All posts under category "Mathematics / Math.DG"
In this note we give a simple, model-independent construction of Chern classes as natural transformations from differential complex K-theory to differential integral cohomology. We verify the expected behaviour of these Chern classes with respect to sums and suspension.
We discuss some properties of the distance functions on Riemannian manifolds and we relate their behavior to the geometry of the manifolds. This leads to alternative proofs of some 'classical' theorems connecting curvature and topology.
We find a necessary and sufficient condition for a compact 7-manifold to admit a $tilde G_2$-structure. As a result we find a sufficient condition for an open 7-manifold to admit a closed 3-form of $tilde G_2$-type.
We construct an analytic multiplicative model of smooth K-theory. We further introduce the notion of a smooth K-orientation of a proper submersion and define the associated push-forward which satisfies functoriality, compatibility with pull-back diagrams, and projection and bordism formulas. We cons
We consider the four-dimensional nonholonomic distribution defined by the 4-potential of the electromagnetic field on the manifold. This distribution has a metric tensor with the Lorentzian signature $(+,-,-,-)$, therefore, the causal structure appears as in the general relativity theory. By means o
We establish the existence of a non-trivial, branched immersion of a closed Riemann surface $Σ$ with constant mean curvature (CMC) $H$ into any closed, orientable 3-manifold $mathcal{M}$, for almost every prescribed value of $H$. The genus of the surface $Σ$ is bounded from above by the Heegaard gen
In this paper, we provide the lower bounds of the first non-zero basic eigenvalue on a closed singular Riemannian manifold $(M,mathcal{F})$ with basic mean curvature that depends on the given non-negative lower bound of the Ricci curvature of $M$ and the diameter of the leaf space $M/mathcal{F}$. Th
We study the geometry and topology of compact submanifolds of Riemannian manifolds with arbitrary codimension that satisfy a pinching condition. The ambient space is not required to be a space form, and the pinching condition involves the length of the second fundamental form and the mean curvature.
Motivated by recent developments in the theory of gravitation, we revisit the idea of topological variations, originally introduced by Wheeler and Hawking, from a rigorous perspective. Starting from a localized version of the Einstein-Hilbert variational principle, we encode the key aspects of the v
Whereas subriemannian geometry usually deals with smooth horizontal distributions, partially hyperbolic dynamical systems provide many examples of subriemannian geometries defined by non-smooth (namely, H'older continuous) distributions. These distributions are of great significance for the behavior
In this paper, we study the L'{e}vy-Milman concentration phenomenon of 1-Lipschitz maps into infinite dimensional metric spaces. Our main theorem asserts that the concentration to an infinite dimensional $ell^p$-ball with the $ell^q$-distance function for $1leq p<qleq +infty$ is equivalent to the co
A well known result on pseudodifferential operators states that the noncommutative residue (Wodzicki residue) of a pseudodifferential projection vanishes. This statement is non-local and implies the regularity of the eta invariant at zero of Dirac type operators. We prove that in a filtered algebra
Motivated by various results on homogeneous geodesics of Riemannian spaces, we study homogeneous trajectories, i.e. trajectories which are orbits of a one-parameter symmetry group, of Lagrangian and Hamiltonian systems. We present criteria under which an orbit of a one-parameter subgroup of a symmet
We prove a discrete Gauss-Bonnet-Chern theorem which states where summing the curvature over all vertices of a finite graph G=(V,E) gives the Euler characteristic of G.
We consider the polynomial representation S(V*) of the rational Cherednik algebra H_c(W) associated to a finite Coxeter group W at constant parameter c. We show that for any degree d of W and nonnegative integer m the space S(V*) contains a single copy of the reflection representation V of W spanned
In his 1979 paper Trotman proves, using the techniques of the Thom transversality theorem, that under some conditions on the dimensions of the manifolds under consideration, openness of the set of maps transverse to a stratification in the strong (Whitney) topology implies that the stratification is
We calculate explicitly the quadratic solution to the WDVV equations corresponds to the quasi-Coxeter conjugacy class $E_8(a_1)$ using the associated classical $W$-algebra.
We develop a formula for the equivariant index of a twisted Dirac operator on a compact globally hyperbolic spacetime with timelike boundary on which a group acts isometrically, subject to APS boundary conditions. The formula is the same as in the Riemannian case: the equivariant index for a group e
We prove that an arbitrary Poisson structure omega^{ij}(u) and an arbitrary closed 3-form T_{ijk}(u) generate the local Poisson structure A^{ij}(u,u_x) = M^i_s(u,u_x)omega^{sj}(u), where M^i_s(u,u_x)(delta^s_j + omega^{sp}(u)T_{pjk}(u)u^k_x) = delta^i_j, on the corresponding loop space. We obtain al
We outline the notions and concepts of the calculus of variational multivectors within the Poisson formalism over the spaces of infinite jets of mappings from commutative (non)graded smooth manifolds to the factors of noncommutative associative algebras over the equivalence under cyclic permutations
We give a parameterization of Alfred Gray's Elliptical Catenoid and Elliptical Hellicoid using Jacobi's elliptic functions. This parameterization avoids some problems present in the original depiction of these surfaces.
Given a family of (almost) disjoint strictly convex subsets of a complete negatively curved Riemannian manifold M, such as balls, horoballs, tubular neighborhoods of totally geodesic submanifolds, etc, the aim of this paper is to construct geodesic rays or lines in M which have exactly once an exact
We define the thin fundamental categorical group ${mathcal P}_2(M,*)$ of a based smooth manifold $(M,*)$ as the categorical group whose objects are rank-1 homotopy classes of based loops on $M$, and whose morphisms are rank-2 homotopy classes of homotopies between based loops on $M$. Here two maps a
Let $G$ be a finite group and $Y$ a $G$-gerbe over an orbifold $B$. A disconnected orbifold $hat{Y}$ and a flat U(1)-gerbe $c$ on $hat{Y}$ is canonically constructed from $Y$. Motivated by a proposal in physics, we study a mathematical duality between the geometry of the $G$-gerbe $Y$ and the geomet
We investigate non-degenerate Lagrangians of the form $$ int f(u_x, u_y, u_t) dx dy dt $$ such that the corresponding Euler-Lagrange equations $ (f_{u_x})_x+ (f_{u_y})_y+ (f_{u_t})_t=0 $ are integrable by the method of hydrodynamic reductions. We demonstrate that the integrability condit
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