Geodesic Submanifolds Research Articles

Lattice point counting in sectors of Hyperbolic 3-space

Let $\Gamma$ be a cocompact discrete subgroup of $\mathrm{PSL}_{2}(\mathbb{C})$ and denote by $\mathcal{H}$ the three dimensional upper half-space. For a $p\in\mathcal{H}$, we count the number of points in the orbit $\Gamma p$, according to their distance, $\operatorname{arccosh} X$, from a totally geodesic hyperplane. The main term in $n$ dimensions was obtained by Herrmann for any subset of a totally geodesic submanifold. We prove a pointwise error term of $O(X^{3/2})$ by extending the method of Huber and Chatzakos-Petridis to three dimensions. By applying Chamizo's large sieve inequalities we obtain the conjectured error term $O(X^{1+\epsilon})$ on average in the spatial aspect. We prove a corresponding large sieve inequality for the radial average and explain why it only improves on the pointwise bound by $1/6$.

Geometric configuration of Riemannian submanifolds of arbitrary codimension

In this paper we study a geometric configuration of submanifolds of arbitrary codimension in an ambient (\(n+q\))-Riemannian manifold \({\overline{M}}^{n+q}\). We obtain relations between the geometry of a q-codimension submanifold \(M^{n}\) along its boundary \(\Sigma ^{n-1}\) and the geometry of its boundary as an hypersuface of a q-codimensional submanifold \(P^{n}\) and the geometry of \(P^{n}\) as a totally umbilical q-codimension submanifold of \({\overline{M}}^{n+q}\). As a consequence of these geometric relations we get that the ellipticity of the generalized Newton transformations implies the tranversality of \(M^{n}\) and \(P^{n}\) along the boundary \(\Sigma ^{n-1}\) when \(P^{n}\) is totally geodesic submanifold of \({\overline{M}}^{n+q}\).

The normal holonomy of $CR$-submanifolds

We study the normal holonomy group, i.e. the holonomy group of the normal connection, of a $CR$-submanifold of a complex space form. We show that the normal holonomy group of a coisotropic submanifold acts as the holonomy representation of a Riemannian symmetric space. In case of a totally real submanifold we give two results about reduction of codimension. We describe explicitly the action of the normal holonomy in the case in which the totally real submanifold is contained in a totally real totally geodesic submanifold. In such a case we prove the compactness of the normal holonomy group.

A possible symplectic framework for Radon-type transforms

Our project is to define Radon-type transforms in symplectic geometry. The chosen framework consists of symplectic symmetric spaces whose canonical connection is of Ricci-type. They can be considered as symplectic analogues of the spaces of constant holomorphic curvature in Kählerian Geometry. They are characterized amongst a class of symplectic manifolds by the existence of many totally geodesic symplectic submanifolds. We present a particular class of Radon type transforms, associating to a smooth compactly supported function on a homogeneous manifold [Formula: see text], a function on a homogeneous space [Formula: see text] of totally geodesic submanifolds of [Formula: see text], and vice versa. We describe some spaces [Formula: see text] and [Formula: see text] in such Radon-type duality with [Formula: see text] a model of symplectic symmetric space with Ricci-type canonical connection and [Formula: see text] an orbit of totally geodesic symplectic submanifolds.

Screen Semi-Slant Lightlike Submanifolds of Indefinite Sasakian Manifolds

In this paper, we introduce the notion of screen semi-slant lightlike submanifolds of indefinite Sasakian manifolds giving characterization theorem with some non-trivial examples of such submanifolds. Integrability conditions of distributions D1, D2 and RadTM on screen semi-slant lightlike submanifolds of indefinite Sasakian manifolds have been obtained. Further we obtain necessary and sufficient conditions for foliations determined by above distributions to be totally geodesic. We also study mixed geodesic screen semi-slant lightlike submanifolds of indefinite Sasakian manifolds and obtain a necessary and sufficient condition for induced connection to be metric connection.

Maximal totally geodesic submanifolds and index of symmetric spaces

Let $M$ be an irreducible Riemannian symmetric space. The index $i(M)$ of $M$ is the minimal codimension of a totally geodesic submanifold of $M$. In [1] we proved that $i(M)$ is bounded from below by the rank $\mathrm{rk}(M)$ of $M$, that is, $\mathrm{rk}(M) \leq i(M)$. In this paper we classify all irreducible Riemannian symmetric spaces $M$ for which the equality holds, that is, $\mathrm{rk}(M) = i(M)$. In this context we also obtain an explicit classification of all nonsemisimple maximal totally geodesic submanifolds in irreducible Riemannian symmetric spaces of noncompact type and show that they are closely related to irreducible symmetric $\mathrm{R}$-spaces. We also determine the index of some symmetric spaces and classify the irreducible Riemannian symmetric spaces of noncompact type with $i(M) \in \{4, 5, 6 \}$.

Parametrizing Shimura subvarieties of $${\mathrm{A}_1}$$ Shimura varieties and related geometric problems

This paper gives a complete parametrization of the commensurability classes of totally geodesic subspaces of irreducible arithmetic quotients of \({X_{a, b} = (\mathbf{H}^2)^a \times (\mathbf{H}^3)^b}\). A special case describes all Shimura subvarieties of type \({\mathrm{A}_1}\) Shimura varieties. We produce, for any \({n\geq 1}\), examples of manifolds/Shimura varieties with precisely n commensurability classes of totally geodesic submanifolds/Shimura subvarieties. This is in stark contrast with the previously studied cases of arithmetic hyperbolic 3-manifolds and quaternionic Shimura surfaces, where the presence of one commensurability class of geodesic submanifolds implies the existence of infinitely many classes.

Hodge Type Theorems for Arithmetic Manifolds Associated to Orthogonal Groups

We show that special cycles generate a large part of the cohomology of locally symmetric spaces associated to orthogonal groups. We prove in particular that classes of totally geodesic submanifolds generate the cohomology groups of degree n of compact congruence p-dimensional hyperbolic manifolds “of simple type” as long as n is strictly smaller than 1 2 [ p 2 ]. We also prove that for connected Shimura varieties associated to O(p, 2) the Hodge conjecture is true for classes of degree < 1 2 [ p+1 2 ]. The proof of our general theorem makes use of the recent endoscopic classification of automorphic representations of orthogonal groups by [3]. As such our results are conditional on the stabilization of the trace formula for the (disconnected) groups GL(N)o〈θ〉 and SO(2n) o 〈θ′〉 (where θ and θ′ are the outer automorphisms), see [3, Hypothesis 3.2]. Unfortunately, at present the stabilization of the trace formula has been proved only for the case of connected groups. The extension needed is part of work in progress by the Paris-Marseille team of automorphic form researchers. For more detail, see the second paragraph of subsection 1.18 below.

Lagrangian submanifolds in the 6-dimensional nearly Kähler manifolds with parallel second fundamental form

In this paper, we study the Lagrangian submanifolds in the homogeneous nearly Kähler S3×S3 with parallel second fundamental form. We first prove that every Lagrangian submanifold with parallel second fundamental form in any 6-dimensional strict nearly Kähler manifold is totally geodesic. Then we give a complete classification of the totally geodesic Lagrangian submanifolds in the homogeneous nearly Kähler S3×S3.

Hyperbolic 4‐manifolds, colourings and mutations

We develop a way of seeing a complete orientable hyperbolic $4$-manifold $\mathcal{M}$ as an orbifold cover of a Coxeter polytope $\mathcal{P} \subset \mathbb{H}^4$ that has a facet colouring. We also develop a way of finding totally geodesic sub-manifolds $\mathcal{N}$ in $\mathcal{M}$, and describing the result of mutations along $\mathcal{N}$. As an application of our method, we construct an example of a complete orientable hyperbolic $4$-manifold $\mathcal{X}$ with a single non-toric cusp and a complete orientable hyperbolic $4$-manifold $\mathcal{Y}$ with a single toric cusp. Both $\mathcal{X}$ and $\mathcal{Y}$ have twice the minimal volume among all complete orientable hyperbolic $4$-manifolds.

Complex and quaternionic hyperbolic Kleinian groups with real trace fields

Let $\Gamma$ be a nonelementary discrete subgroup of SU(n,1) or Sp(n,1). We show that if the trace field of $\Gamma$ is contained in $\mathbb R$, $\Gamma$ preserves a totally geodesic submanifold of constant negative sectional curvature. Furthermore if $\Gamma$ is irreducible, $\Gamma$ is a Zariski dense irreducible discrete subgroup of SO(n,1) up to conjugation. This is an analog of a theorem of Maskit for general semisimple Lie groups of rank $1$.

On Geodesic Paracontact CR-lightlike Submanifolds

In present paper we investigate geodesic paracontact CR-lightlike submanifolds of para-Sasakian manifolds. Also some geometric results on paracontact screen CR-lightlike submanifolds are given.

The relative nullity of complex submanifolds and the Gauss map

We give a short and geometric proof, based on Jacobi fields, of a theorem of K. Abe that asserts that the relative index of nullity is trivial for complete non-totally geodesic complex projective submanifolds. Using this idea we prove a splitting theorem for complex Euclidean submanifolds with a non-trivial relative index of nullity

Contactomorphism group with the $$L^2$$ L 2 metric on stream functions

Here we investigate some geometric properties of the contactomorphism group \(\mathcal {D}_\theta (M)\) of a compact contact manifold with the \(L^2\) metric on the stream functions. Viewing this group as a generalization to the \(\mathcal {D}(S^1)\), the diffeomorphism group of the circle, we show that its sectional curvature is always non-negative and that the Riemannian exponential map is not locally \(C^1\). Lastly, we show that the quantomorphism group is a totally geodesic submanifold of \(\mathcal {D}_\theta (M)\) and talk about its Riemannian submersion onto the symplectomorphism group of the Boothby-Wang quotient of M.

On the index of symmetric spaces

Abstract Let M be an irreducible Riemannian symmetric space. The index of M is the minimal codimension of a (nontrivial) totally geodesic submanifold of M. We prove that the index is bounded from below by the rank of the symmetric space. We also classify the irreducible Riemannian symmetric spaces whose index is less than or equal to 3.

Isometric immersions with prefixed second order geometry in minimal codimension

We analyze the existence of isometric immersions of submanifolds of space forms, preserving the second fundamental form, or a convenient projection of it, into totally geodesic submanifolds of lower dimensions. We characterize some normal fields (that we call Codazzi and Ricci fields) whose global existence ensures such an isometric reduction of the codimension. We relate the existence of these fields to the behaviour of the curvature locus at each point of the submanifold and analyze conditions for the existence of isometric immersions into spheres and the vanishing of the normal curvature.

Totally geodesic subalgebras in 2-step nilpotent Lie algebras

We describe totally geodesic subalgebras of a metric 2-step nilpotent Lie algebra $\n$. We prove that a totally geodesic subalgebra of $\n$ is either abelian and flat or can be decomposed as a direct sum determined by the curvature transformation. In addition, we give conditions under which a totally geodesic submanifold of a simply connected 2-step nilpotent Lie group is a totally geodesic subgroup. We follow Eberlein's 1994 paper in which he imposes the condition of nonsingularity on $\n$. We remove this restriction and illustrate the distinction between the nonsingular case and the unrestricted case.

Some inequalities for warped product pseudo-slant submanifolds of nearly Kenmotsu manifolds

In this paper, we study non-trivial warped product pseudo-slant submanifolds of nearly Kenmotsu manifolds. In the beginning, we obtain some lemmas and then develop the general sharp inequalities for mixed totally geodesic warped products pseudo-slant submanifolds. The equality cases are also considered.

Magnetic curves in Sasakian manifolds

In this paper we classify the magnetic trajectories corresponding to contact magnetic fields in Sasakian manifolds of arbitrary dimension. Moreover, when the ambient is a Sasakian space form, we prove that the codimension of the curve may be reduced to 2. This means that the magnetic curve lies on a 3-dimensional Sasakian space form, embedded as a totally geodesic submanifold of the Sasakian space form of dimension (2n+1).

Totally contact umbilical slant lightlike submanifolds of indefinite Kenmotsu manifolds

In this paper, we study totally contact umbilical slant lightlike submanifolds of indefinite Kenmotsu manifolds. We prove that there does not exist totally contact umbilical proper slant lightlike submanifold in indefinite Kenmotsu manifolds other than totally contact geodesic proper slant lightlike submanifold. We also prove that there does not exist totally contact umbilical proper slant lightlike submanifold of indefinite Kenmotsu space forms. Finally, we give some characterization theorems on minimal slant lightlike submanifolds of indefinite Kenmotsu manifolds.