Related papers: Projections with fixed difference: a Hopf-Rinow th…

The problem in question is whether the quotient space of a compact linear group is a topological manifold and whether it is a homological manifold. In the paper, the case of an infinite group with commutative connected component is…

In this paper, we introduce the metric operator for a compact homogeneous Finsler space, and use it to investigate the geodesic orbit property. We define the notion of standard homogeneous $(\alpha_1,\cdots,\alpha_s)$-metric which…

I consider the class of surfaces $X$ over algebraically closed fields with numerical invariants given in the title. In characteristic zero, this class contains fake projective planes which were introduced by David Mumford. I prove that in…

Let $p:X\rightarrow X/A$ be a quotient map, where $A$ is a subspace of $X$. We explore conditions under which $p_*(\pi_1^{qtop}(X,x_0))$ is dense in $\pi_1^{qtop}(X/A,*))$, where the fundamental groups enjoy the natural quotient topology…

We develop a general structure theory for compact homogeneous Riemannian manifolds in relation to the co-index of symmetry. We will then use these results to classify irreducible, simply connected, compact homogeneous Riemannian manifolds…

Let X be a Hausdorff quotient of a standard space (that is of a locally compact separable metric space). It is shown that the following are equivalent: (i) X is the image of an irreducible quotient map from a standard space; (ii) X has a…

A special linear Lie group over the real number field and the quarternion field admits a projectivley flat affine connection. We show that parabolic subgroups are autoparallel submanifolds and give a criterion the induced connection is…

Let $X$ be a locally 1-connected metric space and $A_1,A_2,...,A_n$ be connected, locally path connected and compact pairwise disjoint subspaces of $X$. In this paper, we show that the quotient space $X/(A_1,A_2,...,A_n)$ obtained from $X$…

A smooth curve on a homogeneous manifold $G/H$ is called a Riemannian equigeo-desic if it is a homogeneous geodesic for any $G$-invariant Riemannian metric. The homogeneous manifold $G/H$ is called Riemannian equigeodesic, if for any $x\in…

The class of statistical manifolds with divisible cubic forms arises from affine differential geometry. We examine the geodesic connectedness of affine connections on this class of statistical manifolds. In information geometry, the…

We propose a simple but effective framework for producing examples of covariant faithfully flat (generalised) Hopf-Galois extensions from a nested pair of quantum homogeneous spaces. Our construction is modelled on the classical situation…

Given any pair $(L,A)$ of Lie algebroids, we construct a differential graded manifold $(L[1]\oplus L/A,Q)$, which we call Fedosov dg manifold. We prove that the cohomological vector field $Q$ constructed on $L[1]\oplus L/A$ by the Fedosov…

We prove the Yau-Tian-Donaldson conjecture for cohomogeneity one manifolds, that is, for projective manifolds equipped with a holomorphic action of a compact Lie group with at least one real hypersurface orbit. Contrary to what seems to be…

Bauer and Catanese \cite{bauercat} have found 4 families of surfaces of general type with $p_g = q = 0$ which are quotients of the product of curves by the action of finite abelian group. We compute integral homology groups of these…

Let $K$ be an infinite field and let $m_1,\ldots,m_n$ be a generalized arithmetic sequence of positive integers, i.e., there exist $h, d, m_1 \in\mathbb{Z}^+$ such that $m_i = h m_1 + (i-1)d$ for all $i \in \{2,\ldots,n\}$. We consider the…

We develop a geometric approach to stable homotopy groups of spheres in the spirit of the work of Pontrjagin and Rokhlin. A new proof of the Hopf Invariant One Theorem by J.F.Adams is obtained in all dimensions except 15 and 31. To prove…

Let $N\subset M$ be a finite Jones' index inclusion of II$_1$ factors, and denote by $U_N\subset U_M$ their unitary groups. In this paper we study the homogeneous space $U_M/U_N$, which is a (infinite dimensional) differentiable manifold,…

A pair $(S,C)$ is called a singular $\mathbb{Q}$-homology plane pair if $S$ is a singular projective surface with only quotient singularities having the same rational homology as $\mathbb{p}^2$ and $C \subset S$ has the same rational…

In this paper we prove that the holonomy group of a simply connected locally projectively flat Finsler manifold of constant curvature is a finite dimensional Lie group if and only if it is flat or it is Riemannian.

We provide specific PDEs for preserved quantities $Q$ in Geometry, as well as a bridge between this and specific PDEs for observables $O$ in Physics. We furthermore prove versions of four other theorems either side of this bridge: the below…