On the holonomy of connections with skew-symmetric torsion

We investigate the holonomy group of a linear metric connection with skew-symmetric torsion. In case of the euclidian space and a constant torsion form this group is always semisimple. It does not preserve any non-degenerated 2-form or any spinor. Suitable integral formulas allow us to prove similar properties in case of a compact Riemannian manifold equipped with a metric connection of skew-symmetric torsion. On the Aloff-Wallach space N(1,1) we construct families of connections admitting parallel spinors. Furthermore, we investigate the geometry of these connections as well as the geometry of the underlying Riemannian metric. Finally, we prove that any 7-dimensional 3-Sasakian manifold admits ²-parameter families of linear metric connections and spinorial connections defined by 4-forms with parallel spinors.Mathematics Subject Classification (2000):53 C 25, 81 T 30We thank Andrzej Trautman for drawing our attention to these papers by Cartan – see [27].     

On fat sub-Riemannian symmetric spaces in codimension three

A sub-Riemannian manifold is a differentiable manifold together with a smooth distribution which is equipped with a Riemannian metric. In this paper we attempt to study sub-Riemannian symmetric spaces (i.e., homogeneous sub-Riemannian manifolds admitting an involutive sub-Riemannian isometry at all points which is a central symmetry when restricted to the distribution) where the associated distribution is a codimension three fat distribution. We obtain a restricted classification theorem in dimension seven and we also construct a class of examples of quaternionic type in varying dimension.     

On the variation of a metric and its application

Some of the variation formulas of a metric were derived in the literatures by using the local coordinates system, In this paper, We give the first and the second variation formulas of the Riemannian curvature tensor, Ricci curvature tensor and scalar curvature of a metric by using the moving frame method. We establish a relation between the variation of the volume of a metric and that of a submanifold. We find that the latter is a consequence of the former. Finally we give an application of these formulas to the variations of heat invariants. We prove that a conformally flat metric g is a critical point of the third heat invariant functional for a compact 4-dimensional manifold M, then （M, g） is either scalar flat or a space form.     

关于切丛和单位切球丛的度量的一个注记

本文在黎曼流形$(M,g)$的切丛$TM$ 上研究与参考文献[10]中平行的一类度量$G$以及相容的近复结构$J$.证明了切丛$TM$关于这些度量和相应的近复结构是局部共形近K\"{a}hler流形,并且把这些结构限制在单位切球丛上得到了切触度量结构的新例子.     

Complex affine distributions

Geometry of affine immersions is the study of hypersurfaces that are invariant under affine transformations. As with the hypersurface theory on the Euclidean space, an affine immersion can induce a torsion-free affine connection and a (pseudo)-Riemannian metric on the hypersurface. Moreover, an affine immersion can induce a statistical manifold, which plays a central role in information geometry. Recently, a statistical manifold with a complex structure is actively studied since it connects information geometry and Kähler geometry. However, a holomorphic complex affine immersion cannot induce such a statistical manifold with a Kähler structure. In this paper, we introduce complex affine distributions, which are non-integrable generalizations of complex affine immersions. We then present the fundamental theorem for a complex affine distribution, and show that a complex affine distribution can induce a statistical manifold with a Kähler structure.     

Lipschitz and path isometric embeddings of metric spaces

We prove that each sub-Riemannian manifold can be embedded in some Euclidean space preserving the length of all the curves in the manifold. The result is an extension of Nash C ¹ Embedding Theorem. For more general metric spaces the same result is false, e.g., for Finsler non-Riemannian manifolds. However, we also show that any metric space of finite Hausdorff dimension can be embedded in some Euclidean space via a Lipschitz map.     

Fisher-Rao geometry of Dirichlet distributions

In this paper, we study the geometry induced by the Fisher-Rao metric on the parameter space of Dirichlet distributions. We show that this space is a Hadamard manifold, i.e. that it is geodesically complete and has everywhere negative sectional curvature. An important consequence for applications is that the Fréchet mean of a set of Dirichlet distributions is uniquely defined in this geometry.     

On the Volume of Unit Tangent Spheres in a Pinsler Manifold

Motivated by some issues which enter into the Gauss-Bonnet-Chern theorem in Finsler geometry, this paper studies the unit tangent sphere (or indicatrix) I_x M at each point x of a Pinsler manifold M. We demonstrate that the volume of I_mM, calculated with respect to a Riemannian metric induced naturally by the Finsler structure, is in general a function of x. This contrasts sharply with the situation in Riemannian geometry. We also express the derivative of such volume function in terms of the second curvature tensor of the Chern connection. In particular, we find that this function is constant on Landsberg spaces (though that constant need not be equal to the value taken by Riemannian manifolds).     

Variational results on flag manifolds: Harmonic maps, geodesics and Einstein metrics

In this paper, we study variational aspects for harmonic maps from M to several types of flag manifolds and the relationship with the rich Hermitian geometry of these manifolds. We consider maps that are harmonic with respect to any invariant metric on each flag manifold. They are called equiharmonic maps. We survey some recent results for the case where M is a Riemann surface or is one dimensional; i.e., we study equigeodesics on several types of flag manifolds. We also discuss some results concerning Einstein metrics on such manifolds.     

Riemannian almost product and para-Hermitian cotangent bundles of general natural lift type

We characterize the almost product and locally product structures of general natural lift type on the cotangent bundle of a Riemannian manifold. We find the conditions under which the cotangent bundle endowed with the constructed almost product (locally product) structure and with a pseudo-Riemannian metric obtained as a general natural lift of the metric from the base manifold, is a Riemannian almost product (locally product) or an (almost) para-Hermitian manifold. Finally, by studying the closedness of the 2-form associated to the obtained (almost) para-Hermitian structure, we characterize the general natural (almost) para-Kählerian structures on the cotangent bundle.     

A new variational characterization of Jacobi fields along geodesies

A classical result of Riemannian geometry states that Jacobi fields along geodesics of a Riemannian manifold (Q, g) can be obtained as geodesies of the so-called «complete lift» of the metric g itself to the tangent bundle TQ. We show that this classical result is in fact a very simple consequence of a completely general theorem of Calculus of Variations.     

Harmonic maps with torsion

In this article we introduce a natural extension of the well-studied equation for harmonic maps between Riemannian manifolds by assuming that the target manifold is equipped with a connection that is metric but has non-vanishing torsion. Such connections have already been classified in the work of Cartan(1924).The maps under consideration do not arise as critical points of an energy functional leading to interesting mathematical challenges. We will perform a first mathematical analysis of these maps which we will call harmonic maps with torsion.     

Geometric Structures on the Complement of a Projective Arrangement

Consider a complex projective space with its Fubini-Study metric. We study certain one parameter deformations of this metric on the complement of an arrangement (= finite union of hyperplanes) whose Levi-Civita connection is of Dunkl type. Interesting examples are obtained from the arrangements defined by finite complex reflection groups. We determine a parameter interval for which the metric is locally of Fubini-Study type, flat, or complex-hyperbolic. We find a finite subset of this interval for which we get a complete orbifold or at least a Zariski open subset thereof, and we analyze these cases in some detail (e.g., we determine their orbifold fundamental group).In this set-up, the principal results of Deligne-Mostow on the Lauricella hypergeometric differential equation and work of Barthel-Hirzebruch-Höfer on arrangements in a projective plane appear as special cases. Along the way we produce in a geometric manner all the pairs of complex reflection groups with isomorphic discriminants, thus providing a uniform approach to work of Orlik-Solomon.     

Metrics of Nonpositive Curvature on Graph-Manifolds and Electromagnetic Fields on Graphs

A 3-dimensional graph-manifold consists of simple blocks that are products of compact surfaces with boundary by the circle. Its global structure may be as complicated as desired and is described by a graph, which can be an arbitrary graph. A metric of nonpositive curvature on such a manifold, if it exists, can be described essentially by a finite number of parameters satisfying a geometrization equation. In the paper, it is shown that this equation is a discrete version of the Maxwell equations of classical electrodynamics, and its solutions, i.e., metrics of nonpositive curvature, are critical configurations of the same sort of action that describes the interaction of an electromagnetic field with a scalar charged field. This analogy is established in the framework of the spectral calculus (noncommutative geometry) of A. Connes. Bibliography: 17 titles.     

Natural almost contact structures and their 3D homogeneous models

We consider a natural condition determining a large class of almost contact metric structures. We study their geometry, emphasizing that this class shares several properties with contact metric manifolds. We then give a complete classification of left‐invariant examples on three‐dimensional Lie groups, and show that any simply connected homogeneous Riemannian three‐manifold admits a natural almost contact structure having g as a compatible metric. Moreover, we investigate left‐invariant CR structures corresponding to natural almost contact metric structures.     

Hamiltonian Systems with Positive Topological Entropy and Conjugate Points

In this article, we consider the geodesic flows induced by the natural Hamiltonian systems $H(x,p)=\frac{1}{2}g^{ij}(x) p_{i}p_{j} + V(x) $ defined on a smooth Riemannian manifold$(M = \mathbb{S}^{1} \times N, g)$, where $\mathbb {S}^{1}$ is the one dimensional torus, N is a compact manifold, g is the Riemannian metric on M and V is a potential function satisfying $V \leq 0$. We prove that under suitable conditions, if the fundamental group $\pi_{1}(N)$ has sub-exponential growth rate, then the Riemannian manifold M with the Jacobi metric $(h-V)g$, i.e., $(M, (h-V)g)$, is a manifold with conjugate points for all h with $0 < h <\delta$, where $\delta$ is a small number.     

Regularization of diffusion tensor MRI via local coordinates

We propose a novel framework for regularization of symmetric positive-definite (SPD) tensors (e.g., diffusion tensors). This framework is based on differential geometry. The space of SPD matrices, P_n, is described as a Riemannian manifold that is parameterized via the Iwasawa coordinate system. Then, distances on P_n are measured in terms of a natural GL (n)-invariant Riemannian metric. Using the Beltrami framework we construct a set of coupled geometric PDEs with respect to the Iwasawa coordinates. Then, by means of the gradient descent method these equations define the regularization flow over P_n. It appears to be that the local coordinate approach via that coordinate system results in very simple numerics that leads to fast convergence of the algorithm. We demonstrate the efficiency of this algorithm on real volumetric DTI datasets. Results of fibers tractography before and afterthe regularization process arepresented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Vrănceanu connections and foliations with bundle-like metrics

We show that the Vrănceanu connection which was initially introduced on non-holonomic manifolds can be used to study the geometry of foliated manifolds. We prove that a foliation is totally geodesic with bundle-like metric if and only if this connection is a metric one. We introduce the notion of a foliated Riemannian manifold of constant transversal Vrănceanu curvature and the notion of a transversal Einstein foliated Riemannian manifold. The geometry of these two classes of manifolds is studied and the relationship between them is determined.     

On doubly warped product Finsler manifolds

In this paper, we introduce horizontal and vertical warped product Finsler manifolds. We prove that every C-reducible or proper Berwaldian doubly warped product Finsler manifold is Riemannian. Then, we find the relation between Riemannian curvatures of doubly warped product Finsler manifold and its components, and consider the cases that this manifold is flat or has scalar flag curvature. We define the doubly warped Sasaki-Matsumoto metric for warped product manifolds and find a condition under which the horizontal and vertical tangent bundles are totally geodesic. We obtain some conditions under which a foliated manifold reduces to a Reinhart manifold. Finally, we study an almost complex structure on the tangent bundle of a doubly warped product Finsler manifold.