Equi-distant

of a set $M$ in a metric space $R$

The boundary of the tubular neighbourhood of $M$ in $R$ consisting of the balls of the same radius $d$ with centres in $M$. If $M$ is a differentiable submanifold $M^k$ in a Riemannian space $V^n$, then the equi-distant of $M$ is given (in a more restricted sense) by the set of end-points of the segments of equal length measured from $M^k$ on the geodesics perpendicular to $M^k$ at the corresponding points. If $V^n$ is complete, then the equi-distant is the image under the exponential mapping of the vectors of constant length $d$ in the normal bundle of $M^k$ in $V^n$. If $V^n$ is not complete, then the equi-distant exists only for sufficiently small values of $d$.

Examples of equi-distants. 1) An equi-distant in the Lobachevskii plane (a hypercycle) is the orthogonal trajectory of the pencil of straight lines perpendicular to some straight line (to a basic line, or basis). The equi-distant consists of two branches situated on different sides from the basis line and concave towards the basis. The curvature of the equi-distant is constant. 2) An equi-distant in the Lobachevskii space is a surface of constant positive exterior curvature.