Natural operators of smooth mappings of manifoldswith metric fields

In this paper we introduce the concepts of both a natural bundle and a natural operator generalized for the case of the category Mf_m × Mf_m of cartesian products of two manifolds and products of local diffeomorphisms. It is shown that any r-th order natural bundle over M × N has a structure of an associated bundle (P^rM × P^rN)Z G_{m^r × Gm^r}]. We consider prolongations of such associated bundles and their reduction with respect to a chosen subgroup. The existence of a bijective correspondence between natural operators of order k and the equivariant mappings of the corresponding type fibers are proved. A basis of invariants of arbitrary order is constructed for natural operators of smooth mappings of manifolds endowed with metric fields or connections, with values in a natural bundle of order one.     

Geometric quantization of an OSp(1/2) coadjoint orbit

In this Letter, we show how the complete geometric quantization extends to specific supersymplectic supermanifolds. More precisely, we extend this procedure to OSp(1/2)-coadjoint orbits, which are graded extensions of elliptic Sp(2, )-coadjoint orbits. Our approach exploits results obtained in a previous work, where the notion of a super-Kähler supermanifold was defined, and the former orbits were shown to be nontrivial examples of such a notion. As their underlying Kähler manifolds, these supermanifolds carry a natural (super-Kähler) polarization, a crucial notion that was so far lacking. Geometric quantization leads here to a nontrivial representation of osp(1/2), which is realized in a space of square integrable holomorphic sections of a super-Hermitian complex line bundle sheaf-with-connection over the homogenous space OSp(1/2)/U(1).     

Local foliations and optimal regularity of Einstein spacetimes

We investigate the local regularity of pointed spacetimes, that is, time-oriented Lorentzian manifolds in which a point and a future-oriented, unit timelike vector (an observer) are selected. Our main result covers the class of Einstein vacuum spacetimes. Under curvature and injectivity bounds only, we establish the existence of a local coordinate chart defined in a ball with definite size in which the metric coefficients have optimal regularity. The proof is based on quantitative estimates for, on one hand, a constant mean curvature (CMC) foliation by spacelike hypersurfaces defined locally near the observer and, on the other hand, the metric in local coordinates that are spatially harmonic in each CMC slice. The results and techniques in this paper should be useful in the context of general relativity for investigating the long-time behavior of solutions to the Einstein equations.     

Connections on Lie algebroids and on derivation-based noncommutative geometry

In this paper, we show how connections and their generalizations on transitive Lie algebroids are related to the notion of connections in the framework of the derivation-based noncommutative geometry. In order to compare the two constructions, we emphasize the algebraic approach of connections on Lie algebroids, using a suitable differential calculus. Two examples allow this comparison: on the one hand, the Atiyah Lie algebroid of a principal fiber bundle and, on the other hand, the space of derivations of the algebra of endomorphisms of an SL(n,C)-vector bundle. Gauge transformations are also considered in this comparison.     

Twisted quiver bundles over almost complex manifolds

In this paper, we study twisted quiver bundle over general almost complex manifolds. A twisted quiver bundle is a set of J-holomorphic vector bundles over an almost complex manifold, labelled by the vertices of a quiver, linked by a set of morphisms twisted by a fixed collection of J-holomorphic vector bundles, labelled by the arrows. We prove a Hitchin–Kobayashi correspondence for twisted quiver bundles over a compact almost Hermitian regularized manifold, relating the existence of solutions to certain gauge equations to an appropriate notion of stability for the corresponding quivers. This result can be seen as a generalization of that in [2], [9].     

Uniqueness of balanced metrics on holomorphic vector bundles

Let E→M$E\to M$ be a holomorphic vector bundle over a compact Kähler manifold (M,ω)$(M,\omega )$. We prove that if E$E$ admits a ω$\omega$-balanced metric (in X. Wang’s terminology (Wang, 2005 [3])) then it is unique. This result together with Biliotti and Ghigi (2008) [14] implies the existence and uniqueness of ω$\omega$-balanced metrics of certain direct sums of irreducible homogeneous vector bundles over rational homogeneous varieties. We finally apply our result to show the rigidity of ω$\omega$-balanced Kähler maps into Grassmannians.     

The existence of homogeneous geodesics in homogeneous pseudo-Riemannian and affine manifolds

It is well known that, in any homogeneous Riemannian manifold, there is at least one homogeneous geodesic through each point. For the pseudo-Riemannian case, even if we assume reductivity, this existence problem is still open. The standard way to deal with homogeneous geodesics in the pseudo-Riemannian case is to use the so-called “Geodesic Lemma”, which is a formula involving the inner product. We shall use a different approach: namely, we imbed the class of all homogeneous pseudo-Riemannian manifolds into the broader class of all homogeneous affine manifolds (possibly with torsion) and we apply a new, purely affine method to the existence problem. In dimension 2, it was solved positively in a previous article by three authors. Our main result says that any homogeneous affine manifold admits at least one homogeneous geodesic through each point. As an immediate corollary, we prove the same result for the subclass of all homogeneous pseudo-Riemannian manifolds.     

Topological and affine classification of complete noncompact flat 4-manifolds

In this paper we give topological and affine classification of complete noncompact flat 4-manifolds. In particular, we show that the number of diffeomorphism classes of them is equal to 44. The affine classification uses the results of [M. Sadowski, Affinely equivalent complete flat manifolds, Cent. Eur. J. Math. 2 (2) (2004) 332–338]. The affine and the topological equivalence classes are the same for flat manifolds not homotopy equivalent to S¹,T²$S^1,T^2$ or the Klein bottle.     

Super line bundles

Super line bundles over supermanifolds are introduced as natural generalizations of line bundles over smooth manifolds. Their classification in terms of their obstruction class and the representation of their Chern class in terms of a connection on the super line bundle are discussed. The case where the base supermanifold is De Witt is analyzed in detail, both in the supersmooth and complex superanalytic case.     

Scalar curvature and holomorphy potentials

A holomorphy potential is a complex valued function whose complex gradient, with respect to some Kähler metric, is a holomorphic vector field. Given k$k$ holomorphic vector fields on a compact complex manifold, form, for a given Kähler metric, a product of the following type: a function of the scalar curvature multiplied by functions of the holomorphy potentials of each of the vector fields. It is shown that the stipulation that such a product be itself a holomorphy potential for yet another vector field singles out critical metrics for a particular functional. This may be regarded as a generalization of the extremal metric variation of Calabi, where k=0$k=0$ and the functional is the square of the L²$L^2$-norm of the scalar curvature. The existence question for such metrics is examined in a number of special cases. Examples are constructed in the case of certain multifactored product manifolds. For the SKR metrics investigated by Derdzinski and Maschler and residing in the complex projective space, it is shown that only one type of nontrivial criticality holds in dimension three and above.     

Determinants of Laplacians in real line bundles over hyperbolic manifolds connected with quantum geometry of membranes

The possibility is discussed of generalizing the Polyakov approach to strings on membranes and the connection of such a generalization with Thurston's classification of three-dimensional geometries. The important ingredients for computing a membrane path integral are the determinants of scalar Laplacians acting in real line bundles over three-dimensional closed manifolds. In the closed bosonic membrane case, such determinants are evaluated for a class of closed 3-manifolds of the H ³/ form with a discrete subgroup of isometries of the three-dimensional Lobachevsky space H ³ and they are expressed in terms of the Selberg zeta function. Some further possible implications of the results obtained are also discussed.     

Axiomatic holonomy maps and generalized Yang-Mills moduli space

This Letter is a follow-up of Barrett, J. W.,Internat. J. Theoret. Phys. 30(9), (1991). Its main goal is to provide an alternative proof of that part of the reconstruction theorem which concerns the existence of a connection. A construction of a connection 1-form is presented. The formula expressing the local coefficients of the connection in terms of the holonomy map is obtained as an immediate consequence of that construction. Thus, the derived formula coincides with that used in Chan, H.-M., Scharbach, P., and Tsou, S. T.,Ann. Physics 166, 396–421 (1986). The reconstruction and representation theorems form a generalization of the fact that the pointed configuration space of the classical Yang-Mills theory is equivalent to the set of all holonomy maps. The point of this generalization is that there is a one-to-one correspondence not only between the holonomy maps and the orbits in the space of connections, but also between all maps M G fulfilling some axioms and all possible equivalence classes ofP(M, G) bundles with connections, where the equivalence relation is defined by a bundle isomorphism in a natural way.     

Surfaces specified by integrable systems of partial differential equations determined by structure equations and Lax pair

A system of evolution equations can be developed from the structure equations for a submanifold embedded in a three-dimensional space. It is seen how these same equations can be obtained from a generalized matrix Lax pair provided a single constraint equation is imposed. This can be done in Euclidean space as well as in Minkowski space. The integrable systems which result from this process can be thought of as generalizing the SO(3)$SO\left(3\right)$ and SO(2,1)$SO(2,1)$ Lax pairs which have been studied previously.     

Inherited structures in deformations of Poisson pencils

In this paper, we study some properties of bi-Hamiltonian deformations of Poisson pencils of hydrodynamic type. More specifically, we are interested in determining those structures of the fully deformed pencils that are inherited through the interaction between structural properties of the dispersionless pencils (in particular exactness or homogeneity) and suitable finiteness conditions on the central invariants (like polynomiality). This approach enables us to gain some information about each term of the deformation to all orders in  ?$?$.     

Integrable systems of partial differential equations determined by structure equations and Lax pair

It is shown how a system of evolution equations can be developed both from the structure equations of a submanifold embedded in three-space as well as from a matrix SO(6) Lax pair. The two systems obtained this way correspond exactly when a constraint equation is selected and imposed on the system of equations. This allows for the possibility of selecting the coefficients in the second fundamental form in a general way.     

Hyper-Kähler induction and self-duality on Riemann surfaces

Universal hyper-Kähler spaces are constructed from Lie groups acting on flat Kähler manifolds. These spaces are used to describe the moduli space of solutions of Hitchin's equation — self-duality equations on a Riemann surface — as the contangent bundle of the moduli space of flat connections on a Riemann surface.     

Deformations of 2k-Einstein structures

It is shown that the space of infinitesimal deformations of 2k-Einstein structures is finite dimensional on compact non-flat space forms. Moreover, spherical space forms are shown to be rigid in the sense that they are isolated in the corresponding moduli space.     

Gauge Theoretical Construction of Non-compact Calabi-Yau Manifolds

We construct the non-compact Calabi-Yau manifolds interpreted as the complex line bundles over the Hermitian symmetric spaces. These manifolds are the various generalizations of the complex line bundle over CP^N−1. Imposing an F-term constraint on the line bundle over CP^N−1, we obtain the line bundle over the complex quadric surface Q^N−2. On the other hand, when we promote the U(1) gauge symmetry in CP^N−1 to the non-abelian gauge group U(M), the line bundle over the Grassmann manifold is obtained. We construct the non-compact Calabi-Yau manifolds with isometries of exceptional groups, which we have not discussed in the previous papers. Each of these manifolds contains the resolution parameter which controls the size of the base manifold, and the conical singularity appears when the parameter vanishes.     

Biminimal properly immersed submanifolds in the Euclidean spaces

We consider a complete nonnegative biminimal   submanifold M$M$ (that is, a complete biminimal submanifold with λ≥0$\lambda \ge 0$) in a Euclidean space E^N${\mathbb{E}}^N$. Assume that the immersion is proper  , that is, the preimage of every compact set in E^N${\mathbb{E}}^N$ is also compact in M$M$. Then, we prove that M$M$ is minimal. From this result, we give an affirmative partial answer to Chen’s conjecture. For the case of λ<0$\lambda <0$, we construct examples of biminimal submanifolds and curves.