On the geometry of the reachability set of vector fields

We study the geometry of the reachability set of a family of vector fields on a C ^∞ manifold. We show that, for each real number T, the T-reachability set is a smooth submanifold of an orbit of codimension zero or one and that, on an arbitrary connected C ^∞ manifold of dimension greater than one, there exists a system of three vector fields such that each 0-reachability set coincides with the manifold itself.     

Vitushkin’s germ theorem for engel-type CR manifolds

We study real analytic CR manifolds of CR dimension 1 and codimension 2 in the three-dimensional complex space. We prove that the germ of a holomorphic mapping between “nonspherical” manifolds can be extended along any path (this is an analog of Vitushkin’s germ theorem). For a cubic model surface (“sphere”), we prove an analog of the Poincaré theorem on the mappings of spheres into ?². We construct an example of a compact “spherical” submanifold in a compact complex 3-space such that the germ of a mapping of the “sphere” into this submanifold cannot be extended to a certain point of the “sphere.”     

Polynomial Models of Degree 5 and Algebras of Their Automorphisms

In classifying and studying holomorphic automorphisms of surfaces, it is often convenient to pass to tangent model surfaces. This method is well developed for surfaces of type (n,K), where K≤ ² ; for such surfaces, tangent quadrics (i.e., surfaces determined by equations of degree 2) with a number of useful properties have been constructed. In recent years, for surfaces of higher codimensions, tangent model surfaces of degrees 3 and 4 with similar properties were constructed. However, this construction imposes new constraints on the codimension. In this paper, the same method is applied to surfaces of even higher codimension. Model surfaces of the fifth degree are constructed. It is shown that all the basic useful properties of model surfaces are preserved, in spite of a number of technical difficulties.     

Controllability in codimension one

Local and global controlability of analytic affine control systems Σ with an arbitrary number of controls are studied, assuming strong accessibility of Σ and codimension one of the Lie algebra T′ generated by the input vector fields, i.e., dim T′ (p) = n − 1 at every p ϵ M. Controls are assumed to have no a priori bound. From the study of the set H where a necessary condition for local controllability at a point is verified and assuming some transversality relations, an easy to verify geometric condition is proved: H is a submanifold and Σ is locally controllable at every point of H outside a codimension one submanifold. A geometric sufficient condition for global controllability on simply connected manifolds is then obtained: if every leaf of T′ intersects the manifold H and some transversality relations (including those involved in the local condition) are verified, Σ is globally controllable.     

On the Geometry of Lagrangian Submanifolds

We prove that a Lagrangian submanifold passes through each point of a symplectic manifold in the direction of arbitrary Lagrangian plane at this point. Generally speaking, such a Lagrangian submanifold is not unique; nevertheless, the set of all such submanifolds in Hermitian extension of a symplectic manifold of dimension greater than 4 for arbitrary initial data contains a totally geodesic submanifold (which we call the s-Lagrangian submanifold) iff this symplectic manifold is a complex space form. We show that each Lagrangian submanifold in a complex space form of holomorphic sectional curvature equal to c is a space of constant curvature c/4. We apply these results to the geometry of principal toroidal bundles.     

Characterizations of Spacelike Submanifolds with Constant Scalar Curvature in the de Sitter Space

In this paper, we deal with complete spacelike submanifolds \(M^n\) immersed in the de Sitter space \(\mathbb S_p^{n+p}\) of index p with parallel normalized mean curvature vector and constant scalar curvature R. Imposing a suitable restriction on the values of R, we apply a maximum principle for the so-called Cheng–Yau operator L, which enables us to show that either such a submanifold must be totally umbilical or it holds a sharp estimate for the norm of its total umbilicity tensor, with equality if and only the submanifold is isometric to a hyperbolic cylinder of the ambient space. In particular, when \(n=2\) this provides a nice characterization of the totally umbilical spacelike surfaces of \(\mathbb {S}^{2+p}_p\) with codimension \(p\ge 2\). Furthermore, we also study the case in which these spacelike submanifold are L-parabolic.     

Embeddings of submanifolds and normal bundles

This paper studies the embeddings of a complex submanifold S inside a complex manifold M; in particular, we are interested in comparing the embedding of S in M with the embedding of S as the zero section in the total space of the normal bundle NS of S in M. We explicitly describe some cohomological classes allowing to measure the difference between the two embeddings, in the spirit of the work by Grauert, Griffiths, and Camacho, Movasati and Sad; we are also able to explain the geometrical meaning of the separate vanishing of these classes. Our results hold for any codimension, but even for curves in a surface we generalize previous results due to Laufert and Camacho, Movasati and Sad.     

Pseudoholomorphic discs attached to CR-submanifolds of almost complex spaces

Let E be a generic real submanifold of an almost complex manifold. The geometry of Bishop discs attached to E is studied in terms of the Levi form of E.     

Special Slant Surfaces and a Basic Inequality

A slant immersion is an isometric immersion of a Riemannian manifold into an almost Hermitian manifold with constant Wirtinger angle. A slant submanifold is called proper if it is neither holomorphic nor totally real. In [2], the author proved that, for any proper slant surface M with slant angle θ in a complex-space-form $?detilde M^2(4?silon)$ with constant holomorphic sectional curvature 4?, the squared mean curvature and the Gauss curvature of M satisfy the following basic inequality: H²(p) ≥ 2K(p) ? 2(1 + 3 cos²θ)?. Every proper slant surface satisfying the equality case of this inequality is special slant. One purpose of this article is to completely classify proper slant surfaces which satisfy the equality case of this inequality. Another purpose of this article is to completely classify special slant surfaces with constant mean curvature. Further results on special slant surfaces are also presented.     

Existence and nonexistence of skew branes

A skew brane is a codimension 2 submanifold in affine space such that the tangent spaces at any two distinct points are not parallel. We show that if an oriented closed manifold has a nonzero Euler characteristic c{\chi}, then it is not a skew brane; generically, the number of oppositely oriented pairs of parallel tangent spaces is not less than c²/4{\chi^2{/4}}. We give a version of this result for immersed surfaces in dimension 4. We construct examples of skew spheres of arbitrary odd dimensions, generalizing the construction of skew loops in 3-dimensional space due to Ghomi and Solomon (2002). We conclude with two conjectures that are theorems in 1-dimensional case.     

Isomorphisme de Dolbeault dans les variétés CR

Let M be a q-concave CR generic C^∞-smooth submanifold of real codimension k in an n-dimensional complex manifold X, then the truncated tangential Cauchy-Riemann complexes τ^≤q−1[ɛ^p,*](M), τ^{≥n−k−q+}2[ɛ^{p, *}](M) of C^∞-smooth forms and τ^≤q−1 [D′^p.*](M), τ^{≥n−k−q+}2[D′^p,*(M)] of currents on M are quasi-isomorphic for 0 ≤ p ≤ n.     

Knots in Riemannian manifolds

In this paper we study submanifolds with nonpositive extrinsic curvature in a positively curved manifold. Among other things we prove that, if ${K\subset (S^n, g)}$ is a totally geodesic submanifold of codimension 2 in a Riemannian sphere with positive sectional curvature where n ≥ 5, then K is homeomorphic to S ^n–2 and the fundamental group of the knot complement ${\pi _1(S^n-K)\cong \mathbb{Z}}$ .     

Sur l'algébrisabilité locale de sous-variétés analytiques réelles génériques de Cn

We prove that the local (pseudo)group of biholomorphisms stabilizing a minimal, finitely nondegenerate real algebraic submanifold in $\text{C}{}^{\mathrm{n}}$ is a real algebraic local Lie group. We deduce necessary conditions for the local algebraizability of real analytic rigid tubes of arbitrary codimension in $\text{C}{}^{\mathrm{n}}$. To cite this article: H. Gaussier, J. Merker, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Angle theorems for the Lagrangian mean curvature flow

We prove that symplectic maps between Riemann surfaces L, M of constant, nonpositive and equal curvature converge to minimal symplectic maps, if the Lagrangian angle for the corresponding Lagrangian submanifold in the cross product space satisfies . If one considers a 4-dimensional K?hler-Einstein manifold of nonpositive scalar curvature that admits two complex structures J, K which commute and assumes that is a compact oriented Lagrangian submanifold w.r.t. J such that the K?hler form w.r.t.K restricted to L is positive and , then L converges under the mean curvature flow to a minimal Lagrangian submanifold which is calibrated w.r.t. . Received: 11 April 2001 / Published online: 29 April 2002     

A notion of robustness and stability of manifolds

Starting from the notion of thickness of Parks we define a notion of robustness for arbitrary subsets of Rk and we investigate its relationship with the notion of positive reach of Federer. We prove that if a set M is robust, then its boundary ∂M is of positive reach and conversely (under very mild restrictions) if ∂M is of positive reach, then M is robust. We then prove that a closed non-empty robust set in Rk (different from Rk) is a codimension zero submanifold of class C¹ with boundary. As a partial converse we show that any compact codimension zero submanifold with boundary of class C² is robust. Using the notion of robustness we prove a kind of stability theorem for codimension zero compact submanifolds with boundary: two such submanifolds, whose boundaries are close enough (in the sense of Hausdorff distance), are diffeomorphic.     

Approximate solutions and micro-regularity in the Denjoy-Carleman classes

We begin with a sequence M of positive real numbers and we consider the Denjoy-Carleman class CM. We show how to construct M-approximate solutions for complex vector fields with CM coefficients. We then use our construction to study micro-local properties of boundary values of approximate solutions in general M-involutive structures of codimension one, where the approximate solution is defined in a wedge whose edge (where the boundary value exists) is a maximally real submanifold. We also obtain a CM version of the Edge-of-the-Wedge Theorem.     

A local and global splitting result for real Kähler Euclidean submanifolds

We show that if a real Kähler Euclidean submanifold is as far as possible of being minimal, then it should split locally as a product of hypersurfaces almost everywhere, possibly in lower codimension. In addition, if the manifold is complete, simply connected and has constant nullity, it should split globally as a product of surfaces in and an Euclidean factor. Several applications are also given.Received: 28 May 2004     

Marked fatgraph complexes and surface automorphisms

Combinatorial aspects of the Torelli–Johnson–Morita theory of surface automorphisms are extended to certain subgroups of the mapping class groups. These subgroups are defined relative to a specified homomorphism from the fundamental group of the surface onto an arbitrary group K. For K abelian, there is a combinatorial theory akin to the classical case, for example, providing an explicit cocycle representing the first Johnson homomophism with target Λ ³ K. Furthermore, the Earle class with coefficients in K is represented by an explicit cocyle.     

Isotropic ppmc immersions

Isometric immersions with parallel pluri-mean curvature (“ppmc”) in euclidean n-space generalize constant mean curvature (“cmc”) surfaces to higher dimensional Kähler submanifolds. Like cmc surfaces they allow a one-parameter family of isometric deformations rotating the second fundamental form at each point. If these deformations are trivial the ppmc immersions are called isotropic. Our main result drastically restricts the intrinsic geometry of such a submanifold: Locally, it must be a symmetric space or a Riemannian product unless the immersion is holomorphic or a superminimal surface in a sphere. We can give a precise classification if the codimension is less than 7. The main idea of the proof is to show that the tangent holonomy is restricted and to apply the Berger-Simons holonomy theorem.     

Local Geometry of Levi-Forms Associated with the Existence of Complex Submanifolds and the Minimality of Generic CR Manifolds

Let M be a generic CR manifold in \BbbC^m+d\Bbb{C}^{m+d} of codimension d, locally given as the common zero set of real-valued functions r ¹,…,r ^d . Given an integer δ=1,…,d, we find a necessary and sufficient condition for M to contain a real submanifold of codimension δ with the same CR structure. We also find a necessary and sufficient condition and several sufficient conditions for M to admit a complex submanifold of complex dimension n, for any n=1,…,m. We use the method of prolongation of an exterior differential system. The conditions are systems of partial differential equations on r ¹,…,r ^d of third order.