A rigidity theorem for homogeneous rational manifolds of rank 1

Sunto Si prova un teorema di rigidità per deformazioni algebriche di varietà omogenee razionali eon secondo numero di Betti uguale a 1 (ad esempio: spazi proiettivi, grassmanniane, quadriche di dimensione 3, grassmanniane dei sottospazi lineari diP ⁿ⁺¹ contenuti in una quadrica Q_n Pⁿ⁺¹).

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This paper was written while the authors were members of the G.N.S.A.G.A. of C.N.R., with a financial support by the M.P.I.     

Bifurcation and local rigidity of homogeneous solutions to the Yamabe problem on spheres

We study existence and non-existence of constant scalar curvature metrics conformal and arbitrarily close to homogeneous metrics on spheres, using variational techniques. This describes all critical points of the Hilbert–Einstein functional on such conformal classes, near homogeneous metrics. Both bifurcation and local rigidity type phenomena are obtained for 1-parameter families of U(n + 1), Sp(n + 1) and Spin(9)-homogeneous metrics.     

The space of spectral measures is a homogeneous reductive space

We prove that the space of spectral measures on a W^*-algebra is a smooth Banach manifold in a natural way and that the action of the group of invertible elements of the algebra by inner automorphisms makes it into a reductive homogeneous space. This gives a geometric structure for the set of normal operators with the same spectrum.     

On the projective geometry of rational homogeneous varieties

We determine the varieties of linear spaces on rational homogeneous varieties, provide explicit geometric models for these spaces, and establish basic facts about the local differential geometry of rational homogeneous varieties. Received: February 23, 2001     

Application of a Bernstein type inequality to rational interpolation in the Dirichlet space

We prove a Bernstein type inequality involving the Bergman and Hardy norms for rational functions in the unit disk mathbb D {mathbb D} that have at most n poles all of which are outside the disk frac1r mathbb D frac{1}{r} {mathbb D} , 0 < r < 1. The asymptotic sharpness of this inequality is shown as n → ∞ and r → 1—. We apply our Bernstein type inequality to an efficient Nevanlinna–Pick interpolation problem in the standard Dirichlet space constrained by the H2-nom. Bibliography: 14 titles.     

Examples of homogeneous vectors on the fiber space of the associated and the tangent bundles

The paper considers the associated bundle ξ = (G × KG/K, ρ _ξ , G/K, G/K) and the tangent bundle τ _G/K = (T _G/K , π _G/K , G/K, R ^m ), and gives special examples of odd dimensional solvable Lie groups equipped with left invariant Riemannian metric. Some conditions about existence of homogeneous geodesic vectors on the fiber space of ξ and τ _G/K are proved.     

On the rigidity of hypersurfaces into space forms

Our purpose is to study the rigidity of complete hypersurfaces immersed into a Riemannian space form. In this setting, first we use a classical characterization of the Euclidean sphere \(\mathbb S ^{n+1}\) due to Obata (J Math Soc Jpn 14:333–340, 1962) in order to prove that a closed orientable hypersurface \(\Sigma ^n\) immersed with null second-order mean curvature in \(\mathbb S ^{n+1}\) must be isometric to a totally geodesic sphere \(\mathbb S ^{n}\) , provided that its Gauss mapping is contained in a closed hemisphere. Furthermore, as suitable applications of a maximum principle at the infinity for complete noncompact Riemannian manifolds due to Yau (Indiana Univ Math J 25:659–670, 1976), we establish new characterizations of totally geodesic hypersurfaces in the Euclidean and hyperbolic spaces. We also obtain a lower estimate of the index of minimum relative nullity concerning complete noncompact hypersurfaces immersed in such ambient spaces.     

Differential invariants of the transformation group of a homogeneous space

We demonstrate that the invariant operators on a homogeneous space generate differential invariants and invariant differentiation operators. The coordinate-free method of this article makes it possible to simply the computations essentially, namely to reduce them to operations of linear algebra. Some examples are exhibited.     

On tangential varieties of rational homogeneous varieties

The sphericality of tangential varieties of rational homogeneousvarieties is determined. The homogeneous coordinate rings andrings of covariants of the tangential varieties of homogenouslyembedded compact Hermitian symmetric spaces (CHSS) are determined.Bounds on the degrees of generators of the ideals of tangentialvarieties of CHSS are given, and more explicit information isobtained about the ideals in certain cases.     

Contractibility of the space of rational maps

We define an algebro-geometric model for the space of rational maps from a smooth curve X to an algebraic group G, and show that this space is homologically contractible. As a consequence, we deduce that the moduli space $\operatorname {Bun}_{G}$ of G-bundles on X is uniformized by the appropriate rational version of the affine Grassmannian, where the uniformizing map has contractible fibers.     

A function space model approach to the rational evaluation subgroups

Let f : U → X be a map from a connected nilpotent space U to a connected rational space X. The evaluation subgroup G _*(U, X; f), which is a generalization of the Gottlieb group of X, is investigated. The key device for the study is an explicit Sullivan model for the connected component containing f of the function space of maps from U to X, which is derived from the general theory of such a model due to Brown and Szczarba (Trans Am Math Soc 349, 4931–4951, 1997). In particular, we show that non Gottlieb elements are detected by analyzing a Sullivan model for the map f and by looking at non-triviality of higher order Whitehead products in the homotopy group of X. The Gottlieb triviality of a fibration in the sense of Lupton and Smith (The evaluation subgroup of a fibre inclusion, 2006) is also discussed from the function space model point of view. Moreover, we proceed to consideration of the evaluation subgroup of the fundamental group of a nilpotent space. In consequence, the first Gottlieb group of the total space of each S ¹-bundle over the n-dimensional torus is determined explicitly in the non-rational case.