arXiv subjects

Marja Kankaanrinta

Publications and source records attributed to Marja Kankaanrinta.

5 recordsLinked to original sources

arXiv2024

On the $n$-transitivity of the group of equivariant diffeomorphisms

Let $G$ be a Lie group and let $M$ be a proper smooth $G$-manifold. If $M$ is connected and $\dim(M)\geq 2$, the group of diffeomorphisms of $M$, that are isotopic to the identity through a compactly supported isotopy, acts $n$-transitively on $M$, for any $n$. In this paper, we prove a version of the $n$-transitivity result for the group of equivariant diffeomorphisms of $M$. As a corollary we obtain a result concerning diffeomorphisms of the orbit space $M/G$. A special case of the result for orbit spaces gives an $n$-transitivity result for orbifold diffeomorphisms that was earlier proved by F. Pasquotto and T. O. Rot.

math.GT↗

arXiv2014

On uniqueness of differential structures on orbifolds

It is known that every ${\rm C}^r$-orbifold, $1\leq r\leq\infty$, has a compatible ${\rm C}^s$-differential structure, for every $s$, where $r< s\leqω$. We prove that if two reduced ${\rm C}^r$-orbifolds, $2\leq r\leqω$, are ${\rm C}^2$-diffeomorphic, then they are ${\rm C}^r$-diffeomorphic. It follows that the compatible ${\rm C}^s$-differential structure on a reduced ${\rm C}^r$-orbifold, $2\leq r<s\leqω$, is unique up to a ${\rm C}^s$-diffeomorphism.

math.GT↗

arXiv2013

On real analytic orbifolds and Riemannian metrics

We begin by showing that every real analytic orbifold has a real analytic Riemannian metric. It follows that every reduced real analytic orbifold can be expressed as a quotient of a real analytic manifold by a real analytic almost free action of a compact Lie group. We then extend a well-known result of Nomizu and Ozeki concerning Riemannian metrics on manifolds to the orbifold setting: Let $X$ be a smooth (real analytic) orbifold and let $α$ be a smooth (real analytic) Riemannian metric on $X$. Then $X$ has a complete smooth (real analytic) Riemannian metric conformal to $α$.

math.GT↗

arXiv2011

A subanalytic triangulation theorem for real analytic orbifolds

Let $X$ be a real analytic orbifold. Then each stratum of $X$ is a subanalytic subset of $X$. We show that $X$ has a unique subanalytic triangulation compatible with the strata of $X$. We also show that every ${\rm C}^r$-orbifold, $1\leq r\leq \infty$, has a real analytic structure. This allows us to triangulate differentiable orbifolds. The results generalize the subanalytic triangulation theorems previously known for quotient orbifolds.

math.GT↗

arXiv2011

On subanalytic subsets of real analytic orbifolds

The purpose of this paper is to define semi- and subanalytic subsets and maps in the context of real analytic orbifolds and to study their basic properties. We prove results analogous to some well-known results in the manifold case. For example, we prove that if $A$ is a subanalytic subset of a real analytic quotient orbifold $X$, then there is a real analytic orbifold $Y$ of the same dimension as $A$ and a proper real analytic map $f\colon Y\to X$ with $f(Y)=A$. We also study images and inverse images of subanalytic sets and show that if $X$ and $Y$ are real analytic orbifolds and if $f\colon X\to Y$ is a subanalytic map, then the inverse image $f^{-1}(B)$ of any subanalytic subset $B$ of $Y$ is subanalytic. If, in addition, $f$ is proper, then also the image $f(A)$ of any subanalytic subset $A$ of $X$ is proper.

math.GT↗