We present an introduction to the geometry of higher order vector and co--vector bundles (including higher order generalizations of the Finsler geometry and Kaluza--Klein gravity) and review the basic results on Clifford and spinor structures on spaces with generic local anisotropy modeled by higher order nonlinear connections. Geometric applications in locally anisotropic gravity and matter field interactions are considered. This article contains a part of material given by these authors and P. Stavrinos in theirs lectures.
The geometry of nonholonomic bundle gerbes, provided with nonlinear connection structure, and nonholonomic gerbe modules is elaborated as the theory of Clifford modules on nonholonomic manifolds which positively fail to be spin. We explore an approach to such nonholonomic Dirac operators and derive the related Atiyah-Singer index formulas. There are considered certain applications in modern gravity and geometric mechanics of such Clifford-Lagrange/ Finsler gerbes and their realizations as nonholonomic Clifford and Riemann-Cartan modules.
We propose a new framework for constructing geometric and physical models on nonholonomic manifolds provided both with Clifford -- Lie algebroid symmetry and nonlinear connection structure. Explicit parametrizations of generic off-diagonal metrics and linear and nonlinear connections define different types of Finsler, Lagrange and/or Riemann-Cartan spaces. A generalization to spinor fields and Dirac operators on nonholonomic manifolds motivates the theory of Clifford algebroids defined as Clifford bundles, in general, enabled with nonintegrable distributions defining the nonlinear connection. …
An almost Clifford and an almost Cliffordian manifold is a $G$--structure based on the definition of Clifford algebras. An almost Clifford manifold based on $\mathcal O:= \cc l (s,t)$ is given by a reduction of the structure group $GL(km, \mathbb R)$ to $GL(m, {\mathcal O})$, where $k=2^{s+t}$ and $m \in \mathbb N$. An almost Cliffordian manifold is given by a reduction of the structure group to $GL(m, \mathcal O) GL(1,\mathcal O)$. We prove that an almost Clifford manifold based on $\mathcal O$ is such that there exists a unique subordinated connection, while the case of an almost Cliffordian…
We survey the geometry of Lagrange and Finsler spaces and discuss the issues related to the definition of curvature of nonholonomic manifolds enabled with nonlinear connection structure. It is proved that any commutative Riemannian geometry (in general, any Riemann--Cartan space) defined by a generic off--diagonal metric structure (with an additional affine connection possessing nontrivial torsion) is equivalent to a generalized Lagrange, or Finsler, geometry modeled on nonholonomic manifolds. This results in the problem of constructing noncommutative geometries with local anisotropy, in parti…
The book contains a collection of works on Riemann-Cartan and metric-affine manifolds provided with nonlinear connection structure and on generalized Finsler-Lagrange and Cartan-Hamilton geometries and Clifford structures modelled on such manifolds. The choice of material presented has evolved from various applications in modern gravity and geometric mechanics and certain generalizations to noncommutative Riemann-Finsler geometry. The authors develop and use the method of anholonomic frames with associated nonlinear connection structure and apply it to a number of concrete problems: constructi…
In this paper we examine a new class of five dimensional (5D) exact solutions in extra dimension gravity possessing Lie algebroid symmetry. The constructions provide a motivation for the theory of Clifford nonholonomic algebroids elaborated in Ref. hep-th/0501217. Such Einstein-Dirac spacetimes are parametrized by generic off--diagonal metrics and nonholonomic frames (vielbeins) with associated nonlinear connection structure. They describe self-consistent propagations of (3D) Dirac wave packets in 5D nonholonomically deformed Taub NUT spacetimes and have two physically distinct properties: Fis…
In this paper, using connections between Clifford-Wolf isometries and Killing vector fields of constant length on a given Riemannian manifold, we classify simply connected Clifford-Wolf homogeneous Riemannian manifolds. We also get the classification of complete simply connected Riemannian manifolds with the Killing property defined and studied previously by J.E. D'Atri and H.K. Nickerson. In the last part of the paper we study properties of Clifford-Killing spaces, that is, real vector spaces of Killing vector fields of constant length, on odd-dimensional round spheres, and discuss numerous c…
This work reconsiders the holomorphic and anti-holomorphic Dirac operators of Hermitian Clifford analysis to determine whether or not they are the natural generalization of the orthogonal Dirac operator to spaces with complex structure. We argue the generalized gradient construction of Stein and Weiss based on representation theory of Lie groups is the natural way to construct such a Dirac-type operator because applied to a Riemannian spin manifold it provides the Atiyah-Singer Dirac operator. This method, however, does not apply to these Hermitian Dirac operators because the representations o…
We introduce the concept of a Clifford-Weyl structure on a conformal manifold, which consists of an even Clifford structure parallel with respect to the tensor product of a metric connection on the Clifford bundle and a Weyl structure on the manifold. We show that the Weyl structure is necessarily closed except for some "generic" low-dimensional instances, where explicit examples of non-closed Clifford-Weyl structures can be constructed.
A generalized Clifford manifold is proposed in which there are coordinates not only for the basis vector generators, but for each element of the Clifford group, including the identity scalar. These new quantities are physically interpreted to represent internal structure of matter (e.g. classical or quantum spin). The generalized Dirac operator must now include differentiation with respect to these higher order geometric coordinates. In a Riemann space, where the magnitude and rank of geometric objects are preserved under displacement, these new terms modify the geodesics. One possible physica…
In this paper we show how to describe the general theory of a linear metric compatible connection with the theory of Clifford valued differential forms. This is done by realizing that for each spacetime point the algebra of Clifford bivectors is isomorphic to the Lie algebra of Sl(2,C). In that way the pullback of the linear connection under a trivialization of the bundle is represented by a Clifford valued 1-form. That observation makes it possible to realize Einstein's gravitational theory can be formulated in a way which is similar to a Sl(2,C) gauge theory. Some aspects of such approach is…
We present an introduction to the geometry of higher order vector and co-vector bundles (including higher order generalizations of the Finsler geometry and Kaluza--Klein gravity) and review the basic results on Clifford and spinor structures on spaces with generic local anisotropy modeled by anholonomic frames with associated nonlinear connection structures. We emphasize strong arguments for application of Finsler like geometries in modern string and gravity theory and noncommutative geometry and noncommutative field theory and gravity.
In this summary of Habilitation Thesis, it is outlined author's 18 years research activity on mathematical physics, geometric methods in particle physics and gravity, modifications and applications (after defending his PhD thesis in 1994). Ten most relevant publications are structured conventionally into three "strategic directions": 1) nonholonomic geometric flows evolutions and exact solutions for Ricci solitons and field equations in (modified) gravity theories; 2) geometric methods in quantization of models with nonlinear dynamics and anisotropic field interactions; 3) (non) commutative ge…
A Dirac bundle is a euclidean bundle over a riemannian manifold $M$ which is a compatible left $C\ell(M)$-module, together with a metric connection also compatible with the Clifford action in a natural way. We prove some vanishing theorems and introduce the twistor equation within this framework. In particular, we exhibit a characterization of solutions for this equation in terms of the Dirac operator $D$ and a suitable Weitzenböck-type curvature operator $\mathcal{R}$. Finally, we analyze the especial case of the Clifford bundle to prove existence of nontrivial solutions of the twistor equati…
We give an account of the classical and integrable geometry of isothermic surfaces in arbitrary co-dimension. We show that the classical transformation theory of Darboux, Bianchi and Calapso goes through unchanged in arbitrary co-dimension as does the connection with the "curved flats" of Ferus and Pedit. Moreover, we identify Darboux transformations with the dressing action of "simple factors" in the sense of Terng and Uhlenbeck. In so doing, we advertise the use of Vahlen's Clifford algebra matrices as an efficient computational tool in conformal geometry.
In this paper we give a construction of Fedosov quantization incorporating the odd variables and an analogous formula to Getzler's pseudodifferential calculus composition formula is obtained. A Fedosov type connection is constructed on the bundle of Weyl tensor Clifford algebras over the cotangent bundle of a Riemannian manifold. The quantum algebra associated with this connection is used to define a deformation of the exterior algebra of Riemannian manifolds.
Here (the last paper in a series of four) we end our presentation of the basics of a systematical approach to the differential geometry of a smooth manifold M (supporting a metric field g and a general connection del) which uses the geometric algebras of multivector and extensors (fields) developed in previous papers. The theory of the Riemann and Ricci fields of the the triple (M,del,g)is investigated to for each particular open set U of M through the introduction of a geometric structure on U, i.e., a triple (U,gamma,g) where gamma is a general connection field on U and g is a metric extenso…
The Hermitian symmetric space $M=\mathrm{EIII}$ appears in the classification of complete simply connected Riemannian manifolds carrying a parallel even Clifford structure. This means the existence of a real oriented Euclidean vector bundle $E$ over it together with an algebra bundle morphism $φ:\mathrm{Cl}^0(E) \rightarrow \mathrm{End}(TM)$ mapping $Λ^2 E$ into skew-symmetric endomorphisms, and the existence of a metric connection on $E$ compatible with $φ$. We give an explicit description of such a vector bundle $E$ as a sub-bundle of $\mathrm{End}(TM)$. From this we construct a canonical di…
We outline a new geometric method of constructing exact solutions of gravitational field equations parametrized by generic off-diagonal metrics, anholonomic frames and possessing, in general, nontrivial torsion and nonmetricity. The formalism of nonlinear connections is elaborated for (pseudo) Riemannian and Einstein-Cartan-Weyl spaces.
We show that, for mechanical system with external forces, the equations of deviations of solution curves of the corresponding Lagrange equations,determine a nonlinear connection on the second order osculator (second order tangent) bundle. In particular, Jacobi equations in Finsler and Riemann spaces determine such a nonlinear connection.
We systematically discuss connections on the spinor bundle of Cahen-Wallach symmetric spaces. A large class of these connections is closely connected to a quadratic relation on Clifford algebras. This relation in turn is associated to the symmetric linear map that defines the underlying space. We present various solutions of this relation. Moreover, we show that the solutions we present provide a complete list with respect to a particular algebraic condition on the parameters that enter into the construction.
A linear connection is associated to a nonlinear connection on a vector bundle by a linearization procedure. Our definition is intrinsic in terms of vector fields on the bundle. For a connection on an affine bundle our procedure can be applied after homogenization and restriction. Several applications in Classical Mechanics are provided.
The construction of a linear connection on a pullback bundle from a connection on a vector bundle is explained in terms of fiberwise linear approximation. This procedure clarifies the geometric meaning of the linearized connection as well as the associated parallel transport and curvature.
This paper studies the behavior of sequences of solutions to Seiberg-Witten like equations for a pair consisting of a Hermitian connection on a line bundle over a 4-dimensional manifold and a section of the self-dual spinor bundle of a complex Clifford module on the manifold. Examples include the cases where the Clifford module is a direct sum of C2 bundles associated to SpinC structures; and the case of the SU(2) Vafa-Witten equations with an Abelian ansatz.
Higher order anisotropic superspaces are constructed as generalized vector superbundles provided with compatible nonlinear connection, distinguished connection and metric structures.
The theory of locally anisotropic superspaces (supersymmetric generalizations of various types of Kaluza--Klein, Lagrange and Finsler spaces) is laid down. In this framework we perform the analysis of construction of the supervector bundles provided with nonlinear and distinguished connections and metric structures. Two models of locally anisotropic supergravity are proposed and studied in details.
For a system of second order differential equations we determine a nonlinear connection that is compatible with a given generalized Lagrange metric. Using this nonlinear connection, we can find the whole family of metric nonlinear connections that can be associated with a system of SODE and a generalized Lagrange structure. For the particular case when the system of SODE and the metric structure are Lagrangian, we prove that the canonic nonlinear connection of the Lagrange space is the only nonlinear connection which is metric and compatible with the symplectic structure of the Lagrange space.…
The main purposes of this article are to extend our previous results on homogeneous sprays to arbitrary (generalized) sprays, to show that locally diffeomorphic exponential maps can be defined for any (generalized) spray, and to give a (possibly nonlinear) covariant derivative for any (possibly nonlinear) connection. In the process, we introduce vertically homogeneous connections. Unlike homogeneous connections, these allow us to include Finsler spaces among the applications. We provide significant support for the prospect of studying nonlinear connections via (generalized) sprays. One of the …
In this paper we present a method by which is obtained a sequence of $k$-semisprays and two sequences of nonlinear connections on the $k$-tangent bundle $T^kM$, starting from a given one. Interesting particular cases appear for Lagrange and Finsler spaces of order $k$.
We review the geometric setting of the field theory with locally anisotropic interactions. The concept of locally anisotropic space is introduced as a general one for various type of extensions of Lagrange and Finsler geometry and higher dimension (Kaluza--Klein type) spaces. The problem of definition of spinors on generalized Finsler spaces is solved in the framework of the geometry of Clifford bundles provided with compatible nonlinear and distinguished connections and metric structures. We construct the spinor differential geometry of locally anisotropic spaces and discuss some related issu…
In this paper we study some geometrical objects (d-tensors, multi-time semisprays of polymomenta and nonlinear connections) on the dual 1-jet vector bundle $J^{1*}(\cal{T}, M)\to \cal{T}\times M$. Some geometrical formulas, which connect the last two geometrical objects, are also derived. Finally, a canonical nonlinear connection produced by a Kronecker $h$-regular multi-time Hamiltonian is given.
A general formulation of zero curvature connections in a principle bundle is presented and some applications are discussed. It is proved that a related connection based on a prolongation in an associated bundle remains zero curvature as well. It is also shown that the connection coefficients can be defined so that the partial differential equation to be studied appears as the curvature term in the structure equations. It is discussed how Lax pairs and Backlund transformations can be formulated for such equations. It is discussed how Lax pairs and Backlund transformations can be formulated for …
The aim of this paper is to generalize the theory of nonlinear connections of Grifone ([3] and [4]). We adopt the point of view of Anona [1] and continue developing the approach established by the first author in [10]. The first part of the work is devoted to the problem of associating to each $L$-regular linear connection on $M$ a nonlinear $L$-connection on $M$. The route we have followed is significantly different from that of Grifone. We introduce an almost-complex and an almost-product structures on $M$ by means of a given $L$-regular linear connection on $M$. The product of these two str…
The aim of this paper is to open the problem of construction of a nonlinear connection $Γ=(M^{(i)}_{(α)β}, N^{(i)}_{(α)j})$ on the jet bundle of first order $J^1(T,M)$, which to be canonically produced by a Kronecker product vertical metrical d-tensor $G^{(α)(β)}_{(i)(j)}=h^{αβ}g_{ij}$, possibly provided by multi-time dependent quadratic Lagrangians coming from various branches of theoretical physics: bosonic strings theory, electrodynamics or elasticity.
