Homogeneous metric

In mathematics, a metric or distance function is a function that defines a distance between each pair of elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set, but not all topologies can be generated by a metric. A topological space whose topology can be described by a metric is called metrizable.

An important source of metrics in differential geometry are metric tensors, bilinear forms that may be defined from the tangent vectors of a differentiable manifold onto a scalar. A metric tensor allows distances along curves to be determined through integration, and thus determines a metric. However, not every metric comes from a metric tensor in this way.

A metric on a set X is a function (called the distance function or simply distance)

where ${\displaystyle [0,\infty )}$ is the set of non-negative real numbers (because distance can't be negative so we can't use ${\displaystyle \mathbb {R} }$), and for all ${\displaystyle x,y,z\in X}$, the following conditions are satisfied:

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