Invariant Riemannian Metrics Research Articles

Gauss map harmonicity and mean curvature of a hypersurface in a homogeneous manifold

We define a Gauss map of an orientable hypersurface in a homogeneous manifold with an invariant Riemannian metric. Our main objective is to extend to this setting some results on the Gauss map of a constant mean curvature hypersurface of an Euclidean space, namely the Ruh?Vilm theorem relating the harmonicity of the Gauss map and the constancy of the mean curvature, and the Hoffman?Osserman?Schoen theorem characterizing the plane and the circular cylinder as the only complete constant mean curvature surfaces whose Gauss image is contained in a closed hemisphere of the sphere.

Invariant f‐structures on the flag manifolds SO(n)/SO(2) × SO(n − 3)

We consider manifolds of oriented flags SO(n)/SO(2) × SO(n − 3)(n ≥ 4) as 4‐ and 6‐symmetric spaces and indicate characteristic conditions for invariant Riemannian metrics under which the canonical f‐structures on these homogeneous Φ‐spaces belong to the classes Kill f, NKf, and G1f of generalized Hermitian geometry.

WITTEN LAPLACIAN ON PINNED PATH GROUP AND ITS EXPECTED SEMICLASSICAL BEHAVIOR

A right invariant Riemannian metric is defined on a pinned path group over a compact Lie group G. The energy function of the path is a Morse function and the critical points are geodesics. We calculate the eigenvalues of the Hessian at the critical points when G=SU(n). On the other hand, there exists a pinned Brownian motion measure νλ with a variance parameter 1/λ on the pinned path group and we can define a Hodge-Kodaira-Witten type operator □λ on L2(νλ)-space of p-forms on the pinned path group. By using the explicit expression of eigenvalues of the Hessian of the energy function, we discuss the asymptotic behavior of the botton of the spectrum of □λ as λ→∞ by a formal semiclassical analysis.

MINIMAL SURFACES IN 3-DIMENSIONAL SOLVABLE LIE GROUPS

The author studies minimal surfaces in 3-dimensional solvable Lie groups with left invariant Riemannian metrics. A Weierstras type integral representation formula for minimal surfaces is obtained.

Degenerations of Lie algebras and geometry of Lie groups

Each point of the variety of real Lie algebras is naturally identified with a left invariant Riemannian metric on a Lie group. We study the interplay between invariant-theoretic and Riemannian aspects of this variety. In particular, using the special critical point behavior of certain natural functional on the variety, we determine all the Lie groups which can be endowed with only one left invariant metric up to isometry and scaling, proving first that they correspond to Lie algebras whose only degeneration is to the abelian one. We also find all the Lie algebras which degenerate to the Lie algebra of the hyperbolic space, and all the possible degenerations for 3-dimensional real Lie algebras, by using well known descriptions of left invariant metrics satisfying some pinching curvature conditions. Finally, as another interaction, the closed S L( n)-orbits on the variety are classified, and explicit curves of Einstein solvmanifolds are provided by using curves of closed orbits of the representation Λ 2S L( m)⊗S L( n).

Finsler Metrics of Constant Positive Curvature on the Lie Group S 3

Guided by the Hopf fibration, a family (indexed by a positive constant K) of right invariant Riemannian metrics on the Lie group S3 is singled out. Using the Yasuda–Shimada paper as an inspiration, a privileged right invariant Killing field of constant length is determined for each K > 1. Each such Riemannian metric couples with the corresponding Killing field to produce a y-global and explicit Randers metric on S3. Employing the machinery of spray curvature and Berwald's formula, it is proved directly that the said Randers metric has constant positive flag curvature K, as predicted by Yasuda–Shimada. It is explained why this family of Finslerian space forms is not projectively flat.

Transitive Isometry Groups of Aspheric Riemannian Manifolds

We consider the isometry groups of Riemannian solvmanifolds and also study a wider class of homogeneous aspheric Riemannian spaces. We clarify the topological structure of these spaces (Theorem 1). We demonstrate that each Riemannian space with a maximally symmetric metric admits an almost simply transitive action of a Lie group with triangular radical (Theorem 2). We apply this result to studying the isometry groups of solvmanifolds and, in particular, solvable Lie groups with some invariant Riemannian metric.

The large scale geometry of nilpotent Lie groups

In this paper, we prove results concerning the large scale geometry of connected, simply connected nonabelian nilpotent Lie groups equipped with left invariant Riemannian metrics. Precisely, we prove that there do not exist quasi-isometric embeddings of such a nilpotent Lie group into either a CAT(0) metric space or an Alexandrov metric space with curvature bounded below. The main technical aspect of this work is the proof of a limited metric differentiability of Lipschitz maps between connected graded nilpotent Lie groups equipped with left invariant Carnot-Caratheodory metrics and complete metric spaces.

Optimal pinching constants of odd dimensional homogeneous spaces

In this article we compute the pinching constants of all invariant Riemannian metrics on the Berger space B13=SU(5)/(Sp(2)×ℤ2S1) and of all invariant U(2)-biinvariant Riemannian metrics on the Aloff–Wallach space W7 1,1=SU(3)/S1 1,1. We prove that the optimal pinching constants are precisely in both cases. So far B13 and W7 1,1 were only known to admit Riemannian metrics with pinching constants.¶We also investigate the optimal pinching constants for the invariant metrics on the other Aloff–Wallach spaces W7 k,l =SU(3)/S1 k,l . Our computations cover the cone of invariant T2-biinvariant Riemannian metrics. This cone contains all invariant Riemannian metrics unless k/l=1. It turns out that the optimal pinching constants are given by a strictly increasing function in k/l∈[0,1]. Thus all the optimal pinching constants are ≤.¶In order to determine the extremal values of the sectional curvature of an invariant Riemannian metric on W7 k,l we employ a systematic technique, which can be applied to other spaces as well. The computation of the pinching constants for B13 is reduced to the curvature computation for two proper totally geodesic submanifolds. One of them is diffeomorphic to ℂℙ3/ℤ2 and inherits an Sp(2)-invariant Riemannian metric, and the other is W7 1,1 embedded as recently found by Taimanov. This approach explains in particular the coincidence of the optimal pinching constants for W7 1,1 and the Berger space B13.

Geometric groups. I

We define a geometry on a group to be an abelian semigroup of symmetric open sets with certain properties. Examples include well-known structures such as invariant Riemannian metrics on Lie groups, hyperbolic groups, and valuations on fields. In this paper we are mostly concerned with geometries where the semigroup is isomorphic to the positive reals, which for Lie groups come from invariant Finsler metrics. We explore various aspects of these geometric groups, including a theory of covering groups for arcwise connected groups, algebraic expressions for invariant metrics and inner metrics, construction of geometries with curvature bounded below, and finding geometrically significant curves in path homotopy classes.

On the riemannian metrics in [formula omitted]n which admit all hyperplanes as minimal hypersurfaces

Inspired by a result of Bekkar (1991), Robert Lutz raised the following problem: determine the riemannian metrics in domains of R n which admit all hyperplanes as minimal hypersurfaces. We solve the problem giving a formula which expresses its solutions in terms of the non-degenerate quadratic first integrals of the geodesic motion in the euclidean space (second-order Killing tensor fields). Then, we prove that for n = 3 the non-flat polynomial solutions of the problem are the left invariant riemannian metrics on the Heisenberg group.

Geometrizing Infinite Dimensional Locally Compact Groups

We study groups having invariant metrics of curvature bounded below in the sense of Alexandrov. Such groups are a generalization of Lie groups with invariant Riemannian metrics, but form a much larger class. We prove that every locally compact, arcwise connected, first countable group has such a metric. These groups may not be (even infinite dimensional) manifolds. We show a number of relationships between the algebraic and geometric structures of groups equipped with such metrics. Many results do not require local compactness.

Compact homogeneous manifolds with integrable invariant distributions, and scalar curvature

It is proved that for a compact simply-connected effective homogeneous space of a connected compact Lie group by a closed subgroup the following conditions are equivalent: (1) Every -invariant distribution on is integrable. (2) The space is of normal type in the sense of Bergery. (3) Every -invariant Riemannian metric on has positive scalar curvature. (4) The space is isomorphic to a direct product of compact simplyconnected strongly isotropy-irreducible homogeneous spaces.

Ricci structure and volume growth for left invariant Riemannian metrics on nilpotent and some solvable Lie groups

We discuss left invariant Riemannian metrics on Lie groups, and the Ricci structures they induce. A computational approach is used to manipulate the curvature tensors, and construct perturbations which preserve the Ricci structure but not the (algebraic) Lie structure. We show that this method can lead to significant changes in the growth of volume. We also show that this approach may be used to reduce the complexity of some curvature computations in Lie groups.

A class ofU(n)-invariant Riemannian metrics on manifolds diffeomorphic to odd-dimensional spheres

A class ofU(n)-invariant Riemannian metrics on manifolds diffeomorphic to odd-dimensional spheres

L(2)-cohomology of orbit spaces

Suppose that a compact Lie group acts on a smooth compact manifold and that the manifold is equipped with an invariant Riemannian metric. This metric induces a metric on the open dense stratum of the quotient space. Then the L (2)-cohomology of the open dense stratum with respect to the induced metric is isomorphic to the intersection cohomology with upper middle perversity of the quotient space.

Markov invariant geometry on manifolds of states

This paper is devoted to certain differential-geometric constructions in classical and noncommutative statistics, invariant with respect to the category of Markov maps, which have recently been developed by Soviet, Japanese, and Danish researchers. Among the topics considered are invariant metrics and invariant characteristics of informational proximity, and lower bounds are found for the uniform topologies that they generate on sets of states. A description is given of all invariant Riemannian metrics on manifolds of sectorial states. The equations of the geodesies for the entire family of invariant linear connections Δ=γΔ, γ∈ IR, are integrated on sets of classical probability distributions. A description is given of the protective structure of all the geodesic curves and totally geodesic submanifolds, which turns out to be a local lattice structure; it is shown to coincide, up to a factor γ(γ−1), with the Riemann-Christoffel curvature tensor.

Actions on Invariant Spheres around Isolated Fixed Points of Actions of Cyclic Groups

Fix a prime number p and let Zp be a cyclic group of order p. We consider a pair (M, 0) consisting of a compact simply connected almost complex manifold M without boundary and a smooth Zp-action 0: ZPX M—>M preserving the almost complex structure of M. We suppose that M is given an invariant Riemannian metric. If a(eM) is an isolated fixed point, then the induced action of Zp on the tangent space at a gives a complex Zp-module Va which has no trivial irreducible factor. Let f : EZp—*BZp be a universal principal Zp-bundle and let f (Vtt) : EZpXzp Va —>BZP be the V0-bundle associated with ?. If a and b are isolated fixed points, we compare the cobordism Euler classes 0(f (Va)) and e(£ (Vb)) which belong to the complex cobordism group MU* (BZP) of the classifying space BZP of Zp. Let .PV be the universal formal group law over MU*9 and write

The space of positive definite matrices and Gromov’s invariant

The space $X_d^n{\text {of}}n \times n$ positive definite matrices with ${\text {determinant}} = 1$ is considered as a subset of ${{\mathbf {R}}^{n(n + 1)/2}}$ with isometries given by $X \to AX{A^t}$ where the determinant of $A = 1$ and $X_d^n$ is given its invariant Riemannian metric. This space has a collection of simplices which are preserved by the isometries and formed by projecting geometric simplices in ${{\mathbf {R}}^{n(n + 1)/2}}$ to the hypersurface $X_d^n$. The main result of this paper is that for each $n$ the volume of all top dimensional simplices of this type has a uniform upper bound. This result has applications to Gromov’s Invariant as defined in William P. Thurston’s notes, The geometry and topology of $3$-manifolds. The result implies that the Gromov Invariant of the fundamental class of compact manifolds which are formed as quotients of $X_d^n$ by discrete subgroups of the isometries is nonzero. This gives the first nontrivial examples of manifolds that have a nontrivial Gromov Invariant but do not have strictly negative curvature or nonvanishing characteristic numbers.

The group of isometries of a left invariant Riemannian metric on a Lie group

The group of isometries of a left invariant Riemannian metric on a Lie group