Let $Z \to Y^{2n+1}$ be the bundle of Legendrian $n$-planes over a contact manifold $Y$. We consider a foliation of $Z$ by canonical lifts of Legendrian submanifolds, called \emph{Legendrian submanifold path geometry}, whose flat model is \[ Sp(n+1, R) \to RP^{2n+1}. \] The equivalence problem provides an $sp(n+1, R)$ valued Cartan connection form that captures the geometry of such foliations.
Two special cases are considered. The first case is characterized by having a well defined conformal class of symmetric $(n+1)$ differentials on the space of leaves of the foliation $X$. The $G$ structure induced on $X$ gives an example of a classical non-metric, irreducible holonomy $GL(n+1,R)$ with representation on $sym^2(R^{n+1})$.
In the second example, we consider a \emph{Legendrian} connection on the contact hyperplane vector bundle over $Y$ whose \emph{geodesic} Legendrian submanifolds give rise to a desired foliation on $Z$. There exists a unique \emph{normal symplectic} connection associated to a Legendrian connection analogous to the normal projective connection for a torsion free affine connection.
For a nonflat example with symmetry, consider a hypersurface $M^n$ in the $(n+1)$ dimensional space form $\bar{M}_c^{n+1}$, $c=1, 0,$ or -1, without any extrinsic symmetry. The images of $M$ under the motion by Iso($\bar{M}_c^{n+1}$), when lifted, generates a Legendrian submanifold path geometry on $Gr(n, \bar{M}_c^{n+1})$.
A contact stationary Legendrian submanifold of $S^{2n+1}$ is a Legendrian submanifold whose volume is stationary under contact deformations. The simplest contact stationary Legendrian submanifold (actually minimal and Legendrian) is the real, equatorial $n$-sphere $S_0$. This paper develops a method for constructing contact stationary (but not minimal) Legendrian submanifolds of $S^{2n+1}$ by gluing together configurations of sufficiently many $U(n+1)$-rotated copies of $S_0$ at isolated points of suitably transverse intersection. The resulting submanifolds are very symmetric; are geometricall…
In this paper we study the deformation theory of submanifolds characterized by a system of differential forms and provide a criterion for deformations of such submanifolds to be unobstructed. We apply this deformation theory to special Legendrian submanifolds in Sasaki-Einstein manifolds. In general, special Legendrian deformations have the obstruction. However, we show that the deformation space of special Legendrian submanifolds is the intersection of two larger smooth deformation spaces of different types. We also prove that any special Legendrian submanifold admits smooth deformations, whi…
In this paper, we prove some differentiable sphere theorems and topological sphere theorems for Lagrangian submanifolds in Kähler manifold and Legendrian submanifolds in Sasaki space form.
In [5], D. Fetcu and C. Oniciuc presented the classification result for biharmonic $\mathcal{C}$-parallel Legendrian submanifolds in $7$-dimensional Sasakian space forms. However, it is incomplete. In this paper, all such submanifolds are explicitly determined.
We introduce the notion of affine Legendrian submanifolds in Sasakian manifolds and define a canonical volume called the $ϕ$-volume as odd dimensional analogues of affine Lagrangian (totally real or purely real) geometry. Then we derive the second variation formula of the $ϕ$-volume to obtain the stability result in some $η$-Einstein Sasakian manifolds. It also implies the convexity of the $ϕ$-volume functional on the space of affine Legendrian submanifolds. Next, we introduce the notion of special affine Legendrian submanifolds in Sasaki-Einstein manifolds as a generalization of that of speci…
We present classifications of totally geodesic and totally umbilical Legendrian submanifolds of $(κ,μ)$-spaces with Boeckx invariant $I \leq -1$. In particular, we prove that such submanifolds must be, up to local isometries, among the examples that we explicitly construct.
We give a procedure to ``average'' canonically $C^1$-close Legendrian submanifolds of contact manifolds. As a corollary we obtain that, whenever a compact group action leaves a Legendrian submanifold almost invariant, there is an invariant Legendrian submanifold nearby.
We show that every toric Sasaki-Einstein manifold $S$ admits a special Legendrian submanifold $L$ which arises as the link ${\rm fix}(τ)\cap S$ of the fixed point set ${\rm fix}(τ)$ of an anti-holomorphic involution $τ$ on the cone $C(S)$. In particular, an irregular toric Sasaki-Einstein manifold $S^{2}\times S^{3}$ has a special Legendrian torus $S^{1}\times S^{1}$. Moreover, we also obtain a special Legendrian submanifold in $\sharp m(S^{2}\times S^{3})$ for each $m\ge 1$.
We consider a generalization of Calabi-Yau structures in the context of $α$-Sasakian manifolds. We study deformations of a special class of Legendrian submanifolds and classify invariant contact Calabi-Yau structures on 5-dimensional nilmanifolds. Finally we generalize to codimension $r$.
In this paper we provide a second variation formula for L-minimal Lagrangian submanifolds in a pseudo-Sasakian manifold. We apply it to the case of Lorentzian-Sasakian manifolds and relate the L-stability of L-minimal Legendrians in a Sasakian M to their L-stability in an associated Lorentzian-Sasakian structure on M.
For a given minimal Legendrian submanifold $L$ of a Sasaki-Einstein manifold we construct two families of eigenfunctions of the Laplacian of $L$ and we give a lower bound for the dimension of the corresponding eigenspace. Moreover, in the case the lower bound is attained, we prove that $L$ is totally geodesic and a rigidity result about the ambient manifold. This is a generalization of a result for the standard Sasakian sphere done by Lê and Wang.
This is a corrected version of our paper published in Osaka Journal of Mathematics 51(2014), 673-693. We correct Theorem~1.1, Proposition~3.3 and their proofs.
We are proposing Tulczyjew's triple for contact dynamics. The most important ingredients of the triple, namely symplectic diffeomorphisms, special symplectic manifolds, and Morse families, are generalized to the contact framework. These geometries permit us to determine so-called generating family (obtained by merging a special contact manifold and a Morse family) for a Legendrian submanifold. Contact Hamiltonian and Lagrangian Dynamics are recast as Legendrian submanifolds of the tangent contact manifold. In this picture, the Legendre transformation is determined to be a passage between two d…
This is mainly a survey article on the recent development of the theory of graph-like Legendrian unfoldings and its applications. The notion of big Legendrian submanifolds was introduced by Zakalyukin for describing the wave front propagations. Graph-like Legendrian unfoldings belong to a special class of big Legendrian submanifolds. Although this is a survey article, some new original results and the corrected proofs of some results are given.
A contact stationary Legendrian submanifold (briefly, CSL submanifold) is a stationary point of the volume functional of Legendrian submanifolds in a Sasakian manifold. Much effort has been paid in the last two decades to construct examples of such manifolds, mainly by geometers using various geometric methods. But we have rare knowledge about their geometric properties till now. Recently, Y. Luo (\cite{ Luo2, Luo1}) proved that a closed CSL surface in $\mathbb{S}^5$ with the square length of its second fundamental form belonging to $[0,2]$ must be totally geodesic or be a flat minimal Legendr…
Rigidity of Lagrangian submanifolds with conformal Maslov form and Legendrian capillary boundary
Beijing University of Civil Engineering and Architecture（北京建筑大学）；Chongqing University of Technology（重庆理工大学）；Tsinghua University（清华大学）；Xinyang Normal University（信阳师范大学）
We study a cone structure ${\mathcal C} \subset {\mathbb P} D$ on a holomorphic contact manifold $(M, D \subset T_M)$ such that each fiber ${\mathcal C}_x \subset {\mathbb P} D_x$ is isomorphic to a Legendrian submanifold of fixed isomorphism type. By characterizing subadjoint varieties among Legendrian submanifolds in terms of contact prolongations, we prove that the canonical distribution on the associated contact G-structure admits a holomorphic horizontal splitting.
In this paper, we study the rigidity theorem of closed minimally immersed Legendrian submanifolds in the unit sphere. Utilizing the maximum principle, we obtain a new characterization of the Calabi torus in the unit sphere which is the minimal Calabi product Legendrian immersion of a point and the totally geodesic Legendrian sphere. We also establish an optimal Simons' type integral inequality in terms of the second fundamental form of three dimensional closed minimal Legendrian submanifolds in the unit sphere.
Unlike Legendrian submanifolds, the deformation problem of coisotropic submanifolds can be obstructed. Starting from this observation, we single out in the contact setting the special class of integral coisotropic submanifolds as the direct generalization of Legendrian submanifolds for what concerns deformation and moduli theory. Indeed, being integral coisotropic is proved to be a rigid condition, and moreover the integral coisotropic deformation problem is unobstructed with discrete moduli space.
Wilson loops in ${\cal N}=4$ supersymmetric Yang-Mills theory correspond at strong coupling to extremal surfaces in $AdS_5$. We study a class of extremal surfaces known as special Legendrian submanifolds. The "hemisphere" corresponding to the circular Wilson loop is an example of a special Legendrian submanifold, and we give another example. We formulate the necessary conditions for the contour on the boundary of $AdS_5$ to be the boundary of the special Legendrian submanifold and conjecture that these conditions are in fact sufficient. We call the solutions of these conditions "special contac…
Y. Eliashberg and N. Mishachev introduced the notion of wrinkled embedding to show that any tangential homotopy can be approximated by a homotopy of topological embeddings with mild singularities. This concept plays an important role in Contact Topology: The loose legendrian h-principle of E. Murphy relies on wrinkled embeddings to manipulate the legendrian front. Similarly, the simplification of legendrian front singularities was proven by D. Álvarez-Gavela by defining the notion of wrinkled legendrian. This paper and its sequel generalise these ideas to general jet spaces. The main theorem i…
We describe a semi-local canonical form for Legendrian foliations on contact manifolds in the neighbourhood of a Legendrian submanifold. This result generalizes local results by Libermann and Pang on Legendrian foliations on contact manifolds, and is analogeous to a semi-local result by Weinstein in the symplectic case. For the proof, we introduce and use a class of cohomology that obstructs the possibility to make a Legendrian foliation flat.
For a Legendrian submanifold $M$ of a Sasaki manifold $N$, we study harmonicity and biharmonicity of the corresponding Lagrangian cone submanifold C(M) of a Kaehler manifold C(N). We show that, if $C(M)$ is biharmonic in C(N), then it is harmonic; and $M$ is proper biharmonic in $N$ if and only if C(M) has a non-zero eigen-section of the Jacobi operator with the eigenvalue $m=dim M$.
Using Legendrian immersions and, in particular, Legendre curves in odd dimensional spheres and anti De Sitter spaces, we provide a method of construction of new examples of Hamiltonian-minimal Lagrangian submanifolds in complex projective and hyperbolic spaces, including explicit one parameter families of embeddings of quotients of certain product manifolds. In addition, new examples of minimal Lagrangian submanifolds in complex projective and hyperbolic spaces also appear. Making use of all of them, we get Hamiltonian-minimal and special Lagrangian cones in complex Euclidean space too.
We introduce a notion of semi-legendrian real submanifold in a complex manifold of dimension 3 and prove that real submanifolds of $\mathbb{C}^2$ can be uniquely lifted to $\mathbb{C}^3$. Then we deduce a complex duality between real submanifolds of $\mathbb{P}^2(\mathbb{C})$.
Homotopy connectedness theorems for complex submanifolds of homogeneous spaces (sometimes referred to as theorems of Barth-Lefshetz type) have been established by a number of authors. Morse Theory on the space of paths lead to an elegant proof of homotopy connectedness theorems for complex submanifolds of Hermitian symmetric spaces. In this work we extend this proof to a larger class of compact complex homogeneous spaces.
