A local classification of the Hermitian manifolds with flat associated connection is given. Hermitian manifolds admitting locally a conformal metric with flat associated connection are characterized by a curvature identity. Locally conformal Kähler manifolds as well as Hermitian surfaces with vanishing associated conformal curvature tensor are characterized as locally conformal to a Kähler manifold of constant holomorphic sectional curvatures.
The minimal Lorentzian surfaces in R 4 2 whose first normal space is twodimensional and whose Gauss curvature K and normal curvature κ satisfy K 2 − κ 2 > 0 are called minimal Lorentzian surfaces of general type. These surfaces admit canonical parameters and with respect to such parameters are determined uniquely up to a motion in R 4 2 by the curvatures K and κ satisfying a system of two natural PDEs. In the present paper we study minimal Lorentzian surfaces in R 4 2 and find a Weierstrass representation with respect to isothermal parameters of any minimal surface with twodimensional first normal space. We also obtain a Weierstrass representation with respect to canonical parameters of any minimal Lorentzian surface of general type and solve explicitly the system of natural PDEs expressing any solution to this system by means of four real functions of one variable.
Minimal surfaces with isothermal parameters admitting Bézier representation were studied by Cosín and Monterde. They showed that, up to an affine transformation, the Enneper surface is the only bi-cubic isothermal minimal surface. Here we study bi-quartic isothermal minimal surfaces and establish the general form of their generating functions in the Weierstrass representation formula. We apply an approach proposed by Ganchev to compute the normal curvature and show that, in contrast to the bi-cubic case, there is a variety of bi-quartic isothermal minimal surfaces. Based on the Bézier representation we establish some geometric properties of the bi-quartic harmonic surfaces. Numerical experiments are visualized and presented to illustrate and support our results.
We prove that an almost Kähler manifold (M, g, J) with dim M ≥ 8 and pointwise constant antiholomorphic sectional curvature is a complex spaceform.
-Introduction and preliminariesLet (M, g, J) be a 2n-dimensional almost Hermitian manifold. A 2-plane α in the tangent space T x M at a point x of M is antiholomorphic if it is orthogonal to Jα.The manifold (M, g, J) has pointwise constant antiholomorphic sectional curvature (p.c.a.s.c.) ν if, at any point x, the Riemannian sectional curvature ν(x) = K x (α) is independent on the choice of the antiholomorphic 2-plane α in T x M.If (g, J) is a Kähler structure, the previous condition means that (M, g, J) is a complex space-form, i.e. a Kähler manifold with constant holomorphic sectional curvature µ = 4ν ([2]). Moreover, the Riemannian curvature tensor R satisfies:ν being a constant function and π 1 , π 2 the tensor fields such that:
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.