Special Submanifolds Research Articles

Generalized Locally Toeplitz matrix-sequences and approximated PDEs on submanifolds: the flat case

In the present paper, we consider a class of elliptic partial differential equations with Dirichlet boundary conditions where the operator is the Laplace-Beltrami operator Δ over , Ω being an open and bounded submanifold of , . We will take into consideration the classical Finite Elements, in the case of Friedrichs-Keller triangulations, leading to sequences of matrices of increasing size. We are interested in carrying out a spectral analysis of the resulting matrix-sequences. The tools for our derivations are mainly taken from the Toeplitz technology and from the rather new theory of Generalized Locally Toeplitz (GLT) matrix-sequences. The current contribution is only quite an initial step, where a general programme is provided, with partial answers leading to further open questions: indeed the analysis is performed on special flat submanifolds and hence there is room for wide generalizations, with a final picture which is still unclear with respect to, e.g. the role of the submanifold curvature.

Decomposition into special submanifolds

AbstractWe study definably complete locally o‐minimal expansions of ordered groups. We propose a notion of special submanifolds with tubular neighborhoods and show that any definable set is decomposed into finitely many special submanifolds with tubular neighborhoods.

An optimized Chen first inequality for special slant submanifolds in Lorentz-Sasakian space forms

An optimized Chen first inequality for special slant submanifolds in Lorentz-Sasakian space forms

Marginal deformations from type IIA supergravity

We study marginal deformations of a class of three-dimensional N=2 SCFTs that admit a holographic dual description in massive type IIA supergravity. We compute the dimension of the conformal manifold in these SCFTs and identify a special submanifold with enhanced flavor symmetry. We show how to apply the TsT transformation of Lunin-Maldacena to construct explicit supersymmetric AdS_44 solutions of massive type IIA supergravity with non-trivial internal fluxes that are the holographic dual of these marginal deformations. Finally, we also briefly comment on a class of RG flows between some of these SCFTs that also admit a holographic realization.

Skew Semi-invariant Submanifolds of Generalized Kenmotsu Manifolds

In this paper, we study a special submanifold of a generalized Kenmotsu manifold. Firstly, we define a skew semi-invariant submanifold of a generalized Kenmotsu manifold and give an example. Later, we obtain some basic results for such submanifolds. Finally, we investigate the geometry of distributions.

On special submanifolds of the Page space

In this paper, we study some classes of submanifolds of codimension one and two in the Page space. These submanifolds are totally geodesic. We also compute their curvature and show that some of them are constant curvature spaces. Finally, we give information on how the Page space is related to some other metrics on the same underlying smooth manifold.

Qualar curvatures of pseudo Riemannian manifolds and pseudo Riemannian submanifolds

Some relations involving the qualar and null sectional curvatures for a pseudo Riemannian manifold are obtained. These curvatures are also investigated for pseudo-Riemannian submanifolds. Obtained relations are discussed for some special submanifolds of a Lorentzian manifold.

On Traces of Operators Associated with Actions of Compact Lie Groups

Given a pair $(M,X)$, where $X$ is a smooth submanifold in a closed smooth manifold $M$, we study the operation, which takes each operator $D$ on the ambient manifold to a certain operator on the submanifold. The latter operator is called the trace of $D$. More precisely, we study traces of operators, associated with actions of compact Lie groups on $M$. We show that traces of such operators are localized at special submanifolds in $X$ and study the structure of the traces on these submanifolds.

Willmore orbits for isometric Lie actions

In this work, we study the Willmore submanifolds in a closed connected Riemannian manifold which are orbits for the isometric action of a compact connected Lie group. We call them homogeneous Willmore submanifolds or Willmore orbits. The criteria for these special Willmore submanifolds is much easier than the general theory which requires a complicated Euler-Lagrange equation. Our main theorem claims, when the orbit type stratification for the group action satisfies certain conditions, then we can find a Willmore orbit in each stratified subset. Some classical examples of special importance, like Willmore torus, Veronese surface, etc., can be interpreted as Willmore orbits and easily verified with our method. Our theorems provide a large number of new examples for Willmore submanifolds, as well as estimates for their numbers which are sharp in some classical cases.

Stabilities of affine Legendrian submanifolds and their moduli spaces

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 special Legendrian submanifolds. Then we show that the moduli space of compact connected special affine Legendrian submanifolds is a smooth Fréchet manifold.

On the sectional curvature of lightlike submanifolds

The main purpose of this paper is to show how to obtain rigidity theorems with the help of curvature invariants in submanifolds of a semi-Riemannian manifold. For this purpose, the bounded sectional curvature is introduced and some special submanifolds of r-lightlike submanifolds of a semi-Riemannian manifold are investigated.

Deformations of special Legendrian submanifolds in Sasaki–Einstein manifolds

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, which are not special Legendrian deformations, given by harmonic 1-forms.

Dirac approach to constrained submanifolds in a double loop group: From Wess-Zumino-Novikov-Witten to Poisson-Lie σ-model

We study the restriction to a family of second class constrained submanifolds in the cotangent bundle of a double Lie group equipped with a 2-cocycle extended symplectic form to build the corresponding Dirac brackets. It is shown that, for 2-cocycle vanishing on each isotropic subspace of the associated Manin triple, the Dirac bracket contains no traces of the cocycle. We also investigate the restriction of the left translation action of the double Lie group on its cotangent bundle, where it fails to be a canonical transformation. However, the Hamiltonian symmetry is restored on some special submanifolds. The main application is to loop groups, showing that a WZNW-type model on the double Lie group with a quadratic Hamilton function in the momentum maps associated with the left translation action on the cotangent bundle with the canonical symplectic form, restricts to a collective system on some special submanifolds. There, the Lagrangian version coincides with the so-called Poisson-Lie σ -model.

Deformations of special Legendrian submanifolds with boundary

Deformations of special Legendrian submanifolds with boundary

An improved Chen–Ricci inequality for special slant submanifolds in Kenmotsu space forms

An improved Chen–Ricci inequality for special slant submanifolds in Kenmotsu space forms

Torus action on S^n and sign changing solutions for conformally invariant equations

— We construct sequences of sign changing solutions for some conformally invariant semilinear elliptic equation which is defined in Sn, when n ≥ 4. The solutions we obtain have large energy and concentrate along some special submanifolds of Sn. For example, when n ≥ 4, we obtain sequences of solutions whose energy concentrates along one great circle or finitely many great circles which are linked (they correspond to Hopf links embedded in S3 × {0} ⊂ Sn). In dimension n ≥ 5, we obtain sequences of solutions whose energy concentrates along a two dimensional torus (which corresponds to a Clifford torus embedded in S3 × {0} ⊂ Sn).

Contact Calabi-Yau manifolds and special Legendrian submanifolds

We consider a generalization of Calabi-Yau structures in the context of 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 $r$.

Volume of Domains in Symmetric Spaces*

The authors derive a formula for the volume of a compact domain in a symmetric space from normal sections through a special submanifold in the symmetric space. This formula generalizes the volume of classical domains as tubes or domains given as motions along the submanifold. Finally, some stereological considerations regarding this formula are provided.

Hamiltonian quantum dynamics with separability constraints

Schroedinger equation on a Hilbert space H , represents a linear Hamiltonian dynamical system on the space of quantum pure states, the projective Hilbert space P H . Separable states of a bipartite quantum system form a special submanifold of P H . We analyze the Hamiltonian dynamics that corresponds to the quantum system constrained on the manifold of separable states, using as an important example the system of two interacting qubits. The constraints introduce nonlinearities which render the dynamics nontrivial. We show that the qualitative properties of the constrained dynamics clearly manifest the symmetry of the qubits system. In particular, if the quantum Hamilton’s operator has not enough symmetry, the constrained dynamics is nonintegrable, and displays the typical features of a Hamiltonian dynamical system with mixed phase space. Possible physical realizations of the separability constraints are discussed.

Affine geometry of special real submanifolds of Cn

In this paper we propose an affine analogue and generalization of the geometry of special Lagrangian submanifolds of Cn.