Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userDirac structures and Nijenhuis operators
137
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We introduce a notion of compatibility between (almost) Dirac structures and (1,1)-tensor fields extending that of Poisson-Nijenhuis structures. We study several properties of the Dirac-Nijenhuis structures thus obtained, including their connection with holomorphic Dirac structures, the geometry of their leaves and quotients, as well as the presence of hierarchies. We also consider their integration to Lie groupoids, which includes the integration of holomorphic Dirac structures as a special case.
rate research
Read More
Adopting the omni-Lie algebroid approach to Dirac-Jacobi structures, we propose and investigate a notion of weak dual pairs in Dirac-Jacobi geometry. Their main motivating examples arise from the theory of multiplicative precontact structures on Lie groupoids. Among other properties of weak dual pairs, we prove two main results. 1) We show that the property of fitting in a weak dual pair defines an equivalence relation for Dirac-Jacobi manifolds. So, in particular, we get the existence of self-dual pairs and this immediately leads to an alternative proof of the normal form theorem around Dirac-Jacobi transversals. 2) We prove the characteristic leaf correspondence theorem for weak dual pairs paralleling and extending analogous results for symplectic and contact dual pairs. Moreover, the same ideas of this proof apply to get a presymplectic leaf correspondence for weak dual pairs in Dirac geometry (not yet present in literature).
Let $S$ be a compact oriented finite dimensional manifold and $M$ a finite dimensional Riemannian manifold, let ${rm Imm}_f(S,M)$ the space of all free immersions $varphi:S to M$ and let $B^+_{i,f}(S,M)$ the quotient space ${rm Imm}_f(S,M)/{rm Diff}^+(S)$, where ${rm Diff}^+(S)$ denotes the group of orientation preserving diffeomorphisms of $S$. In this paper we prove that if $M$ admits a parallel $r$-fold vector cross product $varphi in Omega ^r(M, TM)$ and $dim S = r-1$ then $B^+_{i,f}(S,M)$ is a formally Kahler manifold. This generalizes Brylinskis, LeBruns and Verbitskys results for the case that $S$ is a codimension 2 submanifold in $M$, and $S = S^1$ or $M$ is a torsion-free $G_2$-manifold respectively.
We give the strong form and weak form of the square of Nijenhuis tensor, study their properties, and give some applications.
The space of vector-valued forms on any manifold is a graded Lie algebra with respect to the Frolicher-Nijenhuis bracket. In this paper we consider multiplicative vector-valued forms on Lie groupoids and show that they naturally form a graded Lie subalgebra. Along the way, we discuss various examples and different characterizations of multiplicative vector-valued forms.
We show how to assign to any immersed torus in $R^3$ or $S^3$ a Riemann surface such that the immersion is described by functions defined on this surface. We call this surface the spectrum or the spectral curve of the torus. The spectrum contains important information about conformally invariant properties of the torus and, in particular, relates to the Willmore functional. We propose a simple proof that for isothermic tori in $R^3$ (this class includes constant mean curvature tori and tori of revolution) the spectrum is invariant with respect to conformal transformations of $R^3$. We show that the spectral curves of minimal tori in $S^3$ introduced by Hitchin and of constant mean curvature tori in $R^3$ introduced by Pinkall and Sterling are particular cases of this general spectrum. The construction is based on the Weierstrass representation of closed surfaces in $R^3$ and the construction of the Floquet--Bloch varieties of periodic differential operators.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
