Mircea Crasmareanu

Ricci solitons in manifolds with quasi-constant curvature

The Eisenhart problem of nding parallel tensors treated already in the framework of quasi-constant curvature manifolds in [Jia] is reconsidered for the symmetric case and the result is interpreted in terms of Ricci solitons. If the generator of the manifold provides a Ricci soliton then this is i) expanding on para-Sasakian spaces with constant scalar curvature and vanishing D-concircular tensor eld and ii) shrinking on a class of orientable quasi-umbilical hypersurfaces of a real projective space=elliptic space form.

GOLDEN- AND PRODUCT-SHAPED HYPERSURFACES IN REAL SPACE FORMS

We define two classes of hypersurfaces in real space forms, golden- and product-shaped, respectively, by imposing the shape operator to be of golden or product type. We obtain the whole families of above hypersurfaces, based on the classification of isoparametric hypersurfaces, as follows: in the golden case all are hyperspheres, a hyperbolic space and a generalized Clifford torus, while for the product case we obtain the unit hypersphere, the hyperplane, a hypersphere and its associated Clifford torus, respectively, according to the type of the ambient space form namely parabolic, hyperbolic or elliptic, respectively.

SLANT CURVES AND PARTICLES IN THREE-DIMENSIONAL WARPED PRODUCTS AND THEIR LANCRET INVARIANTS

Slant curves are introduced in three-dimensional warped products with Euclidean factors; these curves are characterized through the scalar product between the normal at the curve and the vertical vector field and an important feature is that the case of constant Frenet curvatures implies a proper mean curvature vector field. A Lancret invariant is obtained and the Legendre curves are analyzed as particular case. An example of a slant curve is given for the exponential warping function; our example illustrates a proper (i.e. not reducible to the two dimensions) case of the Lancret Theorem of

SUB-WEYL GEOMETRY AND ITS LINEAR CONNECTIONS

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Weakly-symmetry of the Sasakian lifts on tangent bundles

-dimensional hyperbolic geometry. We point out an eventually relationship with the geometry of relativistic models. 10.1017/S0004972712000809

DIRAC STRUCTURES FROM LIE INTEGRABILITY

The aim of this paper is to study from the point of view of linear connections the data with M a smooth (n+p)-dimensional real manifold, an n-dimensional manifold semi-Riemannian distribution on M, the conformal structure generated by g and W a Weyl substructure: a map such that W(ḡ) = W(g) - du, ḡ = eug;u ∈ C∞(M). Compatible linear connections are introduced as a natural extension of similar notions from Weyl geometry and such a connection is unique if a symmetry condition is imposed. In the foliated case the local expression of this unique connection is obtained. The notion of Vranceanu connection is introduced for a pair (Weyl structure, distribution) and it is computed for the tangent bundle of Finsler spaces, particularly Riemannian, choosing as distribution the vertical bundle of tangent bundle projection and as one-form the Cartan form.

THE GEOMETRY OF TANGENT CONJUGATE CONNECTIONS

The weakly symmetry of the Sasakian lift G of a Riemannian metric g is characterized in terms of ∞atness for g and G. The cases of recurrent or pseudo- symmetric G studied by Binh and Tamassy are obtained in particular.

Holomorphic last multipliers on complex manifolds

We prove that a pair (F = vector sub-bundle of TM, its annihilator) yields an almost Dirac structure which is Dirac if and only if F is Lie integrable. Then a flat Ehresmann connection on a fiber bundle ξ yields two complementary, but not orthogonally, Dirac structures on the total space M of ξ. These Dirac structures are also Lagrangian sub-bundles with respect to the natural almost symplectic structure of the big tangent bundle of M. The tangent bundle in Riemannian geometry is discussed as particular case and the 3-dimensional Heisenberg space is illustrated as example. More generally, we study the Bianchi–Cartan–Vranceanu metrics and their Hopf bundles.

Parallel second-order tensors on Vaisman manifolds

The notion of conjugate connection is introduced in the almost tangent geometry and its properties are studied from a global point of view. Two variants for this type of connections are also considered in order to find the linear connections making parallel a given almost tangent structure. 2000 AMS Classification: 53C15; 53C05; 53C10; 53C07.

Wick–Tzitzeica solitons and their Monge–Ampér equation

The goal of this paper is to study the theory of last multipliers in the framework of complex manifolds with a fixed holomorphic volume form. The motivation of our study is based on the equivalence between a holomorphic ODE system and an associated real ODE system and we are interested how we can relate holomorphic last multipliers with real last multipliers. Also, we consider some applications of our study for holomorphic gradient vector fields on holomorphic Riemannain manifolds as well as for holomorphic Hamiltonian vector fields and holomorphic Poisson bivector fields on holomorphic Poisson manifolds.