Nonnegatively curved fixed point homogeneous 5-manifolds

Let G be a compact Lie group acting effectively by isometries on a compact Riemannian manifold M with nonempty fixed point set Fix(M, G). We say that the action is fixed point homogeneous if G acts transitively on a normal sphere to some component of Fix(M, G), equivalently, if Fix(M, G) has codimension one in the orbit space of the action. We classify up to diffeomorphism closed, simply connected 5-manifolds with nonnegative sectional curvature and an effective fixed point homogeneous isometric action of a compact Lie group.     

Lattices and infinite-dimensional forms. ‘The Lattice Method’

Hermitean vector spaces E of infinite dimensions are considered. Let G be a subgroup of the orthogonal group of E acting on a set M. The Lattice Method is a technique for classifying the orbits in M under G. We discuss the method in abstract terms and we illustrate it by means of three classification results showing that it is decisive to do a considerable amount of explicit calculations with vector subspace lattices.     

Wave Invariants of the Spectrum of the G-Invariant Laplacian and the Basic Spectrum of a Riemannian Foliation

Given a compact boundaryless Riemannian manifold upon which a compact Lie group G acts by isometries, recall that the G-invariant Laplacian is the restriction of the ordinary Laplacian on functions to the space of functions which are constant along the orbits of the action. By considering the wave trace of the invariant Laplacian and the connection between G manifolds and Riemannian foliations, invariants of the spectrum of the G-invariant Laplacian can be computed. These invariants include the lengths of certain geodesic arcs which are orthogonal to the principal orbits and contained in the open dense set of principal orbits are associated to the singularities of the wave trace of the G-invariant spectrum. If the action admits finite orbits, then the invariants also include the lengths of certain geodesics arcs connecting the finite orbit to itself. Under additional hypotheses, we obtain partial wave traces. As an application, a partial trace formula for Riemannian foliations with bundle-like metrics is also presented, as well as several special cases where better results are available.     

Simplicity of stable principal sheaves

Let M be a compact connected Kähler manifold, and let Gbe a connected complex reductive linear algebraic group. Weprove that a principal G-sheaf on M admits an admissible Einstein–Hermitianconnection if and only if the principal G-sheaf is polystable.Using this it is shown that the holomorphic sections of theadjoint vector bundle of a stable principal G-sheaf on M aregiven by the center of the Lie algebra of G. The Bogomolov inequalityis shown to be valid for polystable principal G-sheaves.     

On uniform local dispersion on a family of G-orbits

Consider a topological space T which is the union of a family of G-orbits, where G is a locally euclidean group G acting on T. On every G-orbit consider a probability which is absolutely continuous with respect to the image measure of the normalized restriction of the Haar measure on some compact neighborhood of the identity in G. Assume that the densities of the probabilities on the orbits have a common upper bound. Let μ be a probability on T which is the integral over the measures on the orbits with respect to some probability μ′ on T. It is shown that this specific kind of integral representation of μ does not depend on the size of the compact neighborhood of the identity in G.     

The Levi problem on strongly pseudoconvex G-bundles

Let G be a unimodular Lie group, X a compact manifold with boundary, and M the total space of a principal bundle G → M → X so that M is also a strongly pseudoconvex complex manifold. In this study, we show that if G acts by holomorphic transformations satisfying a local property, then the space of square-integrable holomorphic functions on M is infinite-dimensional.     

Homogeneous Riemannian manifolds of positive Ricci curvature

We prove that a homogeneous effective spaceM=G/H, whereG is a connected Lie group andH⊂G is a compact subgroup, admits aG-invariant Riemannian metric of positive Ricci curvature if and only if the spaceM is compact and its fundamental group π₁(M) is finite (in this case any normal metric onG/H is suitable). This is equivalent to the following conditions: the groupG is compact and the largest semisimple subgroupLG⊂G is transitive onG/H. Furthermore, ifG is nonsemisimple, then there exists aG-invariant fibration ofM over an effective homogeneous space of a compact semisimple Lie group with the torus as the fiber. Translated fromMatematicheskie Zametki, Vol. 58, No. 3, pp. 334–340, September, 1995.     

Stable minimal hypersurfaces in locally symmetric spaces

LetM be a complete non‐compact stable minimal hypersurface in a locally symmetric space N of nonnegative Ricci curvature. We prove that if the integral of square norm of the second fundamental form is finite, i.e., ∫_M |A |² dv < ∞, then M must be totally geodesic. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Exceptional Compact Symmetric Spaces <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript> and <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript>/<Emphasis Type="Italic">SO</Emphasis>(4) as Tubes

In the present paper we discuss in detail the cohomogeneity one isometric actions of the Lie groups SU(3) × SU(3) and SU(3) on the exceptional compact symmetric spaces G₂ and G₂/SO(4), respectively. We show that the principal orbits coincide with the tubular hypersurfaces around the totally geodesic singular orbits, and the symmetric spaces G₂ and G₂/SO(4) can be thought of as compact tubes around SU(3) and P², respectively. Moreover, we determine the radii of these tubes and describe the shape operators of the principal orbits. Finally, we apply these results to compute the volumes of the two symmetric spaces.The author was partially supported by the Hungarian National Science and Research Foundation OTKA T032478.     

The Exceptional Compact Symmetric Spaces G 2 and G 2/ SO(4) as Tubes

In the present paper we discuss in detail the cohomogeneity one isometric actions of the Lie groups SU(3) × SU(3) and SU(3) on the exceptional compact symmetric spaces G₂ and G₂/SO(4), respectively. We show that the principal orbits coincide with the tubular hypersurfaces around the totally geodesic singular orbits, and the symmetric spaces G₂ and G₂/SO(4) can be thought of as compact tubes around SU(3) and P², respectively. Moreover, we determine the radii of these tubes and describe the shape operators of the principal orbits. Finally, we apply these results to compute the volumes of the two symmetric spaces.     

Holomorphic maps of Kähler manifolds with constrained energy density

We consider holomorphic and antiholomorphic maps of Kähler manifoldsM andN withM compact. In view of bounds on the Ricci curvature ofM and the holomorphic bisectional curvature ofN, the energy density of the map is constrained to satisfy certain inequalities. One inequality implies that the map is constant. Another specifies the image ofM as a totally geodesic real surface of constant Gaussian curvature inN.     

Finsler manifolds without conjugate points and with integral Ricci curvature

We prove that the integral of the Ricci curvature on the unit tangent bundle SM of a complete Finsler manifold M without conjugate points is nonpositive and vanishes only if M is flat, provided that the Ricci curvature on SM has an integrable positive or negative part.     

Finite group actions on compact negatively curved manifolds

We show that some diameters of orbits of a non-trivial finite group G acting effectively on a compact connected negatively curved manifold M have a universal bound depending only on the dimension n of the underlying manifold. It can also be viewed as a rigidity result.     

Kähler structure on moduli spaces of principal bundles

Let M be a moduli space of stable principal G-bundles over a compact Kähler manifold (X,ωX), where G is a reductive linear algebraic group defined over C. Using the existence and uniqueness of a Hermite-Einstein connection on any stable G-bundle P over X, we have a Hermitian form on the harmonic representatives of H¹(X,ad(P)), where ad(P) is the adjoint vector bundle. Using this Hermitian form a Hermitian structure on M is constructed; we call this the Petersson-Weil form. The Petersson-Weil form is a Kähler form, a fact which is a consequence of a fiber integral formula that we prove here. The curvature of the Petersson-Weil Kähler form is computed. Some further properties of this Kähler form are investigated.     

Estimating L ∞ Norms by L 2k Norms for Functions on Orbits

Abstract. Let G be a compact group acting in a real vector space V . We obtain a number of inequalities relating the L ^∞ norm of a matrix element of the representation of G with its L ^2k norm for a positive integer k . As an application, we obtain approximation algorithms to find the maximum absolute value of a given multivariate polynomial over the unit sphere (in which case G is the orthogonal group) and for the assignment problem of degree d , a hard problem of combinatorial optimization generalizing the quadratic assignment problem (in which case G is the symmetric group).     

Courants invariants par une action propre

The purpose of this paper is to characterise the invariant sections-distributions by a proper action. More precisely, we show that if G is a connected Lie group acting on a differentiable vector bundle E→V such that the induced action on V is proper, then the topological vector space of the G-invariant linear functionals (on the space of C ^∞ sections with compact support) equipped with the induced weak-topology (resp. the strong-topology), is isomorphic to the weak (resp. strong) topological dual of the space (of all G-invariant sections σ with compact quotient supp(σ)/G) equipped with a suitable topology; this coincides with the usual C ^∞-topology if the orbit space is compact, and with the Schwartz-topology if the group G is compact. Received: 8 June 1998 / Revised version: 22 September 1998     

Nonnegatively curved fixed point homogeneous manifolds in low dimensions

Let G be a compact Lie group acting isometrically on a compact Riemannian manifold M with nonempty fixed point set M ^G . We say that M is fixed-point homogeneous if G acts transitively on a normal sphere to some component of M ^G . Fixed-point homogeneous manifolds with positive sectional curvature have been completely classified. We classify nonnegatively curved fixed-point homogeneous Riemannian manifolds in dimensions 3 and 4 and determine which nonnegatively curved simply-connected 4-manifolds admit a smooth fixed-point homogeneous circle action with a given orbit space structure.     

The Chern character of a transversally elliptic symbol and the equivariant index

Let G be a compact Lie group acting on a compact manifold M. In this article, we associate to a G-transversally elliptic symbol on M a G-invariant generalized function on G, constructed in terms of equivariant closed differential forms on the cotangent bundle T ^* M. Oblatum 24-VII-1995     

Equivariant Lefschetz number of differential operators

Let G be a compact Lie group acting on a compact complex manifold M by holomorphic transformations. We prove a trace density formula for the G-Lefschetz number of a holomorphic differential operator on M. We generalize the recent results of Engeli and the first author to orbifolds.     

Null 2-type submanifolds of the Euclidean space E 5 with non-parallel mean curvature vector

Let M be a 3-dimensional submanifold of the Euclidean space E⁵ such that M is not of 1-type. We show that if M is flat and of null 2-type with constant mean curvature and non-parallel mean curvature vector then the normal bundle is flat. We also prove that M is an open portion of a 3-dimensional helical cylinder if and only if M is flat and of null 2-type with constant mean curvature and non-parallel mean curvature vector.