R. Zimmer proved that, on a compact manifold, a foliation with a dense leaf, a suitable leafwise Riemannian symmetric metric and a transverse Lie structure has arithmetic holonomy group. In this work we improve such result for totally geodesic foliations by showing that the manifold itself is arithmetic. This also gives a positive answer, for some special cases, to a conjecture of E. Ghys.
Let $Γ$ be a lattice in $\mathrm{SO}_0(n, 1)$. We prove that if the associated locally symmetric space contains infinitely many maximal totally geodesic subspaces of dimension at least $2$, then $Γ$ is arithmetic. This answers a question of Reid for hyperbolic $n$-manifolds and, independently, McMullen for hyperbolic $3$-manifolds. We prove these results by proving a superrigidity theorem for certain representations of such lattices. The proof of our superrigidity theorem uses results on equidistribution from homogeneous dynamics and our main result also admits a formulation in that language.
We prove transverse Weitzenböck identities for the horizontal Laplacians of a totally geodesic foliation. As a consequence, we obtain nullity theorems for the de Rham cohomology assuming only the positivity of curvature quantities transverse to the leaves. Those curvature quantities appear in the adiabatic limit of the canonical variation of the metric.
In 86, Ranjan questioned whether a submersion $π: G\to B$ from a compact simple Lie group with bi-invariant metric is a coset foliation or not, provided the submersion is Riemannian with totally geodesic fibers. Here we answer this question affirmatively, even when the submersion is defined only in an open subset of $G$ (assuming suitable hypothesis).
We classify hypersurfaces of the Minkowski space $Ł^{n+1}$ that carry a totally geodesic foliation with complete leaves of codimension one. We prove that such a hypersurface is ruled, or a partial tube over a curve or contains a two or three dimensional strip. Moreover, if the hypersurface is embedded then it is a partial tube over a curve.
In the current paper,under the transverse Ricci flow on a totally geodesic Riemannian foliation, we prove two types of differential Harnack inequalities (Li-Yau gradient estimate) for the positive solutions of the heat equation associated with the time dependent horizontal Laplacian operators. We also get a time dependent version of the generalized curvature dimension inequality. As a consequence of aforementioned results, we also get parabolic Harnack inequalities and heat kernel upper bounds.
We classify the hypersurfaces of Euclidean space that carry a totally geodesic foliation with complete leaves of codimension one. In particular, we show that rotation hypersurfaces with complete profiles of codimension one are characterized by their warped product structure. The local version of the problem is also considered.
In this paper we give a complete local parametric classification of the hypersurfaces with dimension at least three of a space form that carry a totally geodesic foliation of codimension one. A classification under the assumption that the leaves of the foliation are complete was given in \cite{drt} for Euclidean hypersurfaces. We prove that there exists exactly one further class of local examples in Euclidean space, all of which have rank two. We also extend the classification under the global assumption of completeness of the leaves for hypersurfaces of the sphere and show that there exist pl…
In this work, we study the geometric properties of spacelike foliations by hypersurfaces on a Lorentz manifold. We investigate conditions for the leaves being stable, totally geodesic or totally umbilical. We consider that $\overline{M}^{n+1}$ is equipped with a timelike closed conformal vector field $ξ$. If the foliation has constant mean curvature, we show that the leaves are stable. When the leaves are compact spacelike hypersurfaces we show that, under certain conditions, its are totally umbilic hypersurfaces. In the case of foliations by complete noncompact hypersurfaces, we using a Maxim…
We develop a Malliavin calculus on the horizontal path space of a totally geodesic Riemannian foliation. As a first application, under suitable assumptions, we prove a log-Sobolev inequality for a natural one-parameter family of infinite-dimensional Ornstein-Uhlenbeck type operators. As a second application, we obtain concentration and tail estimates for the horizontal Brownian motion of the foliation.
In this Note we establish a relation between sections in globally generated holomorphic vector bundles on Kähler manifolds, isotropic with respect to a non-degenerate quadratic form, and totally geodesic foliations on Euclidean open domains. We find a geometric condition for a totally geodesic foliation to originate in a holomorphic vector bundle. For codimension-two foliations, this description recovers of P. Baird and J. C. Wood. The universal objects that play a key role are the orthogonal Grassmannians.
We develop a variational theory of geodesics for the canonical variation of the metric of a totally geodesic foliation. As a consequence, we obtain comparison theorems for the horizontal and vertical Laplacians. In the case of Sasakian foliations, we show that sharp horizontal and vertical comparison theorems for the sub-Riemannian distance may be obtained as a limit of horizontal and vertical comparison theorems for the Riemannian distances approximations.
In this paper we present a certain class of geodesic vector fields of the double-twisted product R X R. Some examples of totally geodesic foliations are given.
These notes are the basis of a course given at the Institut Henri Poincare in September 2014. We survey some recent results related to the geometric analysis of hypoelliptic diffusion operators on totally geodesic Riemannian foliations. We also give new applications to the study of hypocoercive estimates for Kolmogorov type operators.
Under suitable conditions, we show that the Euler characteristic of a foliated Riemannian manifold can be computed only from curvature invariants which are transverse to the leaves. Our proof uses the hypoelliptic sub-Laplacian on forms recently introduced by two of the authors.
The aim of this paper is to classify compact, simply connected Kähler manifolds which admit totally geodesic, holomorphic complex homothetic foliation by curves.
We study Bakry-Emery type estimates for the Laplace-Beltrami operator of a totally geodesic foliation. In particular, we are interested in situations for which the $Γ_2$ operator may not be bounded from below but the horizontal Bakry-Emery curvature is. As we prove it, under a bracket generating condition, this weaker condition is enough to imply several functional inequalities for the heat semigroup including the Wang-Harnack inequality and the log-Sobolev inequality. We also prove that, under proper additional assumptions, the generalized curvature dimension inequality introduced by Baudoin-…
We study Riemannian foliations with complex leaves on Kaehler manifolds. The tensor T, the obstruction to the foliation be totally geodesic, is interpreted as a holomorphic section of a certain vector bundle. This enables us to give classification results when the manifold is compact.
We prove a family of new Weitzenböck formulas on a Riemannian foliation with totally geodesic leaves. These Weitzenböck formulas are naturally parametrized by the canonical variation of the metric. As a consequence, under natural geometric conditions, the horizontal Laplacian satisfies a generalized curvature dimension inequality. Among other things, this curvature dimension inequality implies Li-Yau estimates for positive solutions of the horizontal heat equation and a sub-Riemannian Bonnet-Myers compactness theorem whose assumptions only rely on the intrinsic geometry of the horizontal distr…
A contact pair on a manifold always admits an associated metric for which the two characteristic contact foliations are orthogonal. We show that all these metrics have the same volume element. We also prove that the leaves of the characteristic foliations are minimal with respect to these metrics. We give an example where these leaves are not totally geodesic submanifolds.
We study totally geodesic codimension 1 smooth foliations on Lorentzian manifold. We are in particular interested by the relations between riemannian flows and geodesic foliations. We prove that, up to a 2-cover, any Seifert bundle admit such a foliation.
In this work, we find an equation that relates the Ricci curvature of a riemannian manifold $M$ and the second fundamental forms of two orthogonal foliations of complementary dimensions, $\mathcal{F}$ and $\mathcal{F}^{\bot}$, defined on $M$. Using this equation, we show a sufficient condition for the manifold M to be locally a riemannian product of the leaves of $\mathcal{F}$ and $\mathcal{F}^{\bot}$, if one of the foliations is totally umbilical. We also prove an integral formula for such foliations.
Let $g_t$ be a smooth 1-parameter family of negatively curved metrics on a closed hyperbolic 3-manifold $M$ starting at the hyperbolic metric. We construct foliations of the Grassmann bundle $Gr_2(M)$ of tangent 2-planes whose leaves are (lifts of) minimal surfaces in $(M,g_t)$. These foliations are deformations of the foliation of $Gr_2(M)$ by (lifts of) totally geodesic planes projected down from the universal cover $\mathbb{H}^3$. Our construction continues to work as long as the sum of the squares of the principal curvatures of the (projections to $M$) of the leaves remains pointwise small…
For a Riemannian submersion from a simple compact Lie group with a bi-invariant metric, we prove the action of its holonomy group on the fibers is transitive. As a step towards classifying Riemannian submersions with totally geodesic fibers, we consider the parameterized surface induced by lifting a base geodesic to points along a geodesic in a fiber. Such a surface is "doubly ruled" (it is ruled by horizontal geodesics and also by vertical geodesics). Its characterizing properties allow us to define "doubly ruled parameterized surfaces" in any Riemannian manifold, independent of Riemannian su…
A flow of metrics, $g_t$, on a manifold is a solution of a differential equation $\dt g = S(g)$, where a geometric functional $S(g)$ is a symmetric $(0,2)$-tensor usually related to some kind of curvature. The mixed sectional curvature of a foliated manifold regulates the deviation of leaves along the leaf geodesics. We introduce and study the flow of metrics on a foliation (called the 'Partial Ricci Flow'), where $S=-2 r$ and $r$ is the partial Ricci curvature of the foliation; in other words, the velocity for a unit vector $X$ orthogonal to the leaf, $-2 r(X,X)$, is the mean value of section…
Let $(M^{n},g)$ be a closed, connected, oriented, $C^{\infty}$, Riemannian, n-manifold with a transversely oriented foliation $\boldkey F$. We show that if $\lbrace X,Y \rbrace$ are basic vector fields, the leaf component of $[X,Y]$, $\Cal{V}[X,Y]$, has vanishing leaf divergence whenever $κ\wedge χ_{\boldkey F}$ is a closed (possibly zero) de Rham cohomology (p+1)-form. Here $κ$ is the mean curvature one-form of the foliation ${\boldkey F}$ and ${χ_{\boldkey F}}$ is its characteristic form. In the codimension-2 case, $κ\wedge χ_{\boldkey F}$ is closed if and only if $κ$ is horizontally closed.…
We consider strict and complete nearly Kaehler manifolds with the canonical Hermitian connection. The holonomy representation of the canonical Hermitian connection is studied. We show that a strict and complete nearly Kaehler is locally a Riemannian product of homogenous nearly Kaehler spaces, twistor spaces over quaternionic Kaehler manifolds and 6-dimensional nearly Kaehler manifolds. As an application we obtain structure results for totally geodesic Riemannian foliations admitting a compatible Kaehler structure. Finally, we obtain a classification result for the homogenous case, reducing a …
Extrinsic Geometric Flow (EGF) for a codimension-one foliation has been recently introduced by authors as deformations of Riemannian metrics subject to quantities expressed in terms of its second fundamental form. In the paper we introduce soliton solutions to EGF and study their geometry for totally umbilical foliations, foliations on surfaces, and when the EGF is produced by the extrinsic Ricci tensor.
We prove that a foliation $(M, F)$ of codimension $q$ on a $n$-dimen\-sio\-nal pseudo-Riemannian manifold is pseudo-Riemannian if and only if any geodesic that is orthogonal at one point to a leaf is orthogonal to every leaf it intersects. We show that on the graph $G = G(F)$ of a pseudo-Riemannian foliation there exists a unique pseudo-Riemannian metric such that canonical projections are pse\-u\-do-Rieman\-ni\-an submersions and the fibres of different projections are orthogonal at common points. Relatively this metric the induced foliation $(G,\mathbb{F})$ on the graph is pseudo-Riemannian …
We consider foliations of the whole three dimensional hyperbolic space H^3 by oriented geodesics. Let L be the space of all the oriented geodesics of H^3, which is a four dimensional manifold carrying two canonical pseudo-Riemannian metrics of signature (2,2). We characterize, in terms of these geometries of L, the subsets M in L that determine foliations of H^3. We describe in a similar way some distinguished types of geodesic foliations of H^3, regarding to which extent they are in some sense trivial in some directions: On the one hand, foliations whose leaves do not lie in a totally geodesi…
Here, a Finsler manifold (M, F) is considered with corresponding curvature tensor, regarded as 2-forms on the bundle of non-zero tangent vectors. Certain subspaces of the tangent spaces of M determined by the curvature are introduced and called k-nullity foliations of the curvature operator. It is shown that if the dimension of foliation is constant then the distribution is involutive and each maximal integral manifold is totally geodesic. Characterization of the k-nullity foliation is given, as well as some results concerning constancy of the flag curvature, and completeness of their integral…
A foliation F on a Riemannian manifold M is homogeneous if its leaves coincide with the orbits of an isometric action on M. A foliation F is polar if it admits a section, that is, a connected closed totally geodesic submanifold of M which intersects each leaf of F, and intersects orthogonally at each point of intersection. A foliation F is hyperpolar if it admits a flat section. These notes are related to joint work with Jose Carlos Diaz-Ramos and Hiroshi Tamaru about hyperpolar homogeneous foliations on Riemannian symmetric spaces of noncompact type. Apart from the classification result which…
We study transversely Lorentzian foliations on the closed 3-manifolds. We classify them under a completeness hypothesis and we deduce the dual classification of codimension 1 geodesically complete timelike totally geodesic foliations. Besides we provide an example of a Lorentzian foliation on a compact 3-manifold which is equicontinuous on a proper dense open subset of the manifold, and hence not transversely complete.
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.
Given a geodesic line $γ$ the hyperbolic space $\mathbb H^n$ we formulate a necessary and sufficient condition for a function along this geodesic which measure the mean curvature of totally umbilical leaves of a foliation orthogonal to $γ$. Then we extend the result to $γ$ being a hypercycle i.e. a geodesic on a hypersurface equidistant from the totally geodesic one.
We study relations between certain totally geodesic foliations of a closed flat manifold and its collapsed Gromov-Hausdorff limits. Our main results explicitly identify such collapsed limits as flat orbifolds, and provide algebraic and geometric criteria to determine whether they are singular.
In this article we study isometric immersions of nearly Kähler manifolds into a space form (specially Euclidean space) and show that every nearly Kähler submanifold of a space form has a totally umbilic foliation whose leafs are 6-dimensional nearly Kähler manifolds. Moreover using this foliation we show that there is no non-homogeneous 6-dimensional nearly Kähler submanifold of a space form. We prove some results towards a classification of nearly Kähler hypersurfaces in standard space forms.
We derive total mean curvature integration formulae of a three co-dimensional foliation $\mathcal{F}^{n}$ on a screen integrable half-lightlike submanifold, $M^{n+1}$ in a semi-Riemannian manifold $\overline{M}^{n+3}$. We give generalized differential equations relating to mean curvatures of a totally umbilical half-lightlike submanifold admitting a totally umbilical screen distribution, and show that they are generalizations of those given by [4].
We develop a method for preserving pseudoholomorphic curves in contact 3-manifolds under surgery along transverse links. This makes use of a geometrically natural boundary value problem for holomorphic curves in a 3-manifold with stable Hamiltonian structure, where the boundary conditions are defined by 1-parameter families of totally real surfaces. The technique is applied here to construct a finite energy foliation for every closed overtwisted contact 3-manifold.
In this paper we study singular riemannian foliations that have sections,i.e., totally geodesic complete immersed submanifolds that meet each leaf orthogonally and whose dimensions are the codimensions of the regular leaves. We prove here that the restriction of the foliation to a slice of a leaf is diffeomorphic to an isoparametric foliation on an open set of an euclidian space. This result gives us local information about the singular foliation and in particular about the singular stratification of the foliation. It also allows us to describe the plaques of the foliation as level sets of a t…
We study the totally null surfaces of the neutral Kaehler metric on certain 4-manifolds. The tangent spaces of totally null surfaces are either self-dual ($α$-planes) or anti-self-dual ($β$-planes) and so we consider $α$-surfaces and $β$-surfaces. The metric of the examples we study, which include the spaces of oriented geodesics of 3-manifolds of constant curvature, are anti-self-dual, and so it is well-known that the $α$-planes are integrable and $α$-surfaces exist. These are holomorphic Lagrangian surfaces, which for the geodesic spaces correspond to totally umbilic foliations of the underl…
We introduce horizontal holonomy groups, which are groups defined using parallel transport only along curves tangent to a given subbundle $D$ of the tangent bundle. We provide explicit means of computing these holonomy groups by deriving analogues of Ambrose-Singer's and Ozeki's theorems. We then give necessary and sufficient conditions in terms of the horizontal holonomy groups for existence of solutions of two problems on foliated manifolds: determining when a foliation can be either (a) totally geodesic or (b) endowed with a principal bundle structure. The subbundle $D$ plays the role of an…
In the present paper we carry on a systematic study of 3-quasi-Sasakian manifolds. In particular we prove that the three Reeb vector fields generate an involutive distribution determining a canonical totally geodesic and Riemannian foliation. Locally, the leaves of this foliation turn out to be Lie groups: either the orthogonal group or an abelian one. We show that 3-quasi-Sasakian manifolds have a well-defined rank, obtaining a rank-based classification. Furthermore, we prove a splitting theorem for these manifolds assuming the integrability of one of the almost product structures. Finally, w…
A singular riemannian foliation on a complete riemannian manifold is said to be riemannian if each geodesic that is perpendicular at one point to a leaf remains perpendicular to every leaf it meets. The singular foliation is said to admit sections if each regular point is contained in a totally geodesic complete immersed submanifold that meets every leaf orthogonally and whose dimension is the codimension of the regular leaves. A typical example of such singular foliation is the partition by orbits of a polar action,e.g. the orbits of the adjoint action of a compact Lie group on itself. We pro…
In this article we investigate a type of totally geodesic map which has its image being a geodesic in an anisotropic Riemannian manifold. We consider its nonlinear stability among the family of wave maps. We first establish the factorization property and then formulate the stability problem into a PDE system in a specially constructed chart of geodesic normal coordinates. With a generalization of the hyperboloidal foliation, we establish the global existence result associate to small initial data for this PDE system, which leads to the geometric stability.
We prove that if the normal distribution of a singular riemannian foliation is integrable, then each leaf of this normal distribution can be extended to be a complete immersed totally geodesic submanifold (called section) which meets every leaf orthogonally. In addition the set of regular points is open and dense in each section. This result generalizes a result of Boualem and solves a problem inspired by a remark of Palais and Terng and a work of Szenthe about polar actions. We also study the singular holonomy of a singular riemannian foliation with sections (s.r.f.s for short) and in particu…
