arXiv research

We consider odd Laplace operators arising in odd symplectic geometry. Approach based on semidensities (densities of weight 1/2) is developed. The role of semidensities in the Batalin--Vilkovisky formalism is explained. In particular, we study the relations between semidensities on an odd symplectic supermanifold and di…

The paper establishes a local systolic inequality for odd-symplectic forms.

Study cohomology of odd symplectic manifolds, linking to Lagrangian submanifolds and BV Laplacians.

Differential forms on an odd symplectic manifold form a bicomplex: one differential is the wedge product with the symplectic form and the other is de Rham differential. In the corresponding spectral sequence the next differential turns out to be the Batalin-Vilkoviski operator.

It is shown that the characteristic vector field associated to a first order PDE has the same form of an infinitesimal generator of an odd-symplectic transformation with contact Hamiltonian the given PDE. It is considered under which condition such PDE has a characteristic vector field commuting with a generator of an …

We recall the main facts about the odd Laplacian acting on half-densities on an odd symplectic manifold and discuss a homological interpretation for it suggested recently by P. {Š}evera. We study the relationship of odd symplectic geometry with classical objects. We show that the Berezinian of a canonical transformatio…

We consider odd Poisson (odd symplectic) structure on supermanifolds induced by an odd symmetric rank $2$ (non-degenerate) contravariant tensor field. We describe the difference between odd Riemannian and odd symplectic structure in terms of the Cartan prolongation of the corresponding Lie algebras, and formulate an an…

We consider semidensities on a supermanifold E with an odd symplectic structure. We define a new $Δ$-operator action on semidensities as the proper framework for Batalin-Vilkovisky formalism. We establish relations between semidensities on E and differential forms on Lagrangian surfaces. We apply these results to Batal…

Study symplectic scalar curvature on supermanifolds.

We construct the odd symplectic structure and the equivariant even (pre)symplectic one from it on the space of differential forms on the Riemann manifold. The Poincare -- Cartan like invariants of the second structure define the equivariant generalizations of the Euler classes on the surfaces.

Let $Σ$ be a connected closed three-manifold, and let $t_Σ$ be the order of the torsion subgroup of $H_1(Σ;\mathbb Z)$. For a contact form $α$ on $Σ$, we denote by $\mathrm{Volume}(α)$ the contact volume of $α$, and by $T_{\min}(α)$ and $T_{\max}(α)$ the minimal period and the maximal period of prime periodic orbits of…

The action of Batalin-Vilkovisky Delta-operator on semidensities in an odd symplectic superspace is defined. This is used for the construction of integral invariants on surfaces embedded in an odd symplectic superspace and for more clear interpretation of the Batalin-Vilkovisky formalism geometry.

The divergence-like operator on an odd symplectic superspace which acts invariantly on a specially chosen odd vector field is considered. This operator is used to construct an odd invariant semidensity in a geometrically clear way. The formula for this semidensity is similar to the formula of the mean curvature of hype…

Study bounds magnetic geodesics on surfaces using systolic inequalities.

Characterizes symplectic and odd-symplectic Grassmannians using VMRT.

We consider odd Laplace operators acting on densities of various weight on an odd Poisson (= Schouten) manifold $M$. We prove that the case of densities of weight 1/2 (half-densities) is distinguished by the existence of a unique odd Laplace operator depending only on a point of an ``orbit space'' of volume forms. This…

Supergeneralization of $\DC P(N)$ provided by even and odd Kählerian structures from Hamiltonian reduction are construct.Operator $Δ$ which used in Batalin-- Vilkovisky quantization formalism and mechanics which are bi-Hamiltonian under corresponding even and odd Poisson brackets are considered.

Study calculates Ricci bounds for special Fano manifolds.

In our previous works, we introduced, for each (super)manifold, a commutative algebra of densities. It is endowed with a natural invariant scalar product. In this paper, we study geometry of differential operators of second order on this algebra. In the more conventional language they correspond to certain operator pen…

We consider the geometry of second order linear operators acting on the commutative algebra of densities on a (super)manifold introduced in our previous work. In the conventional language, operators on the algebra of densities correspond to operator pencils. This algebra has a natural invariant scalar product. We consi…

A well-known result of A. Vaintrob characterizes Lie algebroids and their morphisms in terms of homological vector fields on supermanifolds. We give an interpretation of Lie bialgebroids and their morphisms in terms of odd symplectic dg-manifolds, building on the approach of D. Roytenberg. This extends naturally to the…

The geometry of supermanifolds provided with $Q$-structure (i.e. with odd vector field $Q$ satisfying $\{ Q,Q\} =0$), $P$-structure (odd symplectic structure ) and $S$-structure (volume element) or with various combinations of these structures is studied. The results are applied to the analysis of Batalin-Vilkovisky ap…

A second order self-adjoint operator $Δ=S\partial^2+U$ is uniquely defined by its principal symbol $S$ and potential $U$ if it acts on half-densities. We analyse the potential $U$ as a compensating field (gauge field) in the sense that it compensates the action of coordinate transformations on the second derivatives in…

The well-known AKSZ construction (for Alexandrov--Kontsevich--Schwarz--Zaboronsky) gives an odd symplectic structure on a space of maps together with a functional $S$ that is automatically a solution for the classical master equation $(S,S)=0$. The input data required for the AKSZ construction consist of a volume eleme…

New interpretation of complex hyperbolic form as Weil-Petersson form.

Evolutionary forms, as well as exterior forms, are skew-symmetric differential forms. But in contrast to the exterior forms, the basis of evolutionary forms is deforming manifolds (with unclosed metric forms). Such forms possess a peculiarity, namely, the closed inexact exterior forms are obtained from that. The closur…

If a closed 3-manifold M supports a closed, nonsingular, irrational 1-form which linearly deforms into contact forms, then M supports a K-contact form. On the 3-torus, a closed nonsingular 1-form deforms linearly into contact forms if and only if it is a fibration 1-form. on any other 2-torus bundle over the circle, ev…

The closure conditions of the inexact exterior differential form and dual form (an equality to zero of differentials of these forms) can be treated as a definition of some differential-geometrical structure. Such a connection discloses the properties and specific features of the differential-geometrical structures. The…

Classifies conformal Killing 3-forms on nearly Kähler manifolds.

The study examines parallel forms on manifolds, focusing on specific dimensions and forms.

The study shows that the second fundamental form is intrinsic under certain conditions in space forms.

New forms generalize Whitney forms with rational coefficients for numerical analysis.

This paper classifies quadratic form parameters over integers and computes their Witt groups.

Paper presents a new flat triangular form for systems.

Abstract: Generalizes multisymplectic forms to vector-valued versions.

The paper finds a contact form on SL(2p) for p > 1.

Researchers solve conformal Killing forms on Kaehler manifolds.

Characterizes Whitney forms on simplices and proves their uniqueness.

Study of tautological forms on curve moduli spaces.

Analytic surgery and gluing formula for torsion forms in fiber bundles.

Paper presents a new triangular form for flat systems.

Special p-forms are forms which have components φ_{μ_1...μ_p} equal to +1,-1 or 0 in some orthonormal basis. A p-form φ\in Λ^p R^d is called democratic if the set of nonzero components {φ_{μ_1...μ_p}} is symmetric under the transitive action of a subgroup of O(d,Z) on the indices {1,...,d}. Knowledge of these symmetry …

The study finds Lagrangian submanifolds in adjoint semisimple orbits for real forms.

Paper transforms torse-forming vector fields into simpler forms.

Researchers found a canonical form for pairs of Hermitian and antilinear operators.

Study on immersions with flat normal bundle in curved spaces.

Paper studies second order symmetric parallel tensors in generalized f.pk-space forms.

Diffeology explores $k$-forms and bundles with more information than traditional differential forms.