The \classical case" is the case in which X is a compact riemannian manifold and D is the (positive de nite) Laplacian. Then (1.1) is the heat equation on X . In this paper we'll look at the special case where X is a riemannian symmetric space of noncompact type. Thus X is a noncompact riemannian manifold with a very large symmetry group G, harmonic analysis on X is understood in terms of the s...

It is well known that ?an almost complex structure? J J2 = ?I on the manifold M called Hermitian manifold? (M, J,G) if G(JX, JY) G(X,Y) and proved (F2M, JD,GD) frame bundle of second order F2M. The term refers to general quadratic structure pJ + qI, where p 0, q ?1. However, this paper aims study equation p, are positive integers, it named as a metallic structure. diagonal lift F2Mis studied sh...

In this paper we describe a reduction process that allows us to define Hamiltonian structures on the manifold of differential invariants of parametrized curves for any homogeneous manifold of the form G/H, with G semisimple. We also prove that equations that are Hamiltonian with respect to the first of these reduced brackets automatically have a geometric realization as an invariant flow of cur...

A polarized Calabi-Yau manifold is a pair (X,ω) of a compact algebraic manifold X with zero first Chern class and a Kähler form ω ∈ H(X,Z). The form ω is called a polarization. Let M be the universal deformation space of (X,ω). M is smooth by a theorem of Tian [5]. By [8], we may assume that each X ′ ∈ M is a Kähler-Einstein manifold. i.e. the associated Kähler metric (g′ αβ ) is Ricci flat. The

Let (Mn, g) be a compact Kähler manifold with nonpositive bisectional curvature. We show that a finite cover is biholomorphic and isometric to a flat torus bundle over a compact Kähler manifold Nk with c1 < 0. This confirms a conjecture of Yau. As a corollary, for any compact Kähler manifold with nonpositive bisectional curvature, the Kodaira dimension is equal to the maximal rank of the Ricci ...

The Hilbert-Smith Conjecture states that if G is a locally compact group which acts effectively on a connected manifold as a topological transformation group, then G is a Lie group. A rather straightforward proof of this conjecture is given. The motivation is work of Cernavskii (“Finite-to-one mappings of manifolds”, Trans. of Math. Sk. 65 (107), 1964.) His work is generalized to the orbit map ...

Let G be a unimodular Lie group, X a compact manifold with boundary, and M be the total space of a principal bundle G → M → X so that M is also a complex manifold satisfying a local subelliptic estimate. In this work, we show that if G acts by holomorphic transformations in M , then the Laplacian = ∂̄∗∂̄ + ∂̄∂̄∗ on M has the following properties: The kernel of restricted to the forms Λ with q > 0 i...

We will show that a statistical manifold $$(M, g, \nabla )$$ has constant curvature if and only it is projectively flat conjugate symmetric manifold, is, the affine connection $$\nabla $$ curvatures satisfies $$R=R^*$$ , where $$R^*$$ of dual ^*$$ . Moreover, we properly convex structures on compact induces $$-1$$ vice versa.

It is proved that the associative differential graded algebra of (polynomial) polyvector fields on a vector space (may be infinite-dimensional) is quasi-isomorphic to the corresponding cohomological Hochschild complex of (polynomial) functions on this vector space as an associative differential graded algebra. This result is an A∞-version of the Formality conjecture of Maxim Kontsevich [K]. 1. ...