such that a -hyperbolic space in the first sense is -hyperbolic in the second, and vice versa. Thus the notion of a hyperbolic space is independent of the chosen definition. Examples The hyperbolic plane is hyperbolic: in fact the incircle of a geodesic triangle is the circle of largest diameter contained in the triangle and every geodesic triangle lies in the interior of an ideal triangle, all of which are isometric with incircles of diameter 2 log 3. Note that in this case the Gromov product also has a simple interpretation in terms of the incircle of a geodesic triangle. In fact the quantity is just the hyperbolic distance from to either of the points of contact of the incircle with the adjacent sides: for from the diagram , so that . The Euclidean plane is not hyperbolic, for example because of the existence of homotheties. Two "degenerate" examples of hyperbolic spaces are spaces with bounded diameter (for example finite or compact spaces) and the real line. Metric trees and more generally real trees are the simplest interesting examples of hyperbolic spaces as they are 0-hyperbolic (i.e. all triangles are tripods). The 1-skeleton of the triangulation by Euclidean equilateral triangles is not hyperbolic (it is in fact quasi-isometric to the Euclidean plane). A triangulation of the plane has a hyperbolic 1-skeleton if every vertex has degree 7 or more. The two-dimensional grid is not hyperbolic (it is quasi-isometric to the Euclidean plane). It is the Cayley graph of the fundamental group of the torus; the Cayley graphs of the fundamental groups of a surface of higher genus is hyperbolic (it is in fact quasi-isometric to the hyperbolic plane). Hyperbolicity and curvature The hyperbolic plane (and more generally any Hadamard manifolds of sectional curvature ) is -hyperbolic. If we scale the Riemannian metric by a factor then the distances are multiplied by and thus we get a space that is -hyperbolic. Since the curvature is multiplied by we see that in this example the more (negatively) curved the space is, the lower the hyperbolicity constant. Similar examples are CAT spaces of negative curvature. With respect to curvature and hyperbolicity it should be noted however that while curvature is a property that is essentially local, hyperbolicity is a large-scale property which does not see local (i.e. happening in a bounded region) metric phenomena. For example, the union of an hyperbolic space with