Geometry of real grassmann manifolds. Part III

A canonical decomposition for an element of the tangent bundle of the Grassmann manifold G _p,n ⁺ in its Plücker model is constructed. This decomposition is used for introducing the notion of stationary angles between oriented planes. The relation with the stationary angles in the nonoriented case is established. An explicit formula for the exponential mapping is given, which allows calculating the diameter and the injectivity radius of the manifold G _p,n ⁺ . The problem of uniqueness of the canonical decomposition is reduced to a similar problem on the decomposition of bivectors that realizes their mass (the latter has earlier been solved by the author). The techniques developed are used for describing the structure of the closure of an arbitrary geodesic in the manifolds G _p,n ⁺ and G_p,n. (For the manifolds G_p,n, this result was announced by Wong without proof.) Bibliography: 11 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 246, 1997, pp. 108–129. Translated by N. Yu. Netsvetaev.     

Geometry of distributions of flags on Grassmann manifolds of a projective space. II

We consider intrinsic normalizations of distributions of flags of type (m, m+1) on the Grassmann manifold ofm-planes of a (2m+2)-dimensional projective space. Translated from Lietuvos Matematikos Rinkinys, Vol. 40, No. 3, pp. 335–349, July–September, 2000. Translated by V. Mackevičius     

Geometry of distributions of flags on Grassmann manifolds of a projective space. I

We consider intrinsic normalizations of distributions of flags of type (m, m+1) on the Grassmann manifold ofm-planes of a (2m+2)-dimensional projective space. Translated from Lietuvos Matematikos Rinkinys, Vol. 40, No. 2, pp. 214–227, April–June, 2000. Translated by V. Mackevičius     

Geometry of generalized Einstein manifolds

A formula linking the horizontal Laplacian $\overline{\mathrm{\Delta }}\phi$ of a function φ on the fibre bundle W of unitary tangent vectors to a Finslerian compact manifold without boundary $(M\text{,}g)$, to the square of a symmetric 2-tensor and Finslerian curvature. From it an estimate, under a certain condition, is obtained for the function $\lambda \text{:}\overline{\mathrm{\Delta }}\phi =\lambda \phi$. If $\lambda =nk$ where k is a positive constant and M simply connected, then M is homeomorphic to an n-sphere. Let $F^{○}(g_t)$ be a deformation of $(M\text{,}g)$ preserving the volume of W. One proves that the critical points $g_0\in F^{○}(g_t)$ of the integral $I(g_t)$ of a certain Finslerian scalar curvature on W define a generalized Einstein manifold. One calculates the second variationals at the critical points first in the general case, then, for an infinitesimal conformal deformation and one shows that in certain cases one has $I^{″}(g_0)?0$. We also study the case when the scalar curvature is non-positive constant. To cite this article: H. Akbar-Zadeh, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Geometry of manifolds with densities

We study the geometry of complete Riemannian manifolds endowed with a weighted measure, where the weight function is of quadratic growth. Assuming the associated Bakry–Émery curvature is bounded from below, we derive a new Laplacian comparison theorem and establish various sharp volume upper and lower bounds. We also obtain some splitting type results by analyzing the Busemann functions. In particular, we show that a complete manifold with nonnegative Bakry–Émery curvature must split off a line if it is not connected at infinity and its weighted volume entropy is of maximal value among linear growth weight functions.     

Geometry of real polynomial mappings

In this paper we study the set of points at which a real polynomial mapping is not proper. Received: 7 May 1999; in final form: 18 September 2000 / Published online: 25 June 2001     

Equivariant cohomology of real flag manifolds

Let P=G/K be a semisimple non-compact Riemannian symmetric space, where G=I₀(P) and K=Gp is the stabilizer of p∈P. Let X be an orbit of the (isotropy) representation of K on Tp(P) (X is called a real flag manifold). Let K₀⊂K be the stabilizer of a maximal flat, totally geodesic submanifold of P which contains p. We show that if all the simple root multiplicities of G/K are at least 2 then K₀ is connected and the action of K₀ on X is equivariantly formal. In the case when the multiplicities are equal and at least 2, we will give a purely geometric proof of a formula of Hsiang, Palais and Terng concerning H^∗(X). In particular, this gives a conceptually new proof of Borel's formula for the cohomology ring of an adjoint orbit of a compact Lie group.     

Totally real bisectional curvature,Bochner-Kaehler and Einstein-Kaehler manifolds

By studying various curvature properties of Kaehler manifolds, we establish many new simple geometric characterizations of Bochner-Kaehler and Einstein-Kaehler spaces and of complex space forms.     

Geometry of holomorphically subprojective Kählerian manifolds

Summary In the previous papers[4, 5, 6] S. Yamaguchi and T. Adati have established the differential geometric properties of holomorphically subprojective K?hlerian manifolds. By the way, in[7] S. Yamaguchi has introduced the notion of K?hlerian torse-forming vector fields. The principal purpose of this paper is to investigate the necessary and sufficient conditions in order that a K?hlerian manifold is the holomorphically subprojective K?hlerian manifold of the first kind, corresponding to T. Adati's conditions of a subprojective Riemannian manifold with respect to torse-forming vector fields. Entrata in Redazione il 7 gennaio 1978.