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We define and study noncommutative generalizations of submanifolds and quotient manifolds, for the derivation-based differential calculus introduced by M.~Dubois-Violette and P.~Michor. We give examples to illustrate these definitions.
A general definition of a bimodule connection in noncommutative geometry has been recently proposed. For a given algebra this definition is compared with the ordinary definition of a connection on a left module over the associated enveloping algebra. The corresponding curvatures are also compared.
A method is proposed for defining an arbitrary number of differential calculi over a given noncommutative associative algebra. As an example the generalized quantum plane is studied. It is found that there is a strong correlation, but not a one-to-one correspondence, between the module structure of the 1-forms and the metric torsion-free connections on it. In the commutative limit the connection remains as a shadow of the algebraic structure of the 1-forms.
One has not any conventional energy-momentum conservation law in Lagrangian field theory, but relations involving different stress-energy-momentum tensors associated with different connections. It is not obvious how to choose the true energy-momentum tensor. This problem is solved in the framework of the multimomentum Hamiltonian formalism which provides the adequate description of constraint field systems. The goal is that, for different solutions of the same constraint field model, one should choose different stress-energy-momentum tensors in general. Gauge theory illustrates this result. The stress-energy-momentum tensors of affine-metric gravity are examined.
The elastic energy functional of a thin elastic rod or sheet is generalized to the case of an M-dimensional manifold in N-dimensional space. We derive potentials for the stress field and curvatures and find the generalized von Karman equations for a manifold in elastic equilibrium. We perform a scaling analysis of an M-1 dimensional ridge in an M = N-1 dimensional manifold. A ridge of linear size X in a manifold with thickness h << X has a width w ~ h^{1/3}X^{2/3} and a total energy E ~ h^{M} (X/h)^{M-5/3}. We also prove that the total bending energy of the ridge is exactly five times the total stretching energy. These results match those of A. Lobkovsky [Phys. Rev. E 53, 3750 (1996)] for the case of a bent plate in three dimensions.
We apply the geometric quantization method with real polarizations to the quantization of a symplectic torus. By quantizing with half-densities we canonically associate to the symplectic torus a projective Hilbert space and prove that the projective factor is expressible in terms of the Maslov-Kashiwara index. As in the quantization of a linear symplectic space, we have two ways of resolving the projective ambiguity: (i) by introducing a metaplectic structure and using half-forms in the definition of the Hilbert space; (ii) by choosing a 4-fold cover of the Lagrangian Grassmannian of the linear symplectic space covering the torus. We show that the Hilbert space constructed through either of these approaches realizes a unitary representation of the integer metaplectic group.
Any discrete differential manifold $M$ (finite set endowed with an algebraic differential calculus) can be represented by appropriate polyhedron ${\cal P}(M)$. This representation demonstrates the adequacy of the calculus of discrete differential manifolds and links this approach with that based on finitary substitutes of continuous spaces introduced by R.D.Sorkin.
The super or Z_2-graded Schouten-Nijenhuis bracket is introduced. Using it, new generalized super-Poisson structures are found which are given in terms of certain graded-skew-symmetric contravariant tensors Λof even order. The corresponding super `Jacobi identities' are expressed by stating that these tensors have zero super Schouten-Nijenhuis bracket with themselves [Λ,Λ]=0. As a particular case, we provide the linear generalized super-Poisson structures which can be constructed on the dual spaces of simple superalgebras with a non-degenerate Killing metric. The su(3,1) superalgebra is given as a generic example.
The definition of the covariant space-time averaging scheme for the objects (tensors, geometric objects, etc.) on differentiable metric manifolds with a volume n-form, which has been proposed for the formulation of macroscopic gravity, is analyzed. An overview of the space-time averaging procedure in Minkowski spacetime is given and comparison between this averaging scheme and that adopted in macroscopic gravity is carried out throughout the paper. Some new results concerning the algebraic structure of the averaging operator are precisely formulated and proved, the main one being that the averaging bilocal operator is idempotent iff it is factorized into a bilocal product of a matrix-valued function on the manifold, taken at a point, by its inverse at another point. The previously proved existence theorems for the averaging and coordination bilocal operators are revisited with more detailed proofs of related results. A number of new results concerning the structure of the volume-preserving averaging operators and the class of proper coordinate systems are given. It is shown, in particular, that such operators are defined on an arbitrary n-dimensional differentiable metric manifold with a volume n-form up to the freedom of (n-1) arbitrary functions of n arguments and 1 arbitrary function of (n-1) arguments. All the results given in this paper are also valid whenever appropriate for affine connection manifolds including (pseudo)-Riemannian manifolds.
3+1 decompositions of differential forms on a Lorentzian manifold (M,g;+ - - -) with respect to arbitrary observer field and the decomposition of the standard operations acting on them are studied, making use of the ideas of the theory of connections on principal bundles. Simple explicit general formulas are given as well as their application to the Maxwell equations.
We consider a parabolic-like systems of differential equations involving geometrical quantities to examine uniformization theorems for two- and three-dimensional closed orientable manifolds. We find that in the two-dimensional case there is a simple gauge theoretic flow for a connection built from a Riemannian structure, and that the convergence of the flow to the fixed points is consistent with the Poincare Uniformization Theorem. We construct a similar system for the three-dimensional case. Here the connection is built from a Riemannian geometry, an SO(3) connection and two other 1-form fields which take their values in the SO(3) algebra. The flat connections include the eight homogeneous geometries relevant to the three-dimensional uniformization theorem conjectured by W. Thurston. The fixed points of the flow include, besides the flat connections (and their local deformations), non-flat solutions of the Yang-Mills equations. These latter "instanton" configurations may be relevant to the fact that generic 3-manifolds do not admit one of the homogeneous geometries, but may be decomposed into "simple 3-manifolds" which do.
We give a construction of the abelian Chern-Simons gauge theory from the point of view of a 2+1 dimensional topological quantum field theory. The definition of the quantum theory relies on geometric quantization ideas which have been previously explored in connection to the nonabelian Chern-Simons theory [JW,ADW]. We formulate the topological quantum field theory in terms of the category of extended 2- and 3-manifolds introduced by Walker [Wa] and prove that it satisfies the axioms of unitary topological quantum field theories formulated by Atiyah [A1].
For any principal bundle $P$, one can consider the subspace of the space of connections on its tangent bundle $TP$ given by the tangent bundle $T{\cal A}$ of the space of connections ${\cal A}$ on $P$. The tangent gauge group acts freely on $T{\cal A}$. Appropriate BRST operators are introduced for quantum field theories that include as fields elements of $T{\cal A}$, as well as tangent vectors to the space of curvatures. As the simplest application, the BRST symmetry of the so-called $BF$-Yang-Mills theory is described and the relevant gauge fixing conditions are analyzed. A brief account on the topological $BF$ theories is also included and the relevant Batalin-Vilkovisky operator is described.
The integrability of multivector fields in a differentiable manifold is studied. Then, given a jet bundle $J^1E\to E\to M$, it is shown that integrable multivector fields in $E$ are equivalent to integrable connections in the bundle $E\to M$ (that is, integrable jet fields in $J^1E$). This result is applied to the particular case of multivector fields in the manifold $J^1E$ and connections in the bundle $J^1E\to M$ (that is, jet fields in the repeated jet bundle $J^1J^1E$), in order to characterize integrable multivector fields and connections whose integral manifolds are canonical lifting of sections. These results allow us to set the Lagrangian evolution equations for first-order classical field theories in three equivalent geometrical ways (in a form similar to that in which the Lagrangian dynamical equations of non-autonomous mechanical systems are usually given). Then, using multivector fields; we discuss several aspects of these evolution equations (both for the regular and singular cases); namely: the existence and non-uniqueness of solutions, the integrability problem and Noether's theorem; giving insights into the differences between mechanics and field theories.
It is shown that the first order (Palatini) variational principle for a generic nonlinear metric-affine Lagrangian depending on the (symmetrized) Ricci square invariant leads to an almost-product Einstein structure or to an almost-complex anti-Hermitian Einstein structure on a manifold. It is proved that a real anti-Hermitian metric on a complex manifold satisfies the Kähler condition on the same manifold treated as a real manifold if and only if the metric is the real part of a holomorphic metric. A characterisation of anti-Kähler Einstein manifolds and almost-product Einstein manifolds is obtained. Examples of such manifolds are considered.
The Einstein/Maxwell equations reduce in the stationary and axially symmetric case to a harmonic map with prescribed singularities phi: R^3Σ-> H^2_C, where Sigma is a subset of the axis of symmetry, and H^2_C is the complex hyperbolic plane. Motivated by this problem, we prove the existence and uniqueness of harmonic maps with prescribed singularities phi: R^nΣ-> H, where Sigma is a submanifold of R^n of co-dimension at least 2, and H is a classical Riemannian globally symmetric space of noncompact type and rank one. This result, when applied to the black hole problem, yields solutions which can be interpreted as equilibrium configurations of multiple co-axially rotating charged black holes held apart by singular struts.
We prove that the maximal development of any spherically symmetric spacetime with collisionless matter (obeying the Vlasov equation) or a massless scalar field (obeying the massless wave equation) and possessing a constant mean curvature $S^1 \times S^2$ Cauchy surface also contains a maximal Cauchy surface. Combining this with previous results establishes that the spacetime can be foliated by constant mean curvature Cauchy surfaces with the mean curvature taking on all real values, thereby showing that these spacetimes satisfy the closed-universe recollapse conjecture. A key element of the proof, of interest in itself, is a bound for the volume of any Cauchy surface $Σ$ in any spacetime satisfying the timelike convergence condition in terms of the volume and mean curvature of a fixed Cauchy surface $Σ_0$ and the maximal distance between $Σ$ and $Σ_0$. In particular, this shows that any globally hyperbolic spacetime having a finite lifetime and obeying the timelike-convergence condition cannot attain an arbitrarily large spatial volume.
We consider sphere bundles P and P' of totally null planes of maximal dimension and opposite self-duality over a 4-dimensional manifold equipped with a Weyl or Riemannian geometry. The fibre product PP' of P and P' is found to be appropriate for the encoding of both the selfdual and the Einstein-Weyl equations for the 4-metric. This encoding is realized in terms of the properties of certain well defined geometrical objects on PP'. The formulation is suitable for both complex- and real-valued metrics. It unifies results for all three possible real signatures. In the purely Riemannian positive definite case it implies the existence of a natural almost hermitian structure on PP' whose integrability conditions correspond to the self-dual Einstein equations of the 4-metric. All Einstein equations for the 4-metric are also encoded in the properties of this almost hermitian structure on PP'.