Riemannian manifolds with a parallel field of complex planes

The purpose of this paper is to discuss Riemannian manifolds which admit a parallel field of complex planes, consisting of vectors of the form , where a,b are real orthogonal vectors of equal length. Using the Nirenberg Frobenius Theorem [12], it follows that these are reducible Riemannian manifolds, whose metric is locally a sum of a Kähler and of a Riemann metric, and we are calling thempartially Kähler manifolds.After a general presentation of these manifolds (including a general presentation of the complex integrable plane fields) we are discussing harmonic forms, Betti numbers, and Dolbeault cohomology. This discussion is based on a theorem of Chern [4], and it provides generalizations of the results of Goldberg [9], as well as some other new results.To Prof. R. Artzy on his 70th Birthday     

On generalized recurrent Riemannian manifolds

The object of the present paper is to study a type of Riemannian manifolds called generalized recurrent manifolds. We have constructed two concrete examples of such a manifold whose scalar curvature is non-zero non-constant. Some other properties have been considered. Among others it is shown that on a generalized recurrent manifold with constant scalar curvature, Weyl-semisymmetry and semisymmetry are equivalent. Sufficient condition for a generalized recurrent manifold to be a special quasi Einstein manifold is obtained.     

Heat semi-group and generalized flows on complete Riemannian manifolds

We will use the heat semi-group to regularize functions and vector fields on Riemannian manifolds in order to develop Di Perna–Lions theory in this setting. Malliavin?s point of view of the bundle of orthonormal frames on Brownian motions will play a fundamental role. As a byproduct we will construct diffusion processes associated to an elliptic operator with singular drift.     

Steepest descent method with a generalized Armijo search for quasiconvex functions on Riemannian manifolds

This paper extends the full convergence of the steepest descent method with a generalized Armijo search and a proximal regularization to solve minimization problems with quasiconvex objective functions on complete Riemannian manifolds. Previous convergence results are obtained as particular cases and some examples in non-Euclidian spaces are given. In particular, our approach can be used to solve constrained minimization problems with nonconvex objective functions in Euclidian spaces if the set of constraints is a Riemannian manifold and the objective function is quasiconvex in this manifold.     

Classification of generalized normal homogeneous Riemannian manifolds of positive Euler characteristic

The authors give a short survey of previous results on generalized normal homogeneous (δ-homogeneous, in other terms) Riemannian manifolds, forming a new proper subclass of geodesic orbit spaces with nonnegative sectional curvature, which properly includes the class of all normal homogeneous Riemannian manifolds. As a continuation and an application of these results, they prove that the family of all compact simply connected indecomposable generalized normal homogeneous Riemannian manifolds with positive Euler characteristic, which are not normal homogeneous, consists exactly of all generalized flag manifolds Sp(l)/U(1)⋅Sp(l−1)=CP2l−1, l?2, supplied with invariant Riemannian metrics of positive sectional curvature with the pinching constants (the ratio of the minimal sectional curvature to the maximal one) in the open interval (1/16,1/4). This implies very unusual geometric properties of the adjoint representation of Sp(l), l?2. Some unsolved questions are suggested.     

Stable projective planes with Riemannian metrics

We consider the question whether the system of lines of a two-dimensional stable plane can be described as the system of geodesics of a Riemannian metric and vice versa; we present two results: A complete two-dimensional Riemannian manifold with the property that every two points are joined by a unique geodesic and its family of geodesics form a stable plane. On the other hand every stable projective plane whose lines are geodesics of a Riemannian metric is isometric to the real projective plane. Combining both results it follows that it is impossible to realize the lines of a non-desarguesian projective plane using the geodesics of a complete Riemannian manifold.     

Pseudo Riemannian manifolds whose generalized Jacobi operator has constant characteristic polynomial

Letg be a non-degenerate innerproduct of signature (p,q) onR ^m . LetGr _r,s (g) be the Grassmanian of planes so the restriction of g to is non-degenerate and has signature (r, s). IfR is an algebraic curvature tensor onR ^m , we define a generalized Jacobi operator onGr _r,s (g) and study when the characteristic polynomial of this operator is constant.Dedicated to Professor Helmut Karzel on his 70th birthdayResearch partially supported by the NSF (USA).Research partially supported by the NFSI (Bulgaria).     

The symmetry axiom in Minkowski planes

We give a synthetic proof that in a symmetric Minkowski plane the rectangle axiom (G) holds. Dedicated to Professor Helmut Karzel     

Riemannian flag manifolds with homogeneous geodesics

A geodesic in a Riemannian homogeneous manifold is called a homogeneous geodesic if it is an orbit of a one-parameter subgroup of the Lie group . We investigate -invariant metrics with homogeneous geodesics (i.e., such that all geodesics are homogeneous) when is a flag manifold, that is, an adjoint orbit of a compact semisimple Lie group . We use an important invariant of a flag manifold , its -root system, to give a simple necessary condition that admits a non-standard -invariant metric with homogeneous geodesics. Hence, the problem reduces substantially to the study of a short list of prospective flag manifolds. A common feature of these spaces is that their isotropy representation has two irreducible components. We prove that among all flag manifolds of a simple Lie group , only the manifold of complex structures in , and the complex projective space admit a non-naturally reductive invariant metric with homogeneous geodesics. In all other cases the only -invariant metric with homogeneous geodesics is the metric which is homothetic to the standard metric (i.e., the metric associated to the negative of the Killing form of the Lie algebra of ). According to F. Podestà and G.Thorbergsson (2003), these manifolds are the only non-Hermitian symmetric flag manifolds with coisotropic action of the stabilizer.

     

Riemannian manifolds with two circulant structures

We consider a three-dimensional Riemannian manifold equipped with two circulant structures—a metric g and a structure q, which is an isometry with respect to g and the third power of q is minus identity. We discuss some curvature properties of this manifold, we give an example of such a manifold and find a condition for q to be parallel with respect to the Riemannian connection of g.     

Riemannian manifolds with structure groupG 2

Summary Riemannian manifolds with structure group G ₂ are 7-dimensional and have a distinguished 3-form. In this paper such manifolds are treated as analogues of almost Hermitian manifolds. Thus S ⁷ has structure group G ₂ just as S ⁶ is an almost Hermitian manifold. We study the covariant derivative of the fundamental 3-form as was done in [GH]for almost Hermitian manifolds.     

Knots in Riemannian manifolds

In this paper we study submanifolds with nonpositive extrinsic curvature in a positively curved manifold. Among other things we prove that, if ${K\subset (S^n, g)}$ is a totally geodesic submanifold of codimension 2 in a Riemannian sphere with positive sectional curvature where n ≥ 5, then K is homeomorphic to S ^n–2 and the fundamental group of the knot complement ${\pi _1(S^n-K)\cong \mathbb{Z}}$ .