In the paper we prove integral formulae for a Riemannian manifold endowed with $k>2$ orthogonal complementary distributions, which generalize well-known formula for $k=2$ and give applications to splitting and isometric immersions of Riemannian manifolds, in particular, multiply warped products, and to hypersurfaces with $k>2$ distinct principal curvatures of constant multiplicities.
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 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.
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 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.
We construct a decomposition of the identity operator on a Riemannian manifold $M$ as a sum of smooth orthogonal projections subordinate to an open cover of $M$. This extends a decomposition of the real line by smooth orthogonal projection due to Coifman, Meyer and Auscher, Weiss, Wickerhauser, and a similar decomposition when $M$ is the sphere by the first two authors.
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).
If one could assume that local coordinates in a Riemannian manifold were orthogonal, then local expressions for differential operators, and curvature computations, would be simplified. It is always possible on 2-manifolds, using geometric normal coordinates or isothermal coordinates. In 1984, Dennis DeTurck and Dean Yang constructed smooth orthogonal coordinates on any Riemannian 3-manifold. In fact, they showed that, at any point in a 3-manifold, and for any orthonormal frame at that point, there is a set of local orthogonal coordinates so that the partial derivatives at that point are that f…
This article introduces the problem of finding intrinsic torsion varieties associated to G-structures on a fixed parallelizable Riemannian manifold. As an illustration, the intrinsic torsion varieties of orthogonal almost product structures are analysed on the Iwasawa manifold.
In the present work we construct a lift of a metric $g$ on a 2-dimensional oriented Riemannian manifold $M$ to a metric $\hat{g}$ on the total space $P$ of the orthonormal frame bundle of $M$. We call this lift the \textit {Wagner lift}. Viktor Vladimirovich Wagner (1908 -1981) proposed a technique to extend a metric defined on a non-holonomic distribution to its prolongation via the Lie brackets. We apply the Wagner construction to the specific case when the distribution is the infinitesimal connection in the orthonormal frame bundle which corresponds to a Levi-Civita connection. We find rela…
On a smooth manifold with distributions ${\cal D}_1$ and ${\cal D}_2$ having trivial intersection, we consider the integral of their mutual curvature, as a functional of Riemannian metrics that make the distributions orthogonal. The mutual curvature is defined as the sum of sectional curvatures of planes spanned by all pairs of vectors from an orthonormal basis, such that one vector of the pair belongs to ${\cal D}_1$ and the second vector belongs to ${\cal D}_2$. As such, it interpolates between the sectional curvature of a plane field (if both distributions are one-dimensional), and the mixe…
In this paper we give a proof of the existence of an orthogonal geodesic chord on a Riemannian manifold homeomorphic to a closed disk and with concave boundary. This kind of study is motivated by the link of the multiplicity problem with the famous Seifert conjecture (formulated in 1948) about multiple brake orbits for a class of Hamiltonian systems at a fixed energy level.
We study the behavior of minimal hypersurfaces in the Schwarzschild $n$-manifolds that intersect the horizon orthogonally along the boundary. We show that a free boundary minimal hypersurface and a totally geodesic hyperplane must intersect when the distance between them is achieved in a bounded region. We also discuss when the Schwarzschild metric is perturbed in a way that its scalar curvature is no longer positive.
We prove that the set of orthogonal separable coordinates on an arbitrary (pseudo-)Riemannian manifold carries a natural structure of a projective variety, equipped with an action of the isometry group. This leads us to propose a new, algebraic geometric approach to the classification of orthogonal separable coordinates by studying the structure of this variety. We give an example where this approach reveals unexpected structure in the well known classification and pose a number of problems arising naturally in this context.
This article shows that for generic choice of Riemannian metric on a smooth manifold $M$ of dimension four, all prime compact parametrized minimal surfaces within $M$ have self-intersections in general position in the following sense: self-intersections are transverse and the two tangent planes at any self-intersection point fail to be complex with respect to any orthogonal complex structure on the ambient manifold $M$. This implies via a result of Sheldon Chang that $H_2(M;{\mathbb Z})$ is generated by homology classes that are represented by imbedded minimal surfaces.
We adapt the Newman-Penrose formalism in general relativity to the setting of three-dimensional Riemannian geometry, and prove the following results. Given a Riemannian 3-manifold without boundary and a smooth unit vector field ${\boldsymbol k}$ with geodesic flow, if an integral curve of ${\boldsymbol k}$ is hypersurface-orthogonal at a point, then it is so at every point along that curve. Furthermore, if ${\boldsymbol k}$ is complete, hypersurface-orthogonal, and satisfies $\text{Ric}({\boldsymbol k},{\boldsymbol k}) \geq 0$, then its divergence must be nonnegative. As an application, we sho…
We develop variation formulas on almost-product (e.g. foliated) pseudo-Riemannian manifolds, and we consider variations of metric preserving orthogonality of the distributions. These formulae are applied to Einstein-Hilbert type actions: the total mixed scalar curvature and the total extrinsic scalar curvature of a distribution. The obtained Euler-Lagrange equations admit a number of solutions, e.g., twisted products, conformal submersions and isoparametric foliations. The paper generalizes recent results about the actions on codimension-one foliations for the case of arbitrary (co)dimension.
In this paper we prove a formula for the analytic index of a basic Dirac-type operator on a Riemannian foliation, solving a problem that has been open for many years. We also consider more general indices given by twisting the basic Dirac operator by a representation of the orthogonal group. The formula is a sum of integrals over blowups of the strata of the foliation and also involves eta invariants of associated elliptic operators. As a special case, a Gauss-Bonnet formula for the basic Euler characteristic is obtained using two independent proofs.
Using recent work of Bettiol, we show that a first-order conformal deformation of Wilking's metric of almost-positive sectional curvature on $S^2\times S^3$ yields a family of metrics with strictly positive average of sectional curvatures of any pair of 2-planes that are separated by a minimal distance in the 2-Grassmanian. A result of Smale's allows us to conclude that every closed simply connected 5-manifold with torsion-free homology and trivial second Stiefel-Whitney class admits a Riemannian metric with a strictly positive average of sectional curvatures of any pair of orthogonal 2-planes…
In these notes we survey basic concepts of affine geometry and their interaction with Riemannian geometry. We give a characterization of affine manifolds which has as counterpart those pseudo-Riemannian manifolds whose Levi-Civita connection is flat. We show that no connected semisimple Lie group admits a left invariant flat affine connection. We characterize flat pseudo-Riemannian Lie groups. For a flat left-invariant pseudo-metric on a Lie group, we show the equivalence between the completeness of the Levi-Civita connection and unimodularity of the group. We emphasize the case of flat left i…
In this article, we prove a series of integral formulae for a codimension-one foliated sub-Riemannian manifold, i.e., a Riemannian manifold $(M,g)$ equipped with a distribution ${\mathcal D}=T{\mathcal F}\oplus\,{\rm span}(N)$, where ${\mathcal F}$ is a foliation of $M$ and $N$ a unit vector field $g$-orthogonal to ${\mathcal F}$. Our integral formulas involve $r$th mean curvatures of ${\mathcal F}$, Newton transformations of the shape operator of ${\mathcal F}$ with respect to $N$ and the curvature tensor of induced connection on ${\mathcal D}$ and generalize some known integral formulas (due…
It is well known that the curvature tensor of a pseudo-Riemannian manifold can be decomposed with respect to the pseudo-orthogonal group into the sum of the Weyl conformal curvature tensor, the traceless part of the Ricci tensor and of the scalar curvature. A similar decomposition with respect to the pseudo-unitary group exists on a pseudo-Kählerian manifold; instead of the Weyl tensor one obtains the Bochner tensor. In the present paper, the known decomposition with respect to the pseudo-orthogonal group of the covariant derivative of the curvature tensor of a pseudo-Riemannian manifold is re…
We define the orthogonal Bakry-Émery tensor as a generalization of the orthogonal Ricci curvature, and then study diameter theorems on Kähler and quaternionic Kähler manifolds under positivity assumption on the orthogonal Bakry-Émery tensor. Moreover, under such assumptions on the orthogonal Bakry-Émery tensor and the holomorphic or quaternionic sectional curvature on a Kähler manifold or a quaternionic Kähler manifold respectively, the Bonnet-Myers type diameter bounds are sharper than in the Riemannian case.
The existence of a recurrent spinor field on a pseudo-Riemannian spin manifold $(M,g)$ is closely related to the existence of a parallel 1-dimensional complex subbundle of the spinor bundle of $(M,g)$. We characterize the following simply connected pseudo-Riemannian manifolds admitting such subbundles in terms of their holonomy algebras: Riemannian manifolds; Lorentzian manifolds; pseudo-Riemannian manifolds with irreducible holonomy algebras; pseudo-Riemannian manifolds of neutral signature admitting two complementary parallel isotropic distributions.
This note is about the interplay between the almost-hermitian and Riemannian geometries of a manifold. These geometries can be seen to interact through curvature. The main result is an obstruction equation to the integrability of almost-complex structures orthogonal with respect to Riemannian metrics with constrained sectional curvature. Several geometric consequences ensue, such as a formula for the norm of the Levi-Civita covariant derivative of a hypothetical orthogonal complex structure. Our results lead to a partial recovery of the well-known fact that the round $6$-sphere $S^6$ is not he…
A pseudo-Riemannian manifold endowed with $k>2$ orthogonal complementary distributions (called a Riemannian almost multi-product structure) appears in such topics as multiply warped products, the webs composed of several foliations, Dupin hypersurfaces and in stu\-dies of the curvature and Einstein equations. In this article, we consider the following two problems on the mixed scalar curvature of a Riemannian almost multi-product manifold with a linear connection: a) integral formulas and applications to splitting of manifolds, b) variation formulas and applications to the mixed Einstein-Hilbe…
A foliation on a Riemannian manifold is hyperpolar if it admits a flat section, that is, a connected closed flat submanifold that intersects each leaf of the foliation orthogonally. In this article we classify the hyperpolar homogeneous foliations on every Riemannian symmetric space of noncompact type.
Our results concern geometry of a manifold endowed with a pair of complementary orthogonal distributions (plane fields) and a time-dependent Riemannian metric. The work begins with formulae concerning deformations of geometric quantities as the Riemannian metric varies conformally along one of the distributions. Then we introduce the Extrinsic Geometric Flow depending on the mean curvature vector field of the distribution, and show existence/uniquenes and convergence of a solution as $t\to\infty$, when the complementary distribution is integrable with compact leaves. We apply the method to the…
A Riemannian manifold endowed with $k\ge2$ complementary pairwise orthogonal distributions is called a Riemannian almost $k$-product manifold. In the article, for the first time, we study the following problem: find a relationship between intrinsic and extrinsic invariants of a Riemannian almost $k$-product manifold isometrically immersed in another Riemannian manifold. For such immersions, we establish an optimal inequality that includes the mixed scalar curvature and the square of the mean curvature. Although Riemannian curvature tensor belongs to intrinsic geometry, a special part called th…
We build new examples of extremal domains with small prescribed volume for the first eigenvalue of the Laplace-Beltrami operator in some Riemannian manifold with boundary. These domains are close to half balls of small radius centered at a nondegenerate critical point of the mean curvature function of the boundary of the manifold, and their boundary intersects the boundary of the manifold orthogonally.
A Riemannian manifold endowed with $k>2$ orthogonal complementary distributions (called here a Riemannian almost $k$-product structure) appears in such topics as multiply warped products, the webs composed of several foliations, and proper Dupin hypersurfaces of real space-forms. In the paper, we consider the mixed scalar curvature of such structure for $k>2$, derive Euler-Lagrange equations for the Einstein-Hilbert type action with respect to adapted variations of metric, and present them in a nice form of Einstein equation.
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…
For positive integers $p$ and $q$ let $G:=\textrm{PSO}(p,q)$ be the projective indefinite special-orthogonal group of signature $(p,q)$. We study counting problems in the Riemannian symmetric space $X_G$ of $G$ and in the pseudo-Riemannian hyperbolic space $\mathbb{H}^{p,q-1}$. Let $S\subset X_G$ be a totally geodesic copy of $X_{\textrm{PSO}(p,q-1)}$. We look at the orbit of $S$ under the action of a projective Anosov subgroup of $G$. For certain choices of such a geodesic copy we show that the number of points in this orbit which are at distance at most $t$ from $S$ is finite and asymptotic …
We define the notion of a smooth pseudo-Riemannian algebraic variety $(X,g)$ over a field $k$ of characteristic $0$, which is an algebraic analogue of the notion of Riemannian manifold and we study, from a model-theoretic perspective, the algebraic differential equation describing the geodesics on $(X,g)$. When $k$ is the field of real numbers, we prove that if the real points of $X$ are Zariski-dense in $X$ and if the real analytification of $(X,g)$ is a compact Riemannian manifold with negative curvature, then the algebraic differential equation describing the geodesics on $(X,g)$ is absolut…
Geodesics become an essential element of the geometry of a semi-Riemannian manifold. In fact, their differences and similarities with the (positive definite) Riemannian case, constitute the first step to understand semi-Riemannian Geometry. The progress in the last two decades has become impressive, being especially relevant the systematic introduction of (infinite-dimensional) variational methods. Our purpose is to give an overview, from refinements of classical results to updated variational settings. First, several properties (and especially completeness) of geodesics in some ambient spaces…
Bi-conformal vector fields and their applications to the characterization of conformally separable pseudo-Riemannian manifolds: New criteria for the existence of conformally flat foliations in pseudo-Riemannian manifolds
