Parallel Torsion Research Articles

A type of nearly parallel G-structures

We study properties of a certain symmetric tensor r induced by the intrinsic torsion of a Riemannian G–structure. This tensor naturally arises in the context of nearly Kähler manifolds and is parallel with respect to the canonical Hermitian connection. In general, we call a G-structure a second order parallel if ∇Gr=0 for a minimal G–connection ∇G. We show correlation with harmonicity of a G–structure and with G–structures with parallel torsion. An example of second order parallel G–structure is, apart from nearly Kähler manifolds and nearly parallel G2 structures, an α-Kenmotsu manifold.

On Hermitian manifolds whose Chern connection is Ambrose-Singer

We consider the class of compact Hermitian manifolds whose Chern connection is Ambrose-Singer, namely, it has parallel torsion and curvature. We prove structure theorems for such manifolds.

Связности с параллельным кососимметрическим кручением на субримановых многообразиях

On a sub-Riemannian manifold M of contact type, is considered an N-connection defined by the pair (, N), where is an interior metric connection, is an endomorphism of the distribution D. It is proved that there exists a unique N-connection such that its torsion is skew-symmetric as a contravariant tensor field. A construction of the endomorphism corresponding to such connection is found. The sufficient conditions for the obtained connection to be a metric connec­tion with parallel torsion are given.

A New Type of Generalized Para-Kähler Spaces and Holomorphically Projective Transformations

In the present paper, we define a new class of generalized para-Kahler spaces by using two different kinds of covariant derivatives. Some properties of the curvature tensors as well as those of the corresponding Ricci tensors of new generalized para-Kahler spaces are pointed out. Also, we consider some tensor fields that are invariant with respect to holomorphically projective mappings of generalized para-Kahler spaces. Finally, we examine some PDE systems for the existence of infinitesimal holomorphically projective transformations of new types of generalized para-Kahler spaces with parallel torsion.

Static balancing of an inverted pendulum with prestressed torsion bars

This paper presents a method for the design of a statically balanced inverted pendulum. The non-linear moment-rotation characteristic of the pendulum's weight is approximated by a piecewise linear characteristic. Each transition is realized by engaging or disengaging one or more torsion bars, by means of mechanical stops. The whole set of torsion bars is located along the hinge axis of the pendulum. A prototype with three parallel torsion bars was built. Experimental evaluation of the prototype revealed a 99% work reduction of the balanced pendulum with respect to the unbalanced one.

The classification of naturally reductive homogeneous spaces in dimensions [formula omitted

We present a new method for classifying naturally reductive homogeneous spaces – i.e. homogeneous Riemannian manifolds admitting a metric connection with skew torsion that has parallel torsion and curvature. This method is based on a deeper understanding of the holonomy algebra of connections with parallel skew torsion on Riemannian manifolds and the interplay of such a connection with the geometric structure on the given Riemannian manifold. It allows to reproduce by easier arguments the known classifications in dimensions 3,4, and 5, and yields as a new result the classification in dimension 6. In each dimension, one obtains a ‘hierarchy’ of degeneracy for the torsion form, which we then treat case by case. For the completely degenerate cases, we obtain results that are independent of the dimension. In some situations, we are able to prove that any Riemannian manifold with parallel skew torsion has to be naturally reductive. We show that a ‘generic’ parallel torsion form defines a quasi-Sasakian structure in dimension 5 and an almost complex structure in dimension 6.

Conformal holonomy, symmetric spaces, and skew symmetric torsion

We consider the question: Can the isotropy representation of an irreducible pseudo-Riemannian symmetric space be realized as a conformal holonomy group? Using recent results by Čap, Gover and Hammerl, we study the representations of SO(2,1), PSU(2,1) and PSp(2,1) as isotropy groups of irreducible symmetric spaces of signature (3,2), (4,4) and (6,8), respectively, describing the geometry induced by a conformal holonomy reduction to the corresponding subgroups. In the case of SO(2,1), we show that conformal manifolds with such a holonomy reduction are always locally conformally flat and hence this group cannot be a conformal holonomy group. This result completes the classification of irreducible conformal holonomy groups in Lorentzian signature. In the case of PSU(2,1), we show that conformal manifolds of signature (3,3) with this holonomy reduction carry, on an open dense subset, a canonical nearly para-Kähler metric with positive Einstein constant. For PSp(2,1) we also show that there is an open dense subset endowed with a canonical Einstein metric in the conformal class. As a result, after restricting to an open dense subset, the conformal holonomy must be a proper subgroup of PSU(2,1) or of PSp(2,1), respectively. These are special cases of an interesting relationship between a class of special conformal holonomy groups, and non-integrable geometries with skew symmetric, parallel torsion, which we also explore. Finally, using a recent result of Graham and Willse we prove the following general non-existence result: for a real-analytic, odd-dimensional conformal manifold, the conformal holonomy group can never be given by the isotropy representation of an irreducible pseudo-Riemannian symmetric space unless the isotropy group is SO0(p+1,q+1).

EINSTEIN MANIFOLDS WITH SKEW TORSION

This paper is devoted to the first systematic investigation of manifolds that are Einstein for a connection ∇ with skew-symmetric torsion. We derive the Einstein equation from a variational principle and prove that, for parallel torsion, any Einstein manifold with skew torsion has constant scalar curvature; and if it is complete of positive scalar ∇-curvature, it is necessarily compact and it has finite first fundamental group π1. The longest part of the paper is devoted to the systematic construction of large families of examples. We discuss when a Riemannian Einstein manifold can be Einstein with skew torsion. We give examples of almost Hermitian, almost metric contact, and G2 manifolds that are Einstein with skew torsion. For example, we prove that any Einstein–Sasaki manifold and any seven-dimensional 3-Sasakian manifolds admit deformations into an Einstein metric with parallel skew torsion.

A NATURAL CONNECTION ON A BASIC CLASS OF RIEMANNIAN PRODUCT MANIFOLDS

A Riemannian manifold M with an integrable almost product structure P is called a Riemannian product manifold. Our investigations are on the manifolds (M, P, g) of the largest class of Riemannian product manifolds, which is closed with respect to the group of conformal transformations of the metric g. This class is an analogue of the class of locally conformal Kähler manifolds in almost Hermitian geometry. In the present paper we study a natural connection D on (M, P, g) (i.e. DP = Dg = 0). We find necessary and sufficient conditions, the curvature tensor of D to have properties similar to the Kähler tensor in Hermitian geometry. We pay attention to the case when D has a parallel torsion. We establish that the Weyl tensors for the connection D and the Levi-Civita connection coincide as well as the invariance of the curvature tensor of D with respect to the usual conformal transformation. We consider the case when D is a flat connection. We construct an example of the considered manifold by a Lie group where D is a flat connection with non-parallel torsion.

A classification of certain almost α-Kenmotsu manifolds

We study $\mathscr{D}$-homothetic deformations of almost α-Kenmotsu structures. We characterize almost contact metric manifolds which are CR-integrable almost α-Kenmotsu manifolds, through the existence of a canonical linear connection, invariant under $\mathscr{D}$-homothetic deformations. If the canonical connection associated to the structure (φ, ξ, η, g) has parallel torsion and curvature, then the local geometry is completely determined by the dimension of the manifold and the spectrum of the operator h′ defined by 2αh′ = ($\mathscr{L}$ξφ) $\circ$ φ. In particular, the manifold is locally equivalent to a Lie group endowed with a left invariant almost α-Kenmotsu structure. In the case of almost α-Kenmotsu (κ, μ)′-spaces, this classification gives rise to a scalar invariant depending on the real numbers κ and α.

A connection with parallel torsion on almost hypercomplex manifolds with Hermitian and anti-Hermitian metrics

Almost hypercomplex manifolds with Hermitian and anti-Hermitian metrics are considered. A linear connection D is introduced such that the structure of these manifolds is parallel with respect to D . Of special interest is the class of the locally conformally equivalent manifolds of the manifolds with covariantly constant almost complex structures and the case when the torsion of D is D -parallel. Curvature properties of these manifolds are studied. An example of 4-dimensional manifolds in the considered basic class is constructed and characterized.

Eigenvalue estimates for Dirac operators in geometries with torsion

On a Riemannian spin manifold (Mn, g), equipped with a non-integrable geometric structure and characteristic connection ▽c with parallel torsion ▽cT c = 0, we can introduce the Dirac operator D1/3, which is constructed by lifting the affine metric connection with torsion 1/3 T c to the spin structure. D1/3 is a symmetric elliptic differential operator, acting on sections of the spinor bundle and can be identified in special cases with Kostant’s cubic Dirac operator or the Dolbeault operator. For compact (Mn, g), we investigate the first eigenvalue of the operator \({\left(D^{1/3} \right)^{2}}\) . As a main tool, we use Weitzenbock formulas, which express the square of the perturbed operator D1/3 + S by the Laplacian of a suitable spinor connection. Here, S runs through a certain class of perturbations. We apply our method to spaces of dimension 6 and 7, in particular, to nearly Kahler and nearly parallel G2-spaces.

Spin (7)-Manifolds with Parallel Torsion Form

Any Spin(7)-manifold admits a metric connection \nabla^c with totally skew-symmetric torsion T^c preserving the underlying structure. We classify those with \nabla^c-parallel T^c\neq0 and non-Abelian isotropy algebra iso(T^c)<spin(7). These are isometric to either Riemannian products or homogeneous naturally reductive spaces, each admitting two \nabla^c-parallel spinor fields.

Almost hermitian structures with parallel torsion

The characteristic connection of an almost hermitian structure is a hermitian connection with totally skew-symmetric torsion. The case of parallel torsion in dimension 6 is of particular interest. In this work, we give a full classification of the algebraic types of the torsion form, and, on the basis of this, undertake a systematic investigation into the possible geometries. Numerous naturally reductive spaces are constructed and classified, and examples on nilmanifolds given.

Almost Hermitian 6-manifolds revisited

A theorem of Kirichenko states that the torsion 3-form of the characteristic connection of a nearly Kähler manifold is parallel. On the other side, any almost Hermitian manifold of type G 1 admits a unique connection with totally skew-symmetric torsion. In dimension 6, we generalize Kirichenko’s theorem and we describe almost Hermitian G 1-manifolds with parallel torsion form. In particular, among them there are only two types of W 3 -manifolds with a non-Abelian holonomy group, namely twistor spaces of four-dimensional self-dual Einstein manifolds and the invariant Hermitian structure on the Lie group SL(2, C) . Moreover, we classify all naturally reductive Hermitian W 3 -manifolds with small isotropy group of the characteristic torsion.

Flat manifolds with parallel torsion

Flat manifolds with parallel torsion

SEESAW NYSTAGMUS. CASE REPORT ELUCIDATING THE MECHANISM.

Seesaw nystagmus is a very rare disturbance of ocular motility. The site of the responsible lesion has not been clearly identified, but a central lesion has been postulated in most previous reports. This case, the eighth noted in the literature,1-5is recorded as it offers a more probable explanation for this form of nystagmus. The original case report by Maddox1contains a superlative description of seesaw nystagmus. The eye movements in the patient described herein are identical to those described by Maddox and consists essentially of a disjunctive vertical pendular movement: One eye rises as the other falls. Simultaneous with the opposed rising and falling of the eyes, there occurs a conjugate parallel torsion. The rising of the right eye and the falling of the left eye are associated with a fine torsional movement to the left; the rising of the left eye and the falling of the