonic Riemannian submersions.
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.
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 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 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.
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 obtain sharp inequalities involving the Ricci curvature and the scalar curvature for anti-invariant Riemannian submersions from Sasakian space forms onto Riemannian manifolds.
We get obstructions to existence of semi-Riemannian submersions with totally umbilic fibres.
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.
We get obstructions to existence of semi-Riemannian submersions with totally umbilic fibres.
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.
It is proved the existence and uniqueness of graphs with prescribed mean curvature in Riemannian submersions fibered by flow lines of a vertical Killing vector field.
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 classify the pseudo-Riemannian biharmonic submersion from a 3-dimensional space form into a surface.
We provide some criteria to $p$-parabolicity of Riemannian submersions. In particular, if $N$ is $p$-parabolic and $π:M\to N$ is a Riemannian submersion with uniformly bounded volume of fibers, then $M$ is also $p$-parabolic. In the case of warped manifolds we characterize $p$-parabolicity in terms of a volume growth condition.
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 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.
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。
For biharmonic maps, there is a famous conjecture named Chen's conjecture. In later paper, Wang and Ou gave an affirmative partial answer to submersion version of Chen's conjecture. In this paper, we give an affirmative partial answer to submersion version of generalized Chen's conjecture, that is, triharmonic Riemannian submersions from a 3-dimensional space form into a surface is harmonic.
We classify pseudo-Riemannian submersions with connected totally geodesic fibres from a real pseudo-hyperbolic space onto a pseudo-Riemannian manifold. Also, we obtain the classification of the pseudo-Riemannian submersions with (para-)complex connected totally geodesic fibres from a (para-)complex pseudo-hyperbolic space onto a pseudo-Riemannian manifold.
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, …
In this paper we construct infinitely many examples of a Riemannian submersion from a simple, compact Lie group $G$ with bi-invariant metric onto a smooth manifold that cannot be a quotient of $G$ by a group action. This partially addresses a question of K. Grove's about Riemannian submersions from Lie groups.
Let $M^n$ be a complete, non-compact and $C^\infty$-smooth Riemannian manifold with nonnegative sectional curvature. Suppose $\Cal S$ is a soul of $M^n$. Then any distance non-increasing retraction $Ψ: M^n \to \Cal S$ must give rise to a $C^\infty$-smooth Riemannian submersion.
In this paper, we will first derive a DDVV-type optimal inequality for real skew-symmetric matrices, then we apply it to establish a Simons-type integral inequality for Riemannian submersions with totally geodesic fibres and Yang-Mills horizontal distributions. In this way, we show phenomenons of duality between Submanifold geometry and Riemannnian submersion, particularly between second fundamental form of a submanifold and integrability tensor of a Riemannian submersion.
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.
In this paper we study a semi-Riemannian submersion from Lorentzian (para)almost contact manifolds and find necessary and sufficient conditions for the characteristic vector field to be vertical or horizontal. We also obtain decomposition theorems for an anti-invariant semi-Riemannian submersion from Lorentzian (para)Sasakian manifolds onto a Lorentzian manifold.
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。
General Casorati inequalities and implications for Riemannian maps and Riemannian submersions
本文推导了Riemannian映射和子浸入的广义Casorati不等式，并探讨了其在多种空间形式下的应用与等号情况。
Harmonic Riemannian submersions between Riemannian symmetric spaces of noncompact type
该研究构造了非紧型对称空间到其一秩全测地子空间的调和黎曼淹没，证明了分裂实半单李群对应黎曼对称空间上存在全局非常值复值调和态射，完成了Gudmundsson猜想的肯定证明。
We introduce semi-invariant Riemannian submersions from almost Hermitian manifolds onto Riemannian manifolds. We give examples, investigate the geometry of foliations which are arisen from the definition of a Riemannian submersion and find necessary-sufficient conditions for total manifold to be locally product Riemannian manifold. We also find necessary and sufficient conditions for a semi-invariant submersion to be totally geodesic. Moreover, we obtain a classification for semi-invariant submersions with totally umbilical fibers and show that such submersions put some restrictions on total m…
λ-biharmonic Riemannian submersions from manifolds with constant sectional curvature
本文研究了泛化双和子浸入的λ-双和子浸入，证明了从具有常截面曲率c的(n+1)维黎曼流形到n维流形的非存在性结果，关键值λ=2(n-1)c起决定性作用，当λ≠2(n-1)c时证明非存在性定理，而当λ=2(n-1)c时在非负曲率情况下证明非存在性定理，负曲率情况下构造了显式例子。
We study submetries between Alexandrov spaces and show how some of the usual features of Riemannian submersions fail due to the lack of smoothness.
In this paper, an optimal inequality involving the delta curvature is exposed. An application of Riemannian submersions dealing meteorology is presented. Some characterizations about the vertical motion and the horizontal divergence are obtained.
In this paper we obtain the structure equation of a contact-complex Riemannian submersion and give some applications of this equation in the study of almost cosymplectic manifolds with Kaehler fibres.
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.
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.
Characterizations for Riemannian submersions to be harmonic or biharmonic are shown. Examples of biharmonic but not harmonic Riemannian submersions are shown.
We consider modified scalar curvature functions for Riemannian manifolds equipped with smooth measures. Given a Riemannian submersion whose fiber transport is measure-preserving up to constants, we show that the modified scalar curvature of the base is bounded below in terms of the scalar curvatures of the total space and fibers. We give an application concerning the scalar curvature of a smooth limit space arising in a bounded curvature collapse.
We find out upper bounds for the first eigenvalue of the stability operator for compact constant mean curvature orientable surfaces immersed in a Riemannian Killing submersion. As a consequence, the strong stability of such surfaces is studied. We also characterize constant mean curvature Hopf tori as the only ones attaining the bound in certain cases.
The present paper deals with the characterization of a new submersion named semi-invariant conformal $ζ^{\perp }$-Riemannian submersion from almost contact metric manifolds onto Riemannian manifolds which is the generalization of some known submersions on Riemannian manifolds. We give important and adequate conditions for such submersions to be totally geodesic and harmonic. Also, few examples are examined for such submersions endowed with horizontal Reeb vector field.
Given a Riemannian submersion, we study the relation between lambda constants introduced by G.Perelman on the base manifold and the total space of a Riemannian submersion. We also discuss the relationship between the first eigenvalues of Laplacians on the base manifold and that of the total space. The quantities on warped products are discussed in detail.
We classify semi-Riemannian submersions with connected totally geodesic fibres from a real pseudo-hyperbolic space onto a semi-Riemannian manifold under the assumption that the dimension of the fibres is less than or equal to three and the metrics induced on fibres are negative definite. Also, we obtain the classification of semi-Riemannian submersions with connected complex totally geodesic fibres from a complex pseudo-hyperbolic space onto a semi-Riemannian manifold under the assumption that the dimension of the fibres is less than or equal to two and the metric induced on fibres are negativ…
A short survey on biharmonic Riemannian submersions
本文综述双调和黎曼浸没（双调和子流形的对偶概念）的研究进展，涵盖其背景、定义与已有成果。
Chen-Ricci and Hineva Inequalities For Riemannian Submersions and Riemannian Maps With Applications
本文首次建立了黎曼子浸入沿垂直分布及垂直和水平分布的里奇曲率上界，同时获得了黎曼映射里奇曲率下界及其几何特征。
Casorati inequalities for Riemannian submersion along mixed distributions and their applications
本文针对黎曼流形间的黎曼浸没，引入了垂直和水平分布的正规化标量曲率和正规化Casorati曲率的Casorati不等式，并从代数和几何角度完全刻画了等号情形，进而应用于实、复和广义Sasakian空间形式。
B.-Y. Chen's inequalities for Riemannian submersion and their applications
本文研究了Riemannian子流形中的B.-Y. Chen不等式，探讨了垂直、水平和混合分布下的不等式，并研究了等号成立的情况。应用包括实、复和广义Sasakian空间形式的子流形。
Considering Riemannian submersions, we find necessary and sufficient conditions for when sub-Riemannian normal geodesics project to curves of constant first geodesic curvature or constant first and vanishing second geodesic curvatures. We describe a canonical extension of the sub-Riemannian metric and study geometric properties of the obtained Riemannian manifold. This work contains several examples illustrating the results.
For Riemannian submersions, we establish some estimates for the spectrum of the total space in terms of the spectrum of the base space and the geometry of the fibers. In particular, for Riemannian submersions of complete manifolds with closed fibers of bounded mean curvature, we show that the spectrum of the base space is discrete if and only if the spectrum of the total space is discrete.
Based on ideas of Pigolla and Setti \cite{PS} we prove that immersed submanifolds with bounded mean curvature of Cartan-Hadamard manifolds are Feller. We also consider Riemannian submersions $π\colon M \to N$ with compact minimal fibers, and based on various criteria for parabolicity and stochastic completeness, see \cite{Grygor'yan}, we prove that $M$ is Feller, parabolic or stochastically complete if and only if the base $N$ is Feller, parabolic or stochastically complete respectively.
Non-canonical variations of Riemannian submersions with totally geodesic fibers
研究通过保持给定黎曼子浸入的纤维全纯性和纤维限制度量不变，但改变水平分布，探讨水平和垂直方向的截面曲率变化，提出使乘积空间的截面曲率正的条件，并分析从K-接触结构获得弱接触度量结构的条件。
As a generalization of anti-invariant Riemannian submersions and Lagrangian Riemannian submersions, we introduce the notions of h-anti-invariant submersions and h-Lagrangian submersions from almost quaternionic Hermitian manifolds onto Riemannian manifolds. We obtain characterizations and investigate some properties: the integrability of distributions, the geometry of foliations, and the harmonicity of such maps. We also find a condition for such maps to be totally geodesic and give some examples of such maps. Finally, we obtain some types of decomposition theorems.
Let $π{:}\,(M,\mathcal{H})\to (B,b)$ be a submersion equipped with a horizontal connection $\cal H$ over a Riemannian manifold $(B,b)$. We present an intrinsic curvature condition that only depends on the pair $(\cal H,b)$. By studying a set of relative flat planes, we prove that a certain class of pairs $(\cal H,b)$ admits a compatible metric with positive sectional curvature only if they are \textit{fat}, verifying Wilhelm's Conjecture in this class.
Killing Mean Curvature Solitons from Riemannian Submersions
针对容许非零Killing向量场的流形，利用黎曼淹没技术将平均曲率孤立子构造问题从PDE简化为ODE，得到双曲空间中旋转子的新实例。
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