Let M be a complete metric space. If C∗(M) admits an isometric shift, then M is separable.

We study the geometry of the space of densities Dens(M), which is the quotient space Diff(M)/Diffμ(M) of the diffeomorphism group of a compact manifold M by the subgroup of volume-preserving diffemorphisms, endowed with a right-invariant homogeneous Sobolev Ḣ-metric. We construct an explicit isometry from this space to (a subset of) an infinite-dimensional sphere and show that the associated Eu...

Let X be a compact metric space which is locally absolutely retract and let ϕ: C(X) → C(Y,M(n)) be a unital homomorphism, where Y is a compact metric space with dim Y ≤ 2. It is proved that there exists a sequence of n continuous maps α(i,m): Y → X (i = 1,2,…,n) and a sequence of sets of mutually orthogonal rank-one projections {p(1,m),p(2,m),…,p(n,m)} C(Y,M(n)) such that [see formula]. This is...

‎in this paper we study the existence of fixed points for mappings defined on complete metric spaces‎, ‎satisfying a general contractive inequality depending on two additional mappings‎.

a projective parameter of a geodesic as solution of certain ode is defined to be a parameter which is invariant under projective change of metric. using projective parameter and poincaré metric, an intrinsic projectively invariant pseudo-distance can be constructed. in the present work, solutions of the above ode are characterized with respect to the sign of parallel ricci tensor on a finsler s...

in this paper, tripled coincidence points of mappings satisfying  -contractive conditions in the framework of partially ordered gb-metric spaces are obtained. our results extend the results of aydi et al. [h. aydi, e. karapnar and w. shatanawi, tripled xed point results in generalized metric space, j. applied math., volume 2012, article id 314279, 10 pages]. moreover, some examples of the mai...

Let M be a metric space with a finite diameter D and a finite normalized measure μM. Let the Hilbert space L2(M) of complex-valued functions be decomposed into a countable (whenM is infinite) or finite (with D+ 1 members whenM is finite) direct sum of mutually orthogonal subspaces L2(M) = V0 ⊕ V1 ⊕ · · · (V0 is the space of constant functions). We denote N = D + 1 ifM is finite and N = ∞ otherw...

In 1912 Bieberbach proved that every compact flat Riemannian manifold M is finitely covered by a flat torus. More precisely, M has the form (F\G)/H where G is a group of translations of Euclidean space, F c G is a discrete subgroup, and H is a finite group of isometries of the space of right cosets F\G. For a proof see e.g. Wolf [18]. The condition that M has a flat Riemannian metric can be sep...

Let S be a surface of finite type; that is, a closed, oriented surface with a finite (possibly empty) set of points removed. In this paper we classify (globally) geodesic rays in the moduli space M(S) of Riemann surfaces, endowed with the Teichmüller metric, and we determine precisely how pairs of rays asymptote. We then use these results to relate two important but disparate topics in the stud...

These equations are the simplest equations for Ricci-flat 4-manifolds. They have been extensively studied in the physics literature on classical relativity, where the solutions represent space-times outside regions of matter which are translation and reflection invariant in the time direction t. However, with the exception of some notable instances, (c.f. Theorem 0.1 below), many of the global ...