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In this paper, we will be concerned with the explicit classification of closed, oriented, simply-connected spin manifolds in dimension eight with vanishing cohomology in the odd dimensions. The study of such manifolds was begun by Stefan M\"uller. In order to understand the structure of these manifolds, we will analyze their minimal handle presentations and describe explicitly to what extent these handle presentations are determined by the cohomology ring and the characteristic classes. It turns out that the cohomology ring and the characteristic classes do not suffice to reconstruct a manifold of the above type completely. In fact, the group ${\rm Aut_0}\bigl(#_{i=1}^b(S^2\times S^5)\bigr)/{\rm Aut}_0\bigl(#_{i=1}^b (S^2\times D^6)\bigr)$ of automorphisms of $#_{i=1}^b(S^2\times S^5)$ which induce the identity on cohomology modulo those which extend to $#_{i=1}^b(S^2\times D^6)$ acts on the set of oriented homeomorphy classes of manifolds with fixed cohomology ring and characteristic classes, and we will be also concerned with describing this group and some facts about the above action.

Leon Green obtained remarkable rigidity results for manifolds of positive scalar curvature with large conjugate radius and/or injectivity radius. Using $C^{k,\alpha}$ convergence techniques, we prove several differentiable stability and sphere theorem versions of these results and apply those also to the study of Einstein manifolds.

Leon Green obtained remarkable rigidity results for manifolds of positive scalar curvature with large conjugate radius and/or injectivity radius. Using $C^{k,\alpha}$ convergence techniques, we prove several differentiable stability and sphere theorem versions of these results and apply those also to the study of Einstein manifolds.

Leon Green obtained remarkable rigidity results for manifolds of positive scalar curvature with large conjugate radius and/or injectivity radius. Using $C^{k,\alpha}$ convergence techniques, we prove several differentiable stability and sphere theorem versions of these results and apply those also to the study of Einstein manifolds.

Leon Green obtained remarkable rigidity results for manifolds of positive scalar curvature with large conjugate radius and/or injectivity radius. Using $C^{k,\alpha}$ convergence techniques, we prove several differentiable stability and sphere theorem versions of these results and apply those also to the study of Einstein manifolds.

Leon Green obtained remarkable rigidity results for manifolds of positive scalar curvature with large conjugate radius and/or injectivity radius. Using $C^{k,\alpha}$ convergence techniques, we prove several differentiable stability and sphere theorem versions of these results and apply those also to the study of Einstein manifolds.

Leon Green obtained remarkable rigidity results for manifolds of positive scalar curvature with large conjugate radius and/or injectivity radius. Using $C^{k,\alpha}$ convergence techniques, we prove several differentiable stability and sphere theorem versions of these results and apply those also to the study of Einstein manifolds.

In this follow-up of our earlier two works D(11.1) (arXiv:1406.0929 [math.DG]) and D(11.2) (arXiv:1412.0771 [hep-th]) in the D-project, we study further the notion of a `differentiable map from an Azumaya/matrix manifold to a real manifold'. A conjecture is made that the notion of differentiable maps from Azumaya/matrix manifolds as defined in D(11.1) is equivalent to one defined through the contravariant ring-homomorphisms alone. A proof of this conjecture for the smooth (i.e. $C^{\infty}$) case is given in this note. Thus, at least in the smooth case, our setting for D-branes in the realm of differential geometry is completely parallel to that in the realm of algebraic geometry, cf.\ arXiv:0709.1515 [math.AG] and arXiv:0809.2121 [math.AG]. A related conjecture on such maps to ${\Bbb R}^n$, as a $C^k$-manifold, and its proof in the $C^{\infty}$ case is also given. As a by-product, a conjecture on a division lemma in the finitely differentiable case that generalizes the division lemma in the smooth case from Malgrange is given in the end, as well as other comments on the conjectures in the general $C^k$ case. We remark that there are similar conjectures in general and theorems in the smooth case for the fermionic/super generalization of the notion.

This paper is a sequel to "Normal forms, stability and splitting of invariant manifolds I. Gevrey Hamiltonians", in which we gave a new construction of resonant normal forms with an exponentially small remainder for near-integrable Gevrey Hamiltonians at a quasi-periodic frequency, using a method of periodic approximations. In this second part we focus on finitely differentiable Hamiltonians, and we derive normal forms with a polynomially small remainder. As applications, we obtain a polynomially large upper bound on the stability time for the evolution of the action variables and a polynomially small upper bound on the splitting of invariant manifolds for hyperbolic tori.

General definitions for causal structures on manifolds of dimension d+1>2 are presented for the topological category and for any differentiable one. Locally, these are given as cone structures via local (pointwise) homeomorphic or diffeomorphic abstraction from the standard null cone variety in R^{d+1}. Weak and strong local cone (LC) structures refer to the cone itself or a manifold thickening of the cone respectively. After introducing cone (C-)causality, a causal complement with reasonable duality properties can be defined. The most common causal concepts of space-times are generalized to the present topological setting. A new notion of precausality precludes inner boundaries within future/past cones. LC-structures, C-causality, a topological causal complement, and precausality may be useful tools in conformal and background independent formulations of (algebraic) quantum field theory and quantum gravity.

In this notes it will be provided a set of techniques which can help one to understand the proof of the Hochschild-Kostant-Rosenberg theorem for differentiable manifolds. Precise definitions of multidiferential operators and polyderivations on an algebra are given, allowing to work on these concepts, when the algebra is an algebra of functions on a differentiable manifold, in a coordinate free description. Also, it will be constructed a cup product on polyderivations which corresponds on (Hochschild) cohomology to wedge product on multivector fields. At the end, a proof of the above mentioned theorem will be given.

In this article we propose a model for stochastic delay differential equation with jumps (SDDEJ) in a differentiable manifold $M$ endowed with a connection $\nabla$. In our model, the continuous part is driven by vector fields with a fixed delay and the jumps are assumed to come from a distinct source of (c\`adl\`ag) noise, without delay. The jumps occur along adopted differentiable curves with some dynamical relevance (with fictitious time) which allow to take parallel transport along them. Using a geometrical approach, in the last section, we show that the horizontal lift of the solution of an SDDEJ is again a solution of an SDDEJ in the linear frame bundle $BM$ with respect to a connection $\nabla^H$ in $BM$.

We give axioms which characterize the local Reidemeister trace for orientable differentiable manifolds. The local Reidemeister trace in fixed point theory is already known, and we provide both uniqueness and existence results for the local Reidemeister trace in coincidence theory.

We give axioms which characterize the local Reidemeister trace for orientable differentiable manifolds. The local Reidemeister trace in fixed point theory is already known, and we provide both uniqueness and existence results for the local Reidemeister trace in coincidence theory.

By using topological methods, mainly the degree of a tangent vector field, we establish multiplicity results for $T$-periodic solutions of parametrized $T$-periodic perturbations of autonomous ODEs on a differentiable manifold $M$. In order to provide insights into the key notion of $T$-resonance, we consider the elementary situations $M = \mathbb{R}$ and $M = \mathbb{R}^2$. So doing, we provide more comprehensive analysis of those cases and find improved conditions.

By using topological methods, mainly the degree of a tangent vector field, we establish multiplicity results for $T$-periodic solutions of parametrized $T$-periodic perturbations of autonomous ODEs on a differentiable manifold $M$. In order to provide insights into the key notion of $T$-resonance, we consider the elementary situations $M = \mathbb{R}$ and $M = \mathbb{R}^2$. So doing, we provide more comprehensive analysis of those cases and find improved conditions.

Let F_0=B,...,F_n be a sequence of differentiable manifolds, G_i a Lie subgroup of diffeomorphisms of F_i, and H_i a subgroup of G_i central in G_i. We suppose also given a locally trivial bundle p_{K_i} over F_{i-1} which typical fiber is K_i the quotient of G_i by H_i. The aim of this paper is to study the differential geometry of the following problem: classify sequences M_n\to...M_1, where each map from M_i to M_{i-1} is a locally trivial fibration which typical fiber is F_i and which transition functions image are elements of G_i. We associate to this problem a tower of gerbes and define for it the notion of connective structure, curvature and holonomy using the notion of free transitive distribution (free TD)

Let M be a differentiable manifold. We say that a tensor field g defined on M is non-regular if g is in some local Lp space or if g is continuous. In this work we define a mollifier smoothing g_t of g that has the following feature: If g is a Riemannian metric of class C2, then the Levi-Civita connection and the Riemannian curvature tensor of g_t converges to the Levi-Civita connection and to the Riemannian curvature tensor of g respectively as t converges to zero. Therefore this mollifier smoothing is a good starting point in order to generalize objects of the classical Riemannian geometry to non-regular Riemannian manifolds. Finally we give some applications of this mollifier smoothing. In particular, we generalize the concept of Lipschitz-Killing curvature measure for some non-regular Riemannian manifolds.

A new differentiable sphere theorem is obtained from the view of submanifold geometry. An important scalar is defined by the scalar curvature and the mean curvature of an oriented complete submanifold $M^n$ in a space form $F^{n+p}(c)$ with $c\ge0$. Making use of the Hamilton-Brendle-Schoen convergence result for Ricci flow and the Lawson-Simons-Xin formula for the nonexistence of stable currents, we prove that if the infimum of this scalar is positive, then $M$ is diffeomorphic to $S^n$. We then introduce an intrinsic invariant $I(M)$ for oriented complete Riemannian $n$-manifold $M$ via the scalar, and prove that if $I(M)>0$, then $M$ is diffeomorphic to $S^n$. It should be emphasized that our differentiable sphere theorem is optimal for arbitrary $n(\ge2)$.

A new differentiable sphere theorem is obtained from the view of submanifold geometry. An important scalar is defined by the scalar curvature and the mean curvature of an oriented complete submanifold $M^n$ in a space form $F^{n+p}(c)$ with $c\ge0$. Making use of the Hamilton-Brendle-Schoen convergence result for Ricci flow and the Lawson-Simons-Xin formula for the nonexistence of stable currents, we prove that if the infimum of this scalar is positive, then $M$ is diffeomorphic to $S^n$. We then introduce an intrinsic invariant $I(M)$ for oriented complete Riemannian $n$-manifold $M$ via the scalar, and prove that if $I(M)>0$, then $M$ is diffeomorphic to $S^n$. It should be emphasized that our differentiable sphere theorem is optimal for arbitrary $n(\ge2)$.

A new differentiable sphere theorem is obtained from the view of submanifold geometry. An important scalar is defined by the scalar curvature and the mean curvature of an oriented complete submanifold $M^n$ in a space form $F^{n+p}(c)$ with $c\ge0$. Making use of the Hamilton-Brendle-Schoen convergence result for Ricci flow and the Lawson-Simons-Xin formula for the nonexistence of stable currents, we prove that if the infimum of this scalar is positive, then $M$ is diffeomorphic to $S^n$. We then introduce an intrinsic invariant $I(M)$ for oriented complete Riemannian $n$-manifold $M$ via the scalar, and prove that if $I(M)>0$, then $M$ is diffeomorphic to $S^n$. It should be emphasized that our differentiable sphere theorem is optimal for arbitrary $n(\ge2)$.

A new differentiable sphere theorem is obtained from the view of submanifold geometry. An important scalar is defined by the scalar curvature and the mean curvature of an oriented complete submanifold $M^n$ in a space form $F^{n+p}(c)$ with $c\ge0$. Making use of the Hamilton-Brendle-Schoen convergence result for Ricci flow and the Lawson-Simons-Xin formula for the nonexistence of stable currents, we prove that if the infimum of this scalar is positive, then $M$ is diffeomorphic to $S^n$. We then introduce an intrinsic invariant $I(M)$ for oriented complete Riemannian $n$-manifold $M$ via the scalar, and prove that if $I(M)>0$, then $M$ is diffeomorphic to $S^n$. It should be emphasized that our differentiable sphere theorem is optimal for arbitrary $n(\ge2)$.

Quantum general relativity may be considered as generally covariant QFT on differentiable manifolds, without any a priori metric structure. The kinematically covariance group acts by general diffeomorphisms on the manifold and by automorphisms on the isotonic net of *-algebras encoding the QFT, while the algebra of observables is covariant under the dynamical subgroup of the general diffeomorphism group. Here, I focus on an algebraic implementation of the dynamical subgroup of dilations. Introducing an small and large scale cutoffs algebraically, their usual a priori conflict with general covariance is avoided. Thereby, a commutant duality between the minimal and maximal algebra is proposed. This allows to extract the modular structure, which is again related to the dilations.

Based on the concept of manifold valued generalized functions we initiate a study of nonlinear ordinary differential equations with singular (in particular: distributional) right hand sides in a global setting. After establishing several existence and uniqueness results for solutions of such equations and flows of singular vector fields we compare the solution concept employed here with the purely distributional setting. Finally, we derive criteria securing that a sequence of smooth flows corresponding to a regularization of a given singular vector field converges to a measurable limiting flow.

The paper outlines procedures for extending the de Casteljau de Boor and Aitken algorithms in such a way as to allow the construction on a Riemannian manifold of curves analogous to Bezier, B-spline, and Lagrange curves. These curves lie in the manifold and respect intrinsic geometry.

We develop a general theory of log spaces, in which one can make sense of the basic notions of logarithmic geometry, in the sense of Fontaine-Illusie-Kato. Many of our general constructions with log spaces are new, even in the algebraic setting. In the differentiable setting, our theory yields a framework for treating manifolds with corners generalizing recent work of Kottke-Melrose. We give a treatment of the theory of fans, which are to monoids as schemes are to rings. By adapting similar results from logarithmic algebraic geometry, we prove a general result on resolution of toric singularities which can be used to resolve singularities of a wide class of "log smooth" spaces.

In this paper a general methodology is introduced for the determination of potential prototype curves used for the drawing of prehistoric wall-paintings. The approach includes a) preprocessing of the wall-paintings contours to properly partition them, according to their curvature, b) choice of prototype curves families, c) analysis and optimization in 4-manifold for a first estimation of the form of these prototypes, d) clustering of the contour parts and the prototypes, to determine a minimal number of potential guides, e) further optimization in 4-manifold, applied to each cluster separately, in order to determine the exact functional form of the potential guides, together with the corresponding drawn contour parts. The introduced methodology simultaneously deals with two problems: a) the arbitrariness in data-points orientation and b) the determination of one proper form for a prototype curve that optimally fits the corresponding contour data. Arbitrariness in orientation has been dealt with a novel curvature based error, while the proper forms of curve prototypes have been exhaustively determined by embedding curvature deformations of the prototypes into 4-manifolds. Application of this methodology to celebrated wall-paintings excavated at Tyrins, Greece and the Greek island of Thera, manifests it is highly probable that these wall-paintings had been drawn by means of geometric guides that correspond to linear spirals and hyperbolae. These geometric forms fit the drawings' lines with an exceptionally low average error, less than 0.39mm. Hence, the approach suggests the existence of accurate realizations of complicated geometric entities, more than 1000 years before their axiomatic formulation in Classical Ages.

In this paper a general methodology is introduced for the determination of potential prototype curves used for the drawing of prehistoric wall-paintings. The approach includes a) preprocessing of the wall-paintings contours to properly partition them, according to their curvature, b) choice of prototype curves families, c) analysis and optimization in 4-manifold for a first estimation of the form of these prototypes, d) clustering of the contour parts and the prototypes, to determine a minimal number of potential guides, e) further optimization in 4-manifold, applied to each cluster separately, in order to determine the exact functional form of the potential guides, together with the corresponding drawn contour parts. The introduced methodology simultaneously deals with two problems: a) the arbitrariness in data-points orientation and b) the determination of one proper form for a prototype curve that optimally fits the corresponding contour data. Arbitrariness in orientation has been dealt with a novel curvature based error, while the proper forms of curve prototypes have been exhaustively determined by embedding curvature deformations of the prototypes into 4-manifolds. Application of this methodology to celebrated wall-paintings excavated at Tyrins, Greece and the Greek island of Thera, manifests it is highly probable that these wall-paintings had been drawn by means of geometric guides that correspond to linear spirals and hyperbolae. These geometric forms fit the drawings' lines with an exceptionally low average error, less than 0.39mm. Hence, the approach suggests the existence of accurate realizations of complicated geometric entities, more than 1000 years before their axiomatic formulation in Classical Ages.

We consider D-branes in string theory and address the issue of how to describe them mathematically as a fundamental object (as opposed to a solitonic object) of string theory in the realm in differential and symplectic geometry. The notion of continuous maps, $k$-times differentiable maps, and smooth maps from an Azumaya/matrix manifold with a fundamental module to a (commutative) real manifold $Y$ is developed. Such maps are meant to describe D-branes or matrix branes in string theory when these branes are light and soft with only small enough or even zero brane-tension. When $Y$ is a symplectic manifold (resp. a Calabi-Yau manifold; a $7$-manifold with $G_2$-holonomy; a manifold with an almost complex structure $J$), the corresponding notion of Lagrangian maps (resp. special Lagrangian maps; associative maps, coassociative maps; $J$-holomorphic maps) are introduced. Indicative examples linking to symplectic geometry and string theory are given. This provides us with a language and part of the foundation required to study themes, new or old, in symplectic geometry and string theory, including (1) $J$-holomorphic D-curves (with or without boundary), (2) quantization and dynamics of D-branes in string theory, (3) a definition of Fukaya category guided by Lagrangian maps from Azumaya manifolds with a fundamental module with a connection, (4) a theory of fundamental matrix strings or D-strings, and (5) the nature of Ramond-Ramond fields in a space-time. The current note D(11.1) is the symplectic/differential-geometric counterpart of the more algebraic-geometry-oriented first two notes D(1) ([L-Y1]) (arXiv:0709.1515 [math.AG]) and D(2) ([L-L-S-Y], with Si Li and Ruifang Song) (arXiv:0809.2121 [math.AG]) in this project.

We develop a family of infinite-dimensional (i.e.~non-parametric) manifolds of probability measures. The latter are defined on underlying Banach spaces, and have densities of class $C_b^k$ with respect to appropriate reference measures. The case $k=\infty$, in which the manifolds are modelled on Fr\'{e}chet spaces, is included. The manifolds admit the Fisher-Rao metric and the dually flat geometry of Amari's $\alpha$-covariant derivatives, for all $\alpha\in R$. By construction, they are $C^\infty$-embedded submanifolds of particular manifolds of finite measures. Unusually for the non-parametric case, the likelihood function associated with a finite sample is a continuous function on each of the manifolds.

The fixed point index of topological fixed point theory is a well studied integer-valued algebraic invariant of a mapping which can be characterized by a small set of axioms. The coincidence index is an extension of the concept to topological (Nielsen) coincidence theory. We demonstrate that three natural axioms are sufficient to characterize the coincidence index in the setting of continuous mappings on oriented differentiable manifolds, the most common setting for Nielsen coincidence theory.