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We formulate several criteria under which the symmetries associated with the Killing and Killing-Yano tensors on the base space can be lifted to the symmetries of the full warped geometry. The procedure is explicitly illustrated on several examples, providing new prototypes of spacetimes admitting such tensors. In particular, we study a warped product of two Kerr-NUT-(A)dS spacetimes and show that it gives rise to a new class of highly symmetric vacuum (with cosmological constant) black hole solutions that inherit many of the properties of the Kerr-NUT-(A)dS geometry., 10 pages, no figures

In this chapter we prove maximal Lp-regularity for various linear parabolic and elliptic problems. These results will be crucial for our study of quasilinear parabolic problems, including those introduced in Chapter 1.

We show that a diffeological bundle gives rise to an exact sequence of internal tangent spaces. We then introduce two new classes of diffeological spaces, which we call weakly filtered and filtered diffeological spaces, whose tangent spaces are easier to understand. These are the diffeological spaces whose categories of pointed plots are (weakly) filtered. We extend the exact sequence one step further in the case of a diffeological bundle with filtered total space and base space. We also show that the tangent bundle $T^H X$ defined by Hector is a diffeological vector space over $X$ when $X$ is filtered or when $X$ is a homogeneous space, and therefore agrees with the dvs tangent bundle introduced by the authors in a previous paper., v3: new results and improvements to exposition; 14 pages; this version to appear in Proceedings of the AMS

The aim of this paper is to describe the obstruction for an almost Lagrangian fibration to be Lagrangian, a problem which is central to the classification of Lagrangian fibrations and, more generally, to understanding the obstructions to carry out surgery of integrable systems, an idea introduced by Zung. It is shown that this obstruction (namely, the homomorphism of Dazord and Delzant) is related to the cup product in cohomology with local coefficients on the base space B of the fibration. The map is described explicitly and some examples are calculated, thus providing the first examples of non-trivial Lagrangian obstructions., 17 pages, to appear in Diff. Geom. Appl

In “Rips complexes and covers in the uniform category” (Brodskiy et al., preprint [4] ) the authors define, following James (1990) [5] , covering maps of uniform spaces and introduce the concept of generalized uniform covering maps. Conditions for the existence of universal uniform covering maps and generalized uniform covering maps are given. This paper extends these results by investigating the existence of these covering maps relative to subgroups of the uniform fundamental group and the fundamental group of the base space.

We consider a degeneration of genus 2 curves, which is op- posite to maximal degeneration in a sense. Such a degeneration of curves yields a variation of mixed Hodge structure with monodromy weight fil- tration. The mixed Hodge structure at each fibre, which is dierent from the limit mixed Hodge structure of Schmid and Steenbrink, can be real- ized as H1 of a noncompact singular elliptic curve. We also prove that the pull back of the above variation of mixed Hodge structure to a dou- ble cover of the base space comes from a family of noncompact singular elliptic curves.

This paper shows that for mechanical systems, the dimension of whose base space is larger than time (there also exist spatial coordinates), the system of equations defining the evolution of the system must be a hyperbolic system of pseudodifferential equations.

Let RP(k) denotethe k-dimensional real projective space. This article deter mines which cobordism classes are represented by the total space of a fibering with pre scribed base space RP (3) × RP (1), RP (2) × RP (1), RP (2) × RP (1) × RP (1) or RP (3) × RP (2).

We show that the universal covering [math] of any compact, negatively curved manifold [math] has an exponential growth rate which is strictly greater than the exponential growth rate of any other normal covering [math] . Moreover, we give an explicit formula estimating the difference between [math] and [math] in terms of the systole of [math] and of other elementary geometric parameters of the base space [math] . Then we discuss some applications of this formula to periodic geodesics, to the bottom of the spectrum and to the critical exponent of normal coverings.

In the present paper we consider Riemannian coverings $(X,g) \to (M,g)$ with residually finite covering group $\Gamma$ and compact base space $(M,g)$. In particular, we give two general procedures resulting in a family of deformed coverings $(X,g_\eps) \to (M,g_\eps)$ such that the spectrum of the Laplacian $\Delta_{(X_\eps,g_\eps)}$ has at least a prescribed finite number of spectral gaps provided $\eps$ is small enough. If $\Gamma$ has a positive Kadison constant, then we can apply results by Br\"uning and Sunada to deduce that $\spec \Delta_{(X,g_\eps)}$ has, in addition, band-structure and there is an asymptotic estimate for the number $N(\lambda)$ of components of $\spec {\laplacian {(X,g_\eps)}}$ that intersect the interval $[0,\lambda]$. We also present several classes of examples of residually finite groups that fit with our construction and study their interrelations. Finally, we mention several possible applications for our results., Comment: final version (26 pages, 2 figures). to appear in Rev. Math. Phys

We prove the Herbert multiple-point formula modulo 2-torsion using the method of the universal immersion.

Let $f,g:\cc^2\ra\cc$ be two quasi-homogeneous polynomials. We compute the $V$-filtration of the restriction of $f$ to any plane curve $C_t=g^{-1}(t)$ and show that the Gorenstein generator $dx\wedge dy/dg$ is a primitive form. Using results of A. Douai and C. Sabbah, we conclude that base space of the miniversal unfolding of $f_t:=f|_{C_t}$ is a Frobenius manifold. At the singular fibre $C_0$ we obtain a non-massive Frobenius manifold., 11 pages

Given a Waelbroeck ring R, we prove that the Grassmannian of a projective finitely generated R -module is a topological manifold modeled on a topological abelian group of R -linear maps. Fibre bundles of fibre type a module as above, over a compact base space B, admitting R -valued partitions of unity, are classified by the homotopy classes of continuous maps on B with values in the respective Grassmannian. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Motivated by the recent results in random metric theory, we establish Clark’s fixed point theorem on complete random normed modules under two kinds of topologies. When the base space of the random normed module is trivial, our results automatically degenerate to the classical case.

Let I = [a, b] a real compact interval and ƒ ∶ I → I a continuous function. Define K c ([a, b]) the class of all non empty compact subinterval of [a, b] and let ƒ c the natural extension of ƒ to K c ([a, b]), that is to say, ƒ c (J) = ƒ(J) for all J ∈ K c ([a, b]). Also, let F c ([a, b]) the class of all fuzzy-intervals with support contained in [a, b] and consider ƒ c the Zadeh's extension of ƒ to F c ([a, b]). The aim of this paper is to show some dynamical properties of ƒ c and ƒ c in relation to the Devaney's conditions for complexity of functions and, in particular, to show the surprising difference between the dynamics on the base space X = [a, b] compared with the dynamics of the interval extension K c ([a, b]) and fuzzy-interval extension F c ([a, b]), which could be useful in the mathematical modelling of real problems.

In this paper, we study the geometry of anti-invariant Riemanniansubmersions from a Kähler manifold onto a Riemannian manifold. Wefirst determine the base space when the total space of ananti-invariant Riemannian submersion is Einstein and then weinvestigate new conditions for anti-invariant Riemannian submersionsto be Clairaut submersions. We also focus on the geometry of Clairaut anti-invariant submersions.

Consider the principal $U(n)$ bundles over Grassmann manifolds $U(n)\rightarrow U(n+m)/U(m) \stackrel{\pi}\rightarrow G_{n,m}$. Given $X \in U_{m,n}(\mathbb{C})$ and a 2-dimensional subspace $\mathfrak{m}' \subset \mathfrak{m} $ $ \subset \mathfrak{u}(m+n), $ assume either $\mathfrak{m}'$ is induced by $X,Y \in U_{m,n}(\mathbb{C})$ with $X^{*}Y = \mu I_n$ for some $\mu \in \mathbb{R}$ or by $X,iX \in U_{m,n}(\mathbb{C})$. Then $\mathfrak{m}'$ gives rise to a complete totally geodesic surface $S$ in the base space. Furthermore, let $\gamma$ be a piecewise smooth, simple closed curve on $S$ parametrized by $0\leq t\leq 1$, and $\widetilde{\gamma}$ its horizontal lift on the bundle $U(n) \rightarrow \pi^{-1}(S) \stackrel{\pi}{\rightarrow} S,$ which is immersed in $U(n) \rightarrow U(n+m)/U(m) \stackrel{\pi}\rightarrow G_{n,m} $. Then $$ \widetilde{\gamma}(1)= \widetilde{\gamma}(0) \cdot ( e^{i \theta} I_n) \quad \text{ or } \quad \widetilde{\gamma}(1)= \widetilde{\gamma}(0), $$ depending on whether the immersed bundle is flat or not, where $A(\gamma)$ is the area of the region on the surface $S$ surrounded by $\gamma$ and $\theta= 2 \cdot \tfrac{n+m}{2n} A(\gamma).$, Comment: arXiv admin note: substantial text overlap with arXiv:1206.3652

In this paper, we construct new characteristic classes of fiber bundles via flat connections with values in infinite-dimensional Lie algberas of derivations. In fact, choosing a fiberwise metric, we construct a chain map to the de Rham complex on the base space, and show that the induced map on cohomology groups is independent of the choices. Moreover, we show that applying the construction to a surface bundle, our construction gives Morita-Mumford-Miller classes., Comment: 16 pages, 1 figure

In this paper we develop a theory of second order connections with a view towards the associated stochastic calculus. Connections in principal fiber bundles are defined as sections of the tangent space of second order differential operators. We prove existence and uniqueness of stochastic horizontal lifts for semimartingales with respect to these connections. Finally, the parallel transport along semimartingales on the base space is studied.

Li and Yorke not only introduced the term “chaos” along with a mathematically rigorous definition of what they meant by it, but also gave a condition for chaos in scalar difference equations, their equally famous “period three implies chaos” result. Generalizations of the Li and Yorke definition of chaos to difference equations in ℝ n are reviewed here as well as higher dimensional conditions ensuring its existence, specifically the “snap-back repeller” condition of Marotto and its counterpart for saddle points. In addition, further generalizations to mappings in Banach spaces and complete metric spaces are considered. These will be illustrated with various simple examples including an application to chaotic dynamics on the metric space (ℰ n , D) of fuzzy sets on the base space ℝ n .

The isometries of the hyperspace of a compact subset of the real line endowed with the generalized Pompeiu metric are considered. It is proved that any such an isometry is generated by an isometry of the base space.

A surface bundle determines a monodromy representation recording the twisting of the fiber under transport around a closed path in the base space. The fascinating relation between these monodromy representations and the Thurston classification of surface automorphisms will be studied. In this note we deal with a simple and interesting case: the fibrations of Fadell and Neuwirth.

Given a Hausdorff space X, we calculate the tightness and the character of the hyperspace CL∅(X) of X, endowed with either the co-compact or the lower Vietoris topology, and give some estimates for the tightness of CL∅(X), endowed with the Fell topology. Some properties related to first-countability and c ountable tightness, such as sequentiality, Frechet property and, less directly, radiality and p seudoradiality, are investigated as well. To carry out our investigation, we also consider on the base space X several cardinal functions, and we compare some of them (which are newly defined or not so well known) with other classical ones, obtaining results and counterexamples which may be of some independent interest.  2004 Elsevier B.V. All rights reserved. MSC: primary 54B20, 54A25; secondary 54D55, 54A20, 54E35

A nonautonomous semidynamical system is a skew-product semi-flow consisting of a cocycle mapping on a state space which is driven by a semidynamical system on a base space. It is shown that the driving system can be extended backwards in time on a compact invariant set, such as a global attractor, as a set-valued semidynamical system. Pullback attraction and invariance with respect to the cocycle mapping are then reformulated in terms of the set-valued backwards extension of the driving system.

The Riemannian submersions of a CR-hypersurfaceMof a Kaehler-Einstein manifoldM˜are studied. IfMis an extrinsic CR-hypersurface ofM˜, then it is shown that the base space of the submersion is also a Kaehler-Einstein manifold.

We study the moduli space of stable sheaves on a reducible projective scheme by use of a suitable stratification of the moduli space. Each stratum is the moduli space of “triples”, which is the main object investigated in this paper. As an application, we can see that the relative moduli space of rank two stable sheaves on quadric surfaces gives a nontrivial example of the relative moduli space which is not flat over the base space.

In this paper, we introduce the notion of fuzzy regular spaces over a base space and investigate some properties of these spaces.

Given a continuous map from a smooth manifold into the base space of a sphere bundle and a lifting, defined outside a regular subset (called the “defect set”), there are obstructions to extend the lifting and there are obstructions to extend the lifting as an ordinary map into the total space. We prove a relation between these two types of obstructions, involving a sort of universal obstruction, and compute this universal obstruction for some concrete examples.

The purpose of this note is to describe the base space of an almost paracontact submersion. Here the base space is an almost para-Hermitian manifold. So, the paper intertwines paracontact and para-Hermitian structures via the theory of submersions.

Given a map $\Phi$ defined on bounded subsets of the (base) metric space $X$ and with bounded sets as its values, one can follow the orbits $A$, $\Phi(A)$, $\Phi^2(A)$, $\ldots$, of nonempty, closed, and bounded sets $A$ in $X$. This is the system $(\Phi, X)$. On the other hand, the same orbits can be viewed as trajectories of points in the hyperspace $X^\sharp$ of nonempty, closed, and bounded subsets of $X$. This is the system $(\Phi, X^\sharp)$. We study the existence and properties of global attractors for both $(\Phi, X)$ and $(\Phi, X^\sharp)$. We give very basic conditions on $\Phi$, stated at the level of the base space $X$, that are necessary and sufficient for the existence of a global attractor for $(\Phi, X)$. Continuity is not among those conditions, but if $\Phi$ is continuous in a certain sense then the attractor and the $\omega$-limit sets are $\Phi$-invariant. If $(\Phi, X)$ has a global attractor, then $(\Phi, X^\sharp)$ has a global attractor as well. Every point of the global attractor of $(\Phi, X^\sharp)$ is a compact set in $X$, and the union of all the points of that attractor is the global attractor of $(\Phi, X)$.

We construct canonical intertwining semi-models with Kobayashi hyperbolic base space for holomorphic self-maps of complex manifolds which are univalent on some absorbing cocompact hyperbolic domain. In particular, in the unit ball we solve the Valiron equation for hyperbolic univalent self-maps and for hyperbolic semigroups., Comment: 39 pages, misprints corrected and some proofs clarified

Motivated by the description of $\mathcal{N}=1$ M-theory compactifications to four-dimensions given by Exceptional Generalized Geometry, we propose a way to geometrize the M-theory fluxes by appropriately relating the compactification space to a higher-dimensional manifold equipped with a torsion-free structure. As a non-trivial example of this proposal, we construct a bijection from the set of $Spin(7)$-structures on an eight-dimensional $S^{1}$-bundle to the set of $G_{2}$-structures on the base space, fully characterizing the $G_{2}$-torsion clases when the total space is equipped with a torsion-free $Spin(7)$-structure. Finally, we elaborate on how the higher-dimensional manifold and its moduli space of torsion-free structures can be used to obtain information about the moduli space of M-theory compactifications., Comment: 24 pages. Typos fixed. Minor clarifications added

In the paper we will introduce show that the $p^3$ power map on $SU(p+t-1)$ is an H-map for $2\leq t\leq p-1$. To do this we will consider a fibration whose base space is $SU(p+t-1)$ with the property that there is a section into the total space. We will then use decomposition methods to identify the fibre and the map from it to the total space. This information will be used to deduce information about $SU(p+t-1)$. In doing this we draw together recent work of Kishimoto and Theriault with more classical work of Cohen and Neisendorfer, and make use of the classical theorems of Hilton and Milnor, James and Barrett., 16 pages