mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the Gromov product is a concept in the theory of

metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...

s named after the mathematician Mikhail Gromov. The Gromov product can also be used to define ''δ''-hyperbolic metric spaces in the sense of Gromov.

Definition

Let (''X'', ''d'') be a metric space and let ''x'', ''y'', ''z'' ∈ ''X''. Then the Gromov product of ''y'' and ''z'' at ''x'', denoted (''y'', ''z'')_''x'', is defined by :

(y, z)_ = \frac1 \big( d(x, y) + d(x, z) - d(y, z) \big).

Motivation

Given three points ''x'', ''y'', ''z'' in the metric space ''X'', by the triangle inequality there exist non-negative numbers ''a'', ''b'', ''c'' such that

d(x,y) = a + b, \ d(x,z) = a + c, \ d(y,z) =  b + c

. Then the Gromov products are

(y,z)_x = a, \ (x,z)_y = b, \ (x,y)_z = c

. In the case that the points ''x'', ''y'', ''z'' are the outer nodes of a

tripod A tripod is a portable three-legged frame or stand, used as a platform for supporting the weight and maintaining the stability of some other object. The three-legged (triangular stance) design provides good stability against gravitational loads ...

then these Gromov products are the lengths of the edges. In the hyperbolic, spherical or euclidean plane, the Gromov product (''A'', ''B'')_''C'' equals the distance ''p'' between ''C'' and the point where the

incircle In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter ...

of the geodesic triangle ''ABC'' touches the edge ''CB'' or ''CA''. Indeed from the diagram , so that . Thus for any metric space, a geometric interpretation of (''A'', ''B'')_''C'' is obtained by isometrically embedding (A, B, C) into the euclidean plane.

Properties

* The Gromov product is symmetric: (''y'', ''z'')_''x'' = (''z'', ''y'')_''x''. * The Gromov product degenerates at the endpoints: (''y'', ''z'')_''y'' = (''y'', ''z'')_''z'' = 0. * For any points ''p'', ''q'', ''x'', ''y'' and ''z'', ::

d(x, y) = (x, z)_ + (y, z)_,

0 \leq (y, z)_ \leq \min \big\,

\big,  (y, z)_ - (y, z)_ \big,  \leq d(p, q),

\big,  (x, y)_ - (x, z)_ \big,  \leq d(y, z).

Points at infinity

Consider

hyperbolic space In mathematics, hyperbolic space of dimension ''n'' is the unique simply connected, ''n''-dimensional Riemannian manifold of constant sectional curvature equal to −1. It is homogeneous, and satisfies the stronger property of being a symme ...

H^''n''. Fix a base point ''p'' and let

x_\infty

and

y_\infty

be two distinct points at infinity. Then the limit ::

\liminf_ (x,y)_p

exists and is finite, and therefore can be considered as a generalized Gromov product. It is actually given by the formula ::

(x_\infty, y_\infty)_ = \log \csc (\theta/2),

where

\theta

is the angle between the

geodesic In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a conn ...

rays

px_\infty

and

py_\infty

δ-hyperbolic spaces and divergence of geodesics

The Gromov product can be used to define ''δ''-hyperbolic spaces in the sense of Gromov.: (''X'', ''d'') is said to be ''δ''-hyperbolic if, for all ''p'', ''x'', ''y'' and ''z'' in ''X'', ::

(x, z)_ \geq \min \big\ - \delta.

In this case. Gromov product measures how long geodesics remain close together. Namely, if ''x'', ''y'' and ''z'' are three points of a ''δ''-hyperbolic metric space then the initial segments of length (''y'', ''z'')_''x'' of geodesics from ''x'' to ''y'' and ''x'' to ''z'' are no further than 2''δ'' apart (in the sense of the

Hausdorff distance In mathematics, the Hausdorff distance, or Hausdorff metric, also called Pompeiu–Hausdorff distance, measures how far two subsets of a metric space are from each other. It turns the set of non-empty set, non-empty compact space, compact subsets o ...

between closed sets).

Notes

References

* * * {{Metric spaces Hyperbolic metric space Metric geometry