Duality and Riemannian cubics

Riemannian cubics are curves used for interpolation in Riemannian manifolds. Applications in trajectory planning for rigid bodiy motion emphasise the group SO(3) of rotations of Euclidean 3-space. It is known that a Riemannian cubic in a Lie group G with bi-invariant Riemannian metric defines a Lie quadratic V in the Lie algebra, and satisfies a linking equation. Results of the present paper include explicit solutions of the linking equation by quadrature in terms of the Lie quadratic, when G is SO(3) or SO(1,2). In some cases we are able to give examples where the Lie quadratic is also given in closed form. A basic tool for constructing solutions is a new duality theorem. Duality is also used to study asymptotics of differential equations of the form , where β₀,β₁ are skew-symmetric 3×3 matrices, and x :ℝ→ SO(3). This is done by showing that the dual of β₀+tβ₁ is a null Lie quadratic. Then results on asymptotics of x follow from known properties of null Lie quadratics. To Charles Micchelli, with warm greetings and deep respect, on his 60th birthday Mathematics subject classifications (2000) 53A17, 53B20, 65D18, 68U05, 70E60.     

A Canonical Compatible Metric for Geometric Structures on Nilmanifolds

Let (N, γ) be a nilpotent Lie group endowed with an invariant geometric structure (cf. symplectic, complex, hypercomplex or any of their ‘almost’ versions). We define a left invariant Riemannian metric on N compatible with γ to be minimal, if it minimizes the norm of the invariant part of the Ricci tensor among all compatible metrics with the same scalar curvature. We prove that minimal metrics (if any) are unique up to isometry and scaling, they develop soliton solutions for the ‘invariant Ricci’ flow and are characterized as the critical points of a natural variational problem. The uniqueness allows us to distinguish two geometric structures with Riemannian data, giving rise to a great deal of invariants.Our approach proposes to vary Lie brackets rather than inner products; our tool is the moment map for the action of a reductive Lie group on the algebraic variety of all Lie algebras, which we show to coincide in this setting with the Ricci operator. This gives us the possibility to use strong results from geometric invariant theory.Communicated by: Nigel Hitchin (Oxford) Mathematics Subject Classifications (2000): Primary: 53D05, 53D55; Secondary: 22E25, 53D20, 14L24, 53C30.     

The word and Riemannian metrics on lattices of semisimple groups

Let G be a semisimple Lie group of rank ⩾2 and Γ an irreducible lattice. Γ has two natural metrics: a metric inherited from a Riemannian metric on the ambient Lie group and a word metric defined with respect to some finite set of generators. Confirming a conjecture of D. Kazhdan (cf. Gromov [Gr2]) we show that these metrics are Lipschitz equivalent. It is shown that a cyclic subgroup of Γ is virtually unipotent if and only if it has exponential growth with respect to the generators of Γ.     

Closed geodesics in compact nilmanifolds

We study the density of closed geodesics property on 2-step nilmanifolds Γ\N, where N is a simply connected 2-step nilpotent Lie group with a left invariant Riemannian metric and Lie algebra ?, and Γ is a lattice in N. We show the density of closedgeodesics property holds for quotients of singular, simply connected, 2-step nilpotent Lie groups N which are constructed using irreducible representations of the compact Lie group SU(2). Received: 8 November 2000 / Revised version: 9 April 2001     

Global rigidity of holomorphic Riemannian metrics on compact complex 3-manifolds

We study compact complex 3-manifolds M admitting a (locally homogeneous) holomorphic Riemannian metric g. We prove the following: (i) If the Killing Lie algebra of g has a non trivial semi-simple part, then it preserves some holomorphic Riemannian metric on M with constant sectional curvature; (ii) If the Killing Lie algebra of g is solvable, then, up to a finite unramified cover, M is a quotient Γ\G, where Γ is a lattice in G and G is either the complex Heisenberg group, or the complex SOL group. S. Dumitrescu was partially supported by the ANR Grant BLAN 06-3-137237.     

Nonholonomic Riemann and Weyl tensors for flag manifolds

On any manifold, any nondegenerate symmetric 2-form (metric) and any nondegenerate skew-symmetric differential form ω can be reduced to a canonical form at any point but not in any neighborhood: the corresponding obstructions are the Riemannian tensor and dω. The obstructions to flatness (to reducibility to a canonical form) are well known for any G-structure, not only for Riemannian or almost symplectic structures. For a manifold with a nonholonomic structure (nonintegrable distribution), the general notions of flatness and obstructions to it, although of huge interest (e.g., in supergravity) were not known until recently, although particular cases have been known for more than a century (e.g., any contact structure is nonholonomically “flat”: it can always be reduced locally to a canonical form). We give a general definition of the nonholonomic analogues of the Riemann tensor and its conformally invariant analogue, the Weyl tensor, in terms of Lie algebra cohomology and quote Premet’s theorems describing these cohomologies. Using Premet’s theorems and the SuperLie package, we calculate the tensors for flag manifolds associated with each maximal parabolic subalgebra of each simple Lie algebra (and in several more cases) and also compute the obstructions to flatness of the G(2)-structure and its nonholonomic superanalogue. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 153, No. 2, pp. 186–219, November, 2007.     

The Chebyshev norm on the lie algebra of the motion group of a compact homogeneous Finsler manifold

In this paper, we prove that the natural metric on the connected component of the unit in the (Lie) motion group of a compact Finsler manifold supplied with its inner metric generates a bi-invariant inner Finsler metric. The latter is defined by the invariant Chebyshev norm on the Lie algebra of generators of 1-parameter motion subgroups on the manifold. This norm is equal to the maximal value of the generator’s length. A δ-homogeneous manifold is characterized by the condition that the canonical projection of the component onto the manifold is a submetry with respect to their inner metrics. The Chebyshev norms for the Euclidean spheres, the Berger spheres, and homogeneous Riemannian metrics on the 3-dimensional complex projective space are found. This gives interesting examples of invariant norms on Lie algebras and a new method for the separating of delta-homogeneous but not normal metrics. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 60, Algebra, 2008.     

On symplectic half-flat manifolds

We construct examples of symplectic half-flat manifolds on compact quotients of solvable Lie groups. We prove that the Calabi-Yau structures are not rigid in the class of symplectic half-flat structures. Moreover, we provide an example of a compact 6-dimensional symplectic half-flat manifold whose real part of the complex volume form is d-exact. Finally we discuss the 4-dimensional case. This work was supported by the Projects M.I.U.R. “Geometric Properties of Real and Complex Manifolds”, “Riemannian Metrics and Differentiable Manifolds” and by G.N.S.A.G.A. of I.N.d.A.M.     

Av-operator on complete foliated Riemannian manifolds

We give a generalization of the result obtained by C. Currás-Bosch. We consider the Av-operator associated to a transverse Killing fieldν on a complete foliated Riemannian manifold (M, ℱ, g). Under a certain assumption, we prove that, for eachx ∈M, (Av) _x belongs to the Lie algebra of the linear holonomy group ψv(x). A special case of our result, the version of the foliation by points, implies the results given by B. Kostant (compact case) and C. Currás-Bosch (non-compact case).     

The laplace operator on normal homogeneous Riemannian manifolds

The article presents an information about the Laplace operator defined on the real-valued mappings of compact Riemannian manifolds, and its spectrum; some properties of the latter are studied. The relationship between the spectra of two Riemannian manifolds connected by a Riemannian submersion with totally geodesic fibers is established. We specify a method of calculating the spectrum of the Laplacian for simply connected simple compact Lie groups with biinvariant Riemannian metrics, by representations of their Lie algebras. As an illustration, the spectrum of the Laplacian on the group SU(2) is found.     

Vector fields dynamics as geodesic motion on Lie groups

We study to what extent vector fields on Lie groups may be considered as geodesic fields. For a given left invariant vector field on a Lie group, we prove there exists a Riemannian metric whose geodesics are its trajectories. When we consider left invariant metrics, differences between the Riemannian and the Lorentzian cases appear, coded by properties of the Lie algebra. To cite this article: G.T. Pripoae, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Locally symmetric left invariant Riemannian metrics on 3‐dimensional Lie groups

In this paper, we use the powerful tool Milnor bases to determine all the locally symmetric left invariant Riemannian metrics up to automorphism, on 3‐dimensional connected and simply connected Lie groups, by solving system of polynomial equations of constants structure of each Lie algebra . Moreover, we show that E ₀(2) is the only 3‐dimensional Lie group with locally symmetric left invariant Riemannian metrics which are not symmetric.     

Complete involutive algebras of functions on cotangent bundles of homogeneous spaces

Homogeneous spaces of all compact Lie groups admit Riemannian metrics with completely integrable geodesic flows by means of C –smooth integrals [9, 10]. The purpose of this paper is to give some constructions of complete involutive algebras of analytic functions, polynomial in velocities, on the (co)tangent bundles of homogeneous spaces of compact Lie groups. This allows us to obtain new integrable Riemannian and sub-Riemannian geodesic flows on various homogeneous spaces, such as Stiefel manifolds, flag manifolds and orbits of the adjoint actions of compact Lie groups. Mathematics Subject Classification (2000): 70H06, 37J35, 53D17, 53D25     

Riemannian geometry of Lie algebroids

We introduce Riemannian Lie algebroids as a generalization of Riemannian manifolds and we show that most of the classical tools and results known in Riemannian geometry can be stated in this setting. We give also some new results on the integrability of Riemannian Lie algebroids.     

Conformal Riemannian Maps between Riemannian Manifolds,Their Harmonicity and Decomposition Theorems

Riemannian maps were introduced by Fischer (Contemp. Math. 132:331–366, 1992) as a generalization isometric immersions and Riemannian submersions. He showed that such maps could be used to solve the generalized eikonal equation and to build a quantum model. On the other hand, horizontally conformal maps were defined by Fuglede (Ann. Inst. Fourier (Grenoble) 28:107–144, 1978) and Ishihara (J. Math. Kyoto Univ. 19:215–229, 1979) and these maps are useful for characterization of harmonic morphisms. Horizontally conformal maps (conformal maps) have their applications in medical imaging (brain imaging)and computer graphics. In this paper, as a generalization of Riemannian maps and horizontally conformal submersions, we introduce conformal Riemannian maps, present examples and characterizations. We show that an application of conformal Riemannian maps can be made in weakening the horizontal conformal version of Hermann’s theorem obtained by Okrut (Math. Notes 66(1):94–104, 1999). We also give a geometric characterization of harmonic conformal Riemannian maps and obtain decomposition theorems by using the existence of conformal Riemannian maps.     

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.     

Obstruction to Positive Curvature on Homogeneous Bundles

Examples of almost-positively and quasi-positively curved spaces of the form M = H\((G, h) × F) were discovered recently [J. Differential Geom. 65:273–287, 2003; Invent. Math. 148:117–141, 2002]. Here h is a left-invariant metric on a compact Lie group G, F is a compact Riemannian manifold on which the subgroup acts isometrically on the left, and M is the orbit space of the diagonal left action of H on (G, h) × F with the induced Riemannian submersion metric. We prove that no new examples of strictly positive sectional curvature exist in this class of metrics. This result generalizes the case F = {point} proven by Geroch [Proc. Amer. Math. Soc. 66(2):321–326, 1977].Supported in part by NSF grant DMS–0303326.     

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.     

Weighted norm inequalities,off-diagonal estimates and elliptic operators. Part IV: Riesz transforms on manifolds and weights

This is the fourth article of our series. Here, we study weighted norm inequalities for the Riesz transform of the Laplace–Beltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Gaussian upper bounds.