The present note deals with the dynamics of metric connections with vectorial torsion, as already described by E. Cartan in 1925. We show that the geodesics of metric connections with vectorial torsion defined by gradient vector fields coincide with the Levi-Civita geodesics of a conformally equivalent metric. By pullback, this yields a systematic way of constructing invariants of motion for such connections from isometries of the conformally equivalent metric, and we explain in as much this result generalizes the Mercator projection which maps sphere loxodromes to straight lines in the plane. An example shows that Beltrami's theorem fails for this class of connections. We then study the system of differential equations describing geodesics in the plane for vector fields which are not gradients, and show among others that the Hopf-Rinow theorem does also not hold in general.
The present note deals with the properties of metric connections $\nabla$ with vectorial torsion $V$ on semi-Riemannian manifolds $(M^n,g)$. We show that the $\nabla$-curvature is symmetric if and only if $V^{\flat}$ is closed, and that $V^\perp$ then defines an $(n-1)$-dimensional integrable distribution on $M^n$. If the vector field $V$ is exact, we show that the $V$-curvature coincides up to global rescaling with the Riemannian curvature of a conformally equivalent metric. We prove that it is possible to construct connections with vectorial torsion on warped products of arbitrary dimension …
We study 5-dimensional Riemannian manifolds that admit an almost contact metric structure. We classify these structures by their intrinsic torsion and review the literature in terms of this scheme. Moreover, we determine necessary and sufficient conditions for the existence of metric connections with vectorial, totally skew-symmetric or traceless cyclic torsion that are compatible with the almost contact metric structure. Finally, we examine explicit examples of almost contact metric 5-manifolds from this perspective.
We show that the absolute value at zero of the Ruelle zeta function defined by the geodesic flow coincides with the higher-dimensional Reidemeister torsion for the unit tangent bundle over a 2-dimensional hyperbolic orbifold and a non-unitary representation of the fundamental group. Our proof is based on the integral expression of the Ruelle zeta function. This integral expression is derived from the functional equation of the Selberg zeta function for a discrete subgroup with elliptic elements in PSL(2;R). We also show that the asymptotic behavior of the higher-dimensional Reidemeister torsio…
We study the irreducible decomposition under Sp(2n, R) of the space of torsion tensors of almost symplectic connections. Then a description of all symplectic quadratic invariants of torsion-like tensors is given. When applied to a manifold M with an almost symplectic structure, these instruments give preliminary insight for finding a preferred linear almost symplectic connection on M . We rediscover Ph. Tondeur's Theorem on almost symplectic connections. Properties of torsion of the vectorial kind are deduced.
We find necessary and sufficient conditions for the foliation defined by level sets of a function f(x_{1},...,x_{n}) to be totally geodesic in a torsion-free connection and apply them to find the conditions for d-webs of hypersurfaces to be geodesic, and in the case of flat connections, for d-webs (d > n) of hypersurfaces to be hyperplanar webs. These conditions are systems of generalized Euler equations, and for flat connections we give an explicit construction of their solutions.
We study the skew-symmetric prolongation of a Lie subalgebra $\g \subseteq \mathfrak{so}(n)$, in other words the intersection $Λ^3 \cap (Λ^1 \otimes \g)$.We compute this space in full generality. Applications include uniqueness results for connections with skew-symmetric torsion and also the proof of the Euclidean version of a conjecture posed in \cite{ofarill} concerning a class of Plücker-type embeddings. We also derive a classification of the metric k-Lie algebras (or Filipov algebras), in positive signature and finite dimension. Prolongations of Lie algebras can also be used to finish the …
In this short note we study flat metric connections with antisymmetric torsion $T \neq 0$. The result has been originally discovered by Cartan/Schouten in 1926 and we provide a new proof not depending on the classification of symmetric spaces. Any space of that type splits and the irreducible factors are compact simple Lie group or a special connection on $S^7$. The latter case is interesting from the viewpoint of $G_2$-structures and we discuss its type in the sense of the Fernandez-Gray classification. Moreover, we investigate flat metric connections of vectorial type.
We investigate the concept of projective equivalence of connections in supergeometry. To this aim, we propose a definition for (super) geodesics on a supermanifold in which, as in the classical case, they are the projections of the integral curves of a vector field on the tangent bundle: the geodesic vector field associated with the connection. Our (super) geodesics possess the same properties as the in the classical case: there exists a unique (super) geodesic satisfying a given initial condition and when the connection is metric, our supergeodesics coincide with the trajectories of a free pa…
Invariant connections and $\nabla$-Einstein structures on isotropy irreducible spaces
Faculty of Science, University of Hradec Kr\'alov\'e, Rokitanskeho 62 Hradec Kr\'alov\'e 50003, Czech Republic；Institute of Mathematics；Statistics, University of Troms , Troms \ 90-37, Norway
本文研究非对称强迷向不可约齐性黎曼流形上的不变仿射、度量联络及带斜挠率的不变度量联络，计算其维数，分类紧李群${\ m U}_{n}$的双不变度量联络，并完成此类空间上带斜挠率的$\ abla$-Einstein不变结构的分类。
For a manifold with an affine connection, we prove formulas which infinitesimally quantify the gap in a certain naturally defined open geodesic quadrilateral associated to a pair of tangent vectors $u$, $v$ at a point of the manifold. We show that the 1st order infinitesimal obstruction to the quadrilateral to close is always zero, the 2nd order infinitesimal obstruction to the quadrilateral to close is $-T(u,v)$ where $T$ is the torsion tensor of the connection, and if $T = 0$ then the 3rd order infinitesimal obstruction to the quadrilateral to close is $(1/2)R(u,v)(u+v)$ in terms of the curv…
Let $\mathcal M= (M,\mathcal O_\mathcal M)$ be a smooth supermanifold with connection $\nabla$ and Batchelor model $\mathcal O_\mathcal M\congΓ_{ΛE^\ast}$. From $(\mathcal M,\nabla)$ we construct a connection on the total space of the vector bundle $E\to{M}$. This reduction of $\nabla$ is well-defined independently of the isomorphism $\mathcal O_\mathcal M \cong Γ_{ΛE^\ast}$. It erases information, but however it turns out that the natural identification of supercurves in $\mathcal M$ (as maps from $ \mathbb R^{1|1}$ to $\mathcal M$) with curves in $E$ restricts to a 1 to 1 correspondence on g…
We investigate octonion product deformations coming from the parallelizable torsion of the 7-sphere $S^7$, obtaining a family of geometries from solutions of the Lagrangian formalism movement equations. This can be achieved by analyzing the spontaneous compactification $M_4\times S^7$, where $M_4$ is a Lorentzian $4$-dimensional manifold. Besides the usual Riemannian geometry and two others proposed by Cartan and Schouten, solutions in geometries with torsion and more general seven-dimensional spaces are obtained. Such formalism may by subsequently derived over the 7-sphere $S^7$, locally give…
We establish a bijective correspondence between affine connections and a class of semi-holonomic jets of local diffeomorphisms of the underlying manifold called symmetry jets in the text. The symmetry jet corresponding to a torsion free connection consists in the family of $2$-jets of the geodesic symmetries. Conversely, any connection is described in terms of the geodesic symmetries by a simple formula involving only the Lie bracket of vector fields. We then formulate, in terms of the symmetry jet, several aspects of the theory of affine connections and obtain geometric and intrinsic descript…
We define and study a random Lie bracket that induces torsion in expectation. Almost all stochastic analysis on manifolds have assumed parallel transport. Mathematically this assumption is very reasonable. However, in many applied geometry and graphics problems parallel transport is not achieved, the "change in coordinates" are not exact due to noise. We formulate a stochastic model on a manifold for which parallel transport does not hold and analyze the consequences of this model with respect to classic quantities studied in Riemannian geometry. We first define a stochastic lie bracket that i…
Here we treat the problem: given a torsion-free connection do its geodesics, as unparametrised curves, coincide with the geodesics of an Einstein metric? We find projective invariants such that the vanishing of these is necessary for the existence of such a metric, and in generic settings the vanishing of these is also sufficient. We also obtain results for the problem of metrisability (without the Einstein condition): We show that the odd Chern type invariants of an affine connection are projective invariants that obstruct the existence of a projectively related Levi-Civita connection. In add…
We introduce the equation of n-dimensional totally geodesic submanifolds of a manifold E as a submanifold of the second order jet space of n-dimensional submanifolds of E. Next we study the geometry of n-Grassmannian equivalent connections, that is linear connections without torsion admitting the same equation of n-dimensional totally geodesic submanifolds. We define the n-Grassmannian structure as the equivalence class of such connections, recovering for n=1 the case of theory of projectively equivalent connections. By introducing the equation of parametrized n-dimensional totally geodesic su…
It is proved that all invariant functions of a complex Finsler manifold can be totally recovered from the torsion and curvature of the connection introduced by Kobayashi for holomorphic vector bundles with complex Finsler structures. Equations of the geodesics and Jacobi fields of a generic complex Finsler manifold, expressed by means Kobayashi's connection, are also derived.
We classify compact surfaces with torsion-free affine connections for which every geodesic is a simple closed curve. In the process, we obtain completely new proofs of all the major results concerning the Riemannian case. In contrast to previous work, our approach is twistor-theoretic, and depends fundamentally on the fact that, up to biholomorphism, there is only one complex structure on CP2.
We describe min-max formulas for the principal eigenvalue of a $V$-drift Laplacian defined by a vector field $V$ on a geodesic ball of a Riemannian manifold $N$. Then we derive comparison results for the principal eigenvalue with the one of a spherically symmetric model space endowed with a radial vector field, under pointwise comparison of the corresponding radial sectional and Ricci curvatures, and of the radial component of the vector fields. These results generalize the known case $V=0$.
We study the twisted Ruelle zeta function $ζ_X(s)$ for smooth Anosov vector fields $X$ acting on flat vector bundles over smooth compact manifolds. In dimension $3$, we prove Fried conjecture, relating Reidemeister torsion and $ζ_X(0)$. In higher dimensions, we show more generally that $ζ_X(0)$ is locally constant with respect to the vector field $X$ under a spectral condition. As a consequence, we also show Fried conjecture for Anosov flows near the geodesic flow on the unit tangent bundle of hyperbolic $3$-manifolds. This gives the first examples of non-analytic Anosov flows and geodesic flo…
In this paper we present the distinguished (d-) Riemannian geometry (in the sense of nonlinear connection, Cartan canonical linear connection, together with its d-torsions and d-curvatures) for a possible Lagrangian inspired by optics in non-uniform media. The corresponding equations of motion are also exposed, and some particular solutions are given. For instance, we obtain as geodesic trajectories some circular helices (depending on an angular velocity ω), certain circles situated in some planes (ones are parallel with xOy, and other ones are orthogonal on xOy), or some straight lines which …
