On isometric and minimal isometric embeddings

In this paper we study critial isometric and minimal isometric embeddings of classes of Riemannian metrics which we call {\it quasi-$k$-curved metrics}. Quasi-$k$-curved metrics generalize the metrics of space forms. We construct explicit examples and prove results about existence and rigidity.Comment: 21 pages, AMSTeX. Significantly changed version of paper originally Titled "On minimal isometric embeddings

Isometric path numbers of graphs

An isometric path between two vertices in a graph $G$ is a shortest path joining them. The isometric path number of $G$, denoted by \ip(G), is the minimum number of isometric paths needed to cover all vertices of $G$. In this paper, we determine exact values of isometric path numbers of complete $r$-partite graphs and Cartesian products of 2 or 3 complete graphs.Comment: 9 page

Global cycle properties in locally isometric graphs

A graph G is locally isometric if the subgraph induced by the neighbourhood of every vertex is an isometric subgraph of G. It is shown that the hamilton cycle problem for locally isometric graphs with maximum degree at most 8 is NP-complete. Structural characterizations of locally isometric graphs, with maximum degree at most 6, that are fully cycle extendable, are established and these results are used to show that locally isometric graphs with maximum degree at most 6 are weakly pancyclic. This proves Ryjacek's conjecture for a subclass of locally connected graphs.Comment: 13 pages three figure

Generalized resolvents of isometric operators in Pontryagin spaces

An isometric operator $V$ in a Pontryagin space $\mathcal{H}$ is called standard, if its domain and the range are nondegenerate subspaces in $\mathcal{H}$. Generalized resolvents of standard isometric operators were described in the paper of A. Dijksma, H. Langer, and H. de Snoo in 1990. In the present paper generalized resolvents of non-standard Pontryagin space isometric operators are described. The method of the proof is based on the notion of boundary triplet of isometric operators in Pontryagin spaces. In the Hilbert space setting the notion of boundary triplet for isometric operators was introduced in the paper of M. Malamud and V. Mogilevskii in 2003

Isometric-path numbers of block graphs

An isometric path between two vertices in a graph G is a shortest path joining them. The isometric-path number of G, denoted by ip(G), is the minimum number of isometric paths required to cover all vertices of G. In this paper, we determine exact values of isometric-path numbers of block graphs. We also give a linear-time algorithm for finding the corresponding paths.Comment: 7 page

C1,αC^{1,\alpha} isometric extensions

In this paper we consider the Cauchy problem for isometric immersions. More precisely, given a smooth isometric immersion of a codimension one submanifold we construct $C^{1,\alpha}$ isometric extensions for any $\alpha<\frac{1}{n(n+1)+1}$ via the method of convex integration.Comment: 25 page

On mm-isometric semigroups, and 22-isometric cogenerators

It is known that a $C_0$-semigroup of Hilbert space operators is $m$-isometric if and only if its generator satisfies a certain condition, which we choose to call $m$-skew-symmetry. This paper contains two main results: We provide a Lumer--Phillips type characterization of generators of $m$-isometric semigroups. This is based on the simple observation that $m$-isometric semigroups are quasicontractive. We also characterize cogenerators of $2$-isometric semigroups. To this end, our main strategy is to construct a functional model for $2$-isometric semigroups with analytic cogenerators. The functional model yields numerous simple examples of $2$-isometric semigroups, but also allows for the construction of a closed, densely defined, $2$-skew-symmetric operator which is not a semigroup generator.Comment: This version has been published with Springer Open Access in Integral Equations and Operator Theory (IEOT). This is a MAJOR REVISION. I wish to discourage anyone from using the previous version of the pape

Quasi-isometric embeddings of symmetric spaces and lattices: reducible case

We study quasi-isometric embeddings of symmetric spaces and non-uniform irreducible lattices in semisimple higher rank Lie groups. We show that any quasi-isometric embedding between symmetric spaces of the same rank can be decomposed into a product of quasi-isometric embeddings into irreducible symmetric spaces. We thus extend earlier rigidity results about quasi-isometric embeddings to the setting of semisimple Lie groups. We also present some examples when the rigidity does not hold, including first examples in which every flat is mapped into multiple flats.Comment: Corrected grammar and typo

The geometry of groups containing almost normal subgroups

A subgroup $H\leq G$ is said to be almost normal if every conjugate of $H$ is commensurable to $H$. If $H$ is almost normal, there is a well-defined quotient space $G/H$. We show that if a group $G$ has type $F_{n+1}$ and contains an almost normal coarse $PD_n$ subgroup $H$ with $e(G/H)=\infty$, then whenever $G'$ is quasi-isometric to $G$, it contains an almost normal subgroup $H'$ that is quasi-isometric to $H$. Moreover, the quotient spaces $G/H$ and $G'/H'$ are quasi-isometric. This generalises a theorem of Mosher-Sageev-Whyte, who prove the case in which $G/H$ is quasi-isometric to a finite valence bushy tree. Using work of Mosher, we generalise a result of Farb-Mosher to show that for many surface group extensions $\Gamma_L$, any group quasi-isometric to $\Gamma_L$ is virtually isomorphic to $\Gamma_L$. We also prove quasi-isometric rigidity for the class of finitely presented $\mathbb{Z}$-by-($\infty$ ended) groups.Comment: 48 page

Global and local boundedness of Polish groups

We present a comprehensive theory of boundedness properties for Polish groups developed with a main focus on Roelcke precompactness (precompactness of the lower uniformity) and Property (OB) (boundedness of all isometric actions on separable metric spaces). In particular, these properties are characterised by the orbit structure of isometric actions on metric spaces and isometric or continuous affine representations on separable Banach spaces