Legendrian Submanifolds Research Articles

Some Pinching Results for Bi-Slant Submanifolds in S-Space Forms

The objective of the present article is to prove two geometric inequalities for submanifolds in S-space forms. First, we establish inequalities for the generalized normalized δ-Casorati curvatures for bi-slant submanifolds in S-space forms and then we derive the generalized Wintgen inequality for Legendrian and bi-slant submanifolds in the same ambient space. We also discuss the equality cases of the inequalities. Further, we provide some immediate geometric applications of the results. Finally, we construct some examples of slant and Legendrian submanifolds, respectively.

Nonequilibrium thermodynamic process with hysteresis and metastable states—A contact Hamiltonian with unstable and stable segments of a Legendre submanifold

In this paper, a dynamical process in a statistical thermodynamic system of spins exhibiting a phase transition is described on a contact manifold, where such a dynamical process is a process that a metastable equilibrium state evolves into the most stable symmetry broken equilibrium state. Metastable and the most stable equilibrium states in the symmetry broken phase or ordered phase are assumed to be described as pruned projections of Legendre submanifolds of contact manifolds, where these pruned projections of the submanifolds express hysteresis and pseudo-free energy curves. Singularities associated with phase transitions are naturally arose in this framework as has been suggested by Legendre singularity theory. Then, a particular contact Hamiltonian vector field is proposed so that a pruned segment of the projected Legendre submanifold is a stable fixed point set in a region of a contact manifold and that another pruned segment is a unstable fixed point set. This contact Hamiltonian vector field is identified with a dynamical process departing from a metastable equilibrium state to the most stable equilibrium one. To show the statements above explicitly, an Ising type spin model with long-range interactions, called the Husimi–Temperley model, is focused, where this model exhibits a phase transition.

Varieties of minimal rational tangents of unbendable rational curves subordinate to contact structures

A nonsingular rational curve $C$ in a complex manifold $X$ whose normal bundle is isomorphic to $\mathcal{O}_{\mathbb{P}^1}(1)^{\oplus p} \oplus \mathcal{O}_{\mathbb{P}^1}^{\oplus q}$ for some nonnegative integers $p$ and $q$ is called an unbendable rational curve on $X$. Associated with it is the variety of minimal rational tangents (VMRT) at a point $x \in C$, which is the germ of submanifolds $\mathcal{C}^C_{x} \subset \mathbb{P} T_{x} X$ consisting of tangent directions of small deformations of $C$ fixing $x$. Assuming that there exists a distribution $D \subset TX$ such that all small deformations of $C$ are tangent to $D$, one asks what kind of submanifolds of projective space can be realized as the VMRT $\mathcal{C}^{C}_{x} \subset \mathbb{P} D_{x}$. When $D \subset TX$ is a contact distribution, a well-known necessary condition is that $\mathcal{C}_{x}^{C}$ should be Legendrian with respect to the induced contact structure on $\mathbb{P} D_{x}$. We prove that this is also a sufficient condition: we construct a complex manifold $X$ with a contact structure $D \subset TX$ and an unbendable rational curve $C \subset X$ such that all small deformations of $C$ are tangent to $D$ and the VMRT $\mathcal{C}^C_{x} \subset \mathbb{P} D_{x}$ at some point $x \in C$ is projectively isomorphic to an arbitrarily given Legendrian submanifold. Our construction uses the geometry of contact lines on the Heisenberg group and a technical ingredient is the symplectic geometry of distributions the study of which has originated from geometric control theory.

A note on Lagrangian intersections and Legendrian Cobordism

Let Λ,Λ′ be a pair of closed Legendrian submanifolds in a closed contact manifold (Y,ξ=Ker(α)) related by a Legendrian cobordism W⊂(C×Y,ξ˜=Ker(−yḍx+α)). In this note, we show that in the hypertight setting, if Λ intersects for reasons of Floer homology a closed pre-Lagrangian P⊂Y which is either weakly exact or monotone, then so does Λ′.

New Characterizations of the Whitney Spheres and the Contact Whitney Spheres

In this paper, based on the classical Yano’s formula, we first establish an optimal integral inequality for compact Lagrangian submanifolds in the complex space forms, which involves the Ricci curvature in the direction \(J\mathbf {H}\) and the norm of the covariant differentiation of the second fundamental form h, where J is the almost complex structure and \(\mathbf {H}\) is the mean curvature vector field. Second and analogously, for compact Legendrian submanifolds in the Sasakian space forms with Sasakian structure \((\varphi ,\xi ,\eta ,g)\), we also establish an optimal integral inequality involving the Ricci curvature in the direction \(\varphi \mathbf {H}\) and the norm of the modified covariant differentiation of the second fundamental form. The integral inequality is optimal in the sense that all submanifolds attaining the equality are completely classified. As direct consequences, we obtain new and global characterizations for the Whitney spheres in complex space forms as well as the contact Whitney spheres in Sasakian space forms. Finally, we show that, just as the Whitney spheres in complex space forms, the contact Whitney spheres in Sasakian space forms are locally conformally flat manifolds with sectional curvatures non-constant.

Bounds for Statistical Curvatures of Submanifolds in Kenmotsu-like Statistical Manifolds

In this article, we obtain certain bounds for statistical curvatures of submanifolds with any codimension of Kenmotsu-like statistical manifolds. In this context, we construct a class of optimum inequalities for submanifolds in Kenmotsu-like statistical manifolds containing the normalized scalar curvature and the generalized normalized Casorati curvatures. We also define the second fundamental form of those submanifolds that satisfy the equality condition. On Legendrian submanifolds of Kenmotsu-like statistical manifolds, we discuss a conjecture for Wintgen inequality. At the end, some immediate geometric consequences are stated.

Lifting the non-isothermal CSTR dynamics to the complete Thermodynamic Phase Space

This paper deals with the contact formulation of the contact geometry of irreversible thermodynamics. Firstly, a family of parameterized contact Hamiltonian functions, defined on the complete Thermodynamic Phase Space, generating non-strict contact vector fields which are equivalent on the associated Legendre submanifold, is proposed using both the Gibbs relation and the Gibbs-Duhem relation. Secondly, the parameterized lifts of the CSTR dynamics to the complete Thermodynamic Phase Space are performed, while guaranteeing the global attractivity of the Legendre submanifold on which the dynamics of the system is living.

Minimal Legendrian submanifolds of $\mathbb S^{9}$ with nonnegative sectional curvature

Minimal Legendrian submanifolds of $\mathbb S^{9}$ with nonnegative sectional curvature

On topologically distinct infinite families of exact Lagrangian fillings

In this note we construct examples of closed connected Legendrian submanifolds in high dimensional contact vector space that admit an arbitrary finite number of topologically distinct infinite families of diffeomorphic, but not Hamiltonian isotopic exact Lagrangian fillings.

Chekanov-Eliashberg $\mathrm{dg}$-algebras for singular Legendrians

The Chekanov-Eliashberg dg-algebra is a holomorphic curve invariant associated to Legendrian submanifolds of a contact manifold. We extend the definition to Legendrian embeddings of skeleta of Weinstein manifolds. Via Legendrian surgery, the new definition gives direct proofs of wrapped Floer cohomology push-out diagrams. It also leads to a proof of a conjectured isomorphism between partially wrapped Floer cohomology and Chekanov-Eliashberg dg-algebras with coefficients in chains on the based loop space.

The Principal Extrinsic and Intrinsic Tangent Directions of GeneralisedWintgen Ideal Legendrian Submanifolds

or Legendrian submanifolds Mn in Sasakian space forms ∼M 2n+1 (c), I. Mihai obtained an inequality relating (intrinsic) the normalised scalar curvature and (extrinsic) squared mean curvature and normalised normal curvature of M in ∼M , characterising also the corresponding equality case. In this paper, it’s shown that (intrinsic) Ricci principal directions and (extrinsic) Casorati principal directions, for generalised Wintgen ideal Legendrian submanifolds Mn in Sasakian space forms ∼M 2n+1 (c), do coincide.

The Legendrian Whitney trick

In this article, we prove a Legendrian Whitney trick which allows for the removal of intersections between codimension-two contact submanifolds and Legendrian submanifolds, assuming such a smooth cancellation is possible. This technique is applied to show the existence h-principle for codimension-two contact embeddings with a prescribed contact structure.

Contact Dynamics: Legendrian and Lagrangian Submanifolds

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 different generators of the same Legendrian submanifold. A variant of contact Tulczyjew’s triple is constructed for evolution contact dynamics.

Classification of Casorati ideal Legendrian submanifolds in Sasakian space forms II

In Lee et al. (2020) [21], the authors of the present article proved two optimal inequalities involving the Casorati curvatures δC(n−1) and δCˆ(n−1) of n-dimensional Legendrian submanifolds in Sasakian space forms and identified the classes of those submanifolds for which the equality cases of both inequalities hold. The aim of this paper is to generalize these results to the case of generalized Casorati curvatures δC(r;n−1) and δCˆ(r;n−1), which are fundamental extrinsic invariants of Riemannian submanifolds originally introduced by Decu et al. (2008) [14] as a natural generalization of δC(n−1) and δCˆ(n−1), where r is any real number such that 0<r<n(n−1) or r>n(n−1), respectively. We also provide examples of submanifolds that are ideal for any given r.

Chen-Ricci Inequalities with a Quarter Symmetric Connection in Generalized Space Forms

In this article, we obtain improved Chen-Ricci inequalities for submanifolds of generalized space forms with quarter-symmetric metric connection, with the help of which we completely characterized the Lagrangian submanifold in generalized complex space form and a Legendrian submanifold in a generalized Sasakian space form. We also discuss some geometric applications of the obtained results.

Sheaves via augmentations of Legendrian surfaces

Given an augmentation for a Legendrian surface in a 1-jet space, $$\Lambda \subset J^1(M)$$ , we explicitly construct an object, $$\mathcal {F} \in \mathbf {Sh}^\bullet _{\Lambda }(M\times \mathbb {R}, \mathbb {K})$$ , of the (derived) category from Shende, Treumann and Zaslow (Invent Math 207(3), 1031–1133 (2017)) of constructible sheaves on $$M\times \mathbb {R}$$ with singular support determined by $$\Lambda $$ . In the construction, we introduce a simplicial Legendrian DGA (differential graded algebra) for Legendrian submanifolds in 1-jet spaces that, based on Rutherford and Sullivan (Cellular Legendrian contact homology for surfaces, Part I, arXiv:1608.02984 .) Rutherford and Sullivan (Internat J Math 30(7):135, 2019) Rutherford and Sullivan (Internat J Math 30(7):111, 2019), is equivalent to the Legendrian contact homology DGA in the case of Legendrian surfaces. In addition, we extend the approach of Shende, Treumann and Zaslow (Invent Math 207(3), 1031–1133 (2017)) for 1-dimensional Legendrian knots to obtain a combinatorial model for sheaves in $$\mathbf {Sh}^\bullet _{\Lambda }(M\times \mathbb {R}, \mathbb {K})$$ in the 2-dimensional case.

R-Almost Newton–Ricci solitons on Legendrian submanifolds of Sasakian space forms

r-Almost Newton–Ricci solitons on Legendrian submanifolds of Sasakian space forms

Contact polarizations and associated metrics in geometric thermodynamics

In this work we show that a Legendre transformation is nothing but a mere change of contact polarization from the point of view of contact geometry. Then, we construct a set of Riemannian and pseudo-Riemannian metrics on a contact manifold by introducing almost contact and para-contact structures and we analyze their isometries. We show that it is not possible to find a class of metric tensors which fulfills two properties: on the one hand, to be polarization independent i.e. the Legendre transformations are the corresponding isometries and, on the other, that it induces a Hessian metric into the corresponding Legendre submanifolds. This second property is motivated by the well known Riemannian structures of the geometric description of thermodynamics which are based on Hessian metrics on the space of equilibrium states and whose properties are related to the fluctuations of the system. We find that to define a Riemannian structure with such properties it is necessary to abandon the idea of an associated metric to an almost contact or para-contact structure. We find that even extending the contact metric structure of the thermodynamic phase space the thermodynamic desiderata cannot be fulfilled.

Local rigidity, contact homeomorphisms, and conformal factors

We show that if the image of a Legendrian submanifold under a contact homeomorphism (i.e. a homeomorphism that is a $C^0$-limit of contactomorphisms) is smooth then it is Legendrian, assuming only positive local lower bounds on the conformal factors of the approximating contactomorphisms. More generally the analogous result holds for coisotropic submanifolds in the sense of arXiv:1306.6367. This is a contact version of the Humili\`ere-Leclercq-Seyfaddini coisotropic rigidity theorem in $C^0$ symplectic geometry, and the proof adapts the author's recent re-proof of that result in arXiv:1912.13043 based on a notion of local rigidity of points on locally closed subsets. We also provide two different flavors of examples showing that a contact homeomorphism can map a submanifold that is transverse to the contact structure to one that is smooth and tangent to the contact structure at a point.

Functorial LCH for immersed Lagrangian cobordisms

For $1$-dimensional Legendrian submanifolds of $1$-jet spaces, we extend the functorality of the Legendrian contact homology DG-algebra (DGA) from embedded exact Lagrangian cobordisms, as in \cite{EHK}, to a class of immersed exact Lagrangian cobordisms by considering their Legendrian lifts as conical Legendrian cobordisms. To a conical Legendrian cobordism $\Sigma$ from $\Lambda_-$ to $\Lambda_+$, we associate an immersed DGA map, which is a diagram $$\alg(\Lambda_+) \stackrel{f}{\rightarrow} \alg(\Sigma) \stackrel{i}{\hookleftarrow} \alg(\Lambda_-), $$ where $f$ is a DGA map and $i$ is an inclusion map. This construction gives a functor between suitably defined categories of Legendrians with immersed Lagrangian cobordisms and DGAs with immersed DGA maps. In an algebraic preliminary, we consider an analog of the mapping cylinder construction in the setting of DG-algebras and establish several of its properties. As an application we give examples of augmentations of Legendrian twist knots that can be induced by an immersed filling with a single double point but cannot be induced by any orientable embedded filling.