We show that, for an affine submersion $π: \mathbf{M}\longrightarrow \mathbf{B}$ with horizontal distribution, $\mathbf{B}$ is a statistical manifold with the metric and connection induced from the statistical manifold $\mathbf{M}$. The concept of conformal submersion with horizontal distribution is introduced, which is a generalization of affine submersion with horizontal distribution. Then proved a necessary and sufficient condition for $(\mathbf{M}, \nabla, g_M)$ to become a statistical manifold for a conformal submersion with horizontal distribution. A necessary and sufficient condition is obtained for the curve $π\circ σ$ to be a geodesic of $\mathbf{B}$, if $σ$ is a geodesic of $\mathbf{M}$ for $π: (\mathbf{M},\nabla) \longrightarrow (\mathbf{B},\nabla^*) $ a conformal submersion with horizontal distribution. Also, we obtained a necessary and sufficient condition for the tangent bundle $T\mathbf{M}$ to become a statistical manifold with respect to the Sasaki lift metric and the complete lift connection.
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.
In this article, conformal submersion with horizontal distribution of Riemannian manifolds is defined which is a generalization of the affine submersion with horizontal distribution. Then, a necessary condition is obtained for the existence of a conformal submersion with horizontal distribution. For the dual connections $\nabla$ and $\overline{\nabla}$ on manifold $\mathbf{M}$ and $\nabla^*$ and $\overline{\nabla}^*$ on manifold $\mathbf{B}$, we show that $π: (\mathbf{M},\nabla) \longrightarrow (\mathbf{B}, \nabla^{*}) $ is a conformal submersion with horizontal distribution if and only if $π:…
We study singularity formation of Kähler-Ricci flow on a Kähler manifold that admits a horizontally homothetic conformal submersion into another Kähler manifold. We will derive necessary and sufficient conditions for the preservation of horizontally homothetic conformal submersion along the flow and establish the formation of type I singularity together with a standard splitting of the Cheeger-Gromov limit. This generalizes the setup of Calabi symmetry that was discussed in \cite{1} and \cite{2}, thus gives new proofs for the results listed there.
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 discuss how to travel along horizontal broken geodesics of a homogenous Finsler submersion, i.e., we study, what in Riemannian geometry was called by Wilking, the dual leaves. More precisely, we investigate the attainable sets $\mathcal{A}_{q}(\mathcal{C})$ of the set of analytic vector fields $\mathcal{C}$ determined by the family of horizontal unit geodesic vector fields $\mathcal{C}$ to the fibers $\mathcal{F}=\{ρ^{-1}(c)\}$ of a homogenous analytic Finsler submersion $ρ: M\to B$. Since reverse of geodesics don't need to be geodesics in Finsler geometry, one can have examp…
In this paper, we study Riemannian, anti-invariant Riemannian and Lagrangian submersions. We prove that the horizontal distribution of a Lagrangian submersion from a Kaehlerian manifold is integrable. We also give some applications of this result. Moreover, we investigate the effect of the submersion to the geometry of its total manifold and its fibers.
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 the notions of h-conformal slant submersions and almost h-conformal slant submersions from almost quaternionic Hermitian manifolds onto Riemannian manifolds as a generalization of Riemannian submersions, horizontally conformal submersions, slant submersions, h-slant submersions, almost h-slant submersion and conformal slant submersions. We investigate several properties of these including the integrability of distributions, the geometry of foliations and the conditions for such maps to be totally geodesic. Further we give some examples of such maps.
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.
As a generalization of Riemannian submersions, horizontally conformal submersions, semi-invariant submersions, h-semi-invariant submersions, almost h-semi-invariant submersions, conformal semi-invariant submersions, we introduce h-conformal semi-invariant submersions and almost h-conformal semi-invariant submersions from almost quaternionic Hermitian manifolds onto Riemannian manifolds. We study their properties: the geometry of foliations, the conditions for total manifolds to be locally product manifolds, the conditions for such maps to be totally geodesic, etc. Finally, we give some example…
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.
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.
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.
Akyol M.A. [Conformal anti-invariant submersions from cosymplectic manifolds, Hacettepe Journal of Mathematics and Statistic, 46(2), (2017), 177-192.] defined and studied conformal anti-invariant submersions from cosymplectic manifolds. The aim of the present paper is to define and study the notion of conformal slant submersions (it means the Reeb vector field $ξ$ is a vertical vector field) from almost contact metric manifolds onto Riemannian manifolds as a generalization of Riemannian submersions, horizontally conformal submersions, slant submersions and conformal anti-invariant submersions.…
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.
We study focal points and Maslov index of a horizontal geodesic $γ:I\to M$ in the total space of a semi-Riemannian submersion $π:M\to B$ by determining an explicit relation with the corresponding objects along the projected geodesic $π\circγ:I\to B$ in the base space. We use this result to calculate the focal Maslov index of a (spacelike) geodesic in a stationary space-time which is orthogonal to a timelike Killing vector field.
We study unit horizontal bundles associated with Riemannian submersions. First we investigate metric properties of an arbitrary unit horizontal bundle equipped with a Riemannian metric of the Cheeger-Gromoll type. Next we examine it from the Gromov-Hausdorff convergence theory point of view, and we state a collapse theorem for unit horizontal bundles associated with a sequence of warped Riemannian submersions.
This article presents the study of almost Hermitian submersion from an almost Yamabe soliton onto an almost Hermitian manifold. A non-trivial example is also mentioned in order to guarantee the existence of such solitons on the total space of almost Hermitian submersions. We mainly focus on Kaehler submersions from Kaehler manifolds which are special case of almost Hermitian submersions. Under certain conditions, we find out that the fibres and the base manifold of such submersions are almost Yamabe soliton. We give the characterizations for an almost Yamabe soliton of a Kaehler submersion to …
Harmonic morphisms are maps between Riemannian manifolds that pull back harmonic functions to harmonic functions. These maps are characterized as horizontally weakly conformal harmonic maps and they have many interesting links and applications to several areas in mathematics (see the book by Baird and Wood for details). In this paper, we study generalized harmonic morphisms which are defined to be maps between Riemannian manifolds that pull back harmonic functions to biharmonic functions. We obtain some characterizations of generalized harmonic morphisms into a Euclidean space and give two met…
We deform a map into a Riemannian manifold that is horizontal with respect to a submersion onto a non-positively curved manifold and satisfies a Chow condition into a harmonic one through a horizontal homotopy.
The main purpose of present paper is to study the affine connection induced from the horizontal lift on the cross-section determined by a vector field in Mn with respect to the adapte frame of .
Weyl derivatives, Weyl-Lie derivatives and conformal submersions are defined, then used to generalize the Jones-Tod correspondence between selfdual 4-manifolds with symmetry and Einstein-Weyl 3-manifolds with an abelian monopole. In this generalization, the conformal symmetry is replaced by a particular kind of conformal submersion with one dimensional fibres. Special cases are studied in which the conformal submersion is holomorphic, affine, or projective. All scalar-flat Kahler metrics with such a holomorphic conformal submersion, and all four dimensional hypercomplex structures with a compa…
We prove that any (real or complex) analytic horizontally conformal submersion from a three-dimensional conformal manifold M to a two-dimensional conformal manifold N can be, locally, `extended' to a unique harmonic morphism from the heaven space of M to N.
We obtain integral formulas for a metric-affine space equipped with two complementary orthogonal distributions. The integrand depends on the Ricci and mixed scalar curvatures and invariants of the second fundamental forms and integrability tensors of the distributions. The formulas under some conditions yield splitting of manifolds (including submersions and twisted products) and provide geometrical obstructions for existence of distributions and foliations (or compact leaves of them).
We show Riemannian geometry could be studied by identifying the tangent bundle of a Riemannian manifold $\mathcal{M}$ with a subbundle of the trivial bundle $\mathcal{M} \times \mathcal{E}$, obtained by embedding $\mathcal{M}$ differentiably in a Euclidean space $\mathcal{E}$. Given such an embedding, we can extend the metric tensor on $\mathcal{M}$ to a (positive-definite) operator-valued function acting on $\mathcal{E}$, giving us an embedded ambient structure. The formulas for the Christoffel symbols and Riemannian curvature in local coordinates have simple generalizations to this setup. Fo…
In this paper, we consider a smooth connected finite-dimensional manifold $M$, an affine connection $\nabla$ with holonomy group $H^{\nabla}$ and $Δ$ a smooth completely non integrable distribution. We define the $Δ$-horizontal holonomy group $H^{\;\nabla}_Δ$ as the subgroup of $H^{\nabla}$ obtained by $\nabla$-parallel transporting frames only along loops tangent to $Δ$. We first set elementary properties of $H^{\;\nabla}_Δ$ and show how to study it using the rolling formalism (\cite{ChitourKokkonen}). In particular, it is shown that $H^{\;\nabla}_Δ$ is a Lie group. Moreover, we study an expl…
In 1939 H. Weyl has introduced the so called intrinsic volumes $V_i(M^n), i=0,\dots,n$, (known also as Lipschitz-Killing curvatures) for any closed smooth Riemannian manifold $M^n$. Given a Riemmanian submersion of compact smooth Riemannian manifolds $M\to B$, $B$ is connected. For $\varepsilon >0$ let us define a new Riemannian metric on $M$ by multiplying the original one by $\varepsilon$ along the vertical directions and keeping it the same along the (orthogonal) horizontal directions. Denote the corresponding Riemannian manifold by $M_\varepsilon$. The main result says that $\lim_{\varepsi…
We introduce a new approach for computing curvature of sub-Riemannian manifolds. Curvature is here meant as symplectic invariants of Jacobi curves of geodesics, as introduced by Zelenko and Li. We describe how they can be expressed using a compatible affine connection and induced tensors, without any restriction on our sub-Riemannian manifold or the choice of connection. In particular, we obtain a universal Bonnet-Myers theorem of Riemannian type. We also give universal formulas for the canonical horizontal frame along each geodesic and an algorithm for computing the curvature in general. Seve…
We establish Thom's jet transversality theorem for regular maps from an affine algebraic manifold to an algebraic manifold satisfying a suitable flexibility condition. It can be considered as the algebraic version of Forstnerič's jet transversality theorem for holomorphic maps from a Stein manifold to an Oka manifold. Our jet transversality theorem implies genericity theorems for regular maps of maximal ranks. As an application, it follows that every connected compact locally flexible manifold is the image of a holomorphic submersion from an affine space. We also show that every algebraically …
Using the notion of Levi form of a smooth distribution, we discuss the local and the global problem of existence of one horizontal section of a smooth vector bundle endowed with a horizontal distribution. The analysis will lead to the formulation of a "one-leaf" analogue of the classical Frobenius integrability theorem in elementary differential geometry. Several applications of the result will be discussed. First, we will give a characterization of symmetric connections arising as Levi-Civita connections of semi-Riemannian metric tensors. Second, we will prove a general version of the classic…
The space of invariant affine connections on every $3$-Sasakian homogeneous manifold of dimension at least $7$ is described. In particular, the remarkable subspaces of invariant affine metric connections, and the subclass with skew-torsion, are also determined. To this aim, an explicit construction of all $3$-Sasakian homogeneous manifolds is exhibited. The unique $3$-Sasakian homogeneous manifolds which admit nontrivial Einstein with skew-torsion invariant affine connections are those of dimension $7$, that is, $\mathbb{S}^7=\mathrm{Sp} (2)/ \mathrm{Sp(1)}$, $\mathbb{R} P^7=\mathrm{Sp}(2)/ \m…
