Lots of sections

Point section set $p$ ${\ displaystyle p}$ $p$ in the Riemannian manifold $M$ ${\ displaystyle M}$ $M$ - a subset of points $\operatorname {C} _{p}\subset M$ ${\ displaystyle \ operatorname {C} _ {p} \ subset M}$ ${\ displaystyle \ operatorname {C} _ {p} \ subset M}$ through which not a single shortest of $p$ ${\ displaystyle p}$ $p$ .

A lot of the section is also called Catlocus , from the English. cut locus .

Content

Examples

Point section set $p$ ${\ displaystyle p}$ standard sphere consists of a point opposite $p$ ${\ displaystyle p}$ .
The set of dividing a point on the surface of an infinite circular cylinder is a straight line parallel to the axis of the cylinder passing along the surface of the cylinder from the side opposite to the selected point.

Properties

A section set is a closed set .
Many sections have zero volume.
Subset $M\backslash \operatorname {C} _{p}$ ${\ displaystyle M \ backslash \ operatorname {C} _ {p}}$ diffeomorphic to the ball.
If between points $p$ ${\ displaystyle p}$ and $q$ ${\ displaystyle q}$ there are two different shortest arcs then $p\in \operatorname {C} _{q}$ ${\ displaystyle p \ in \ operatorname {C} _ {q}}$ and $q\in \operatorname {C} _{p}$ ${\ displaystyle q \ in \ operatorname {C} _ {p}}$ .
If a $p\in \operatorname {C} _{q}$ ${\ displaystyle p \ in \ operatorname {C} _ {q}}$ and shortest $\gamma$ ${\ displaystyle \ gamma}$ between points $p$ ${\ displaystyle p}$ and $q$ ${\ displaystyle q}$ is unique, then they are conjugate on the continuation $\gamma$ ${\ displaystyle \ gamma}$ .
If a $M$ {\ displaystyle M} Is an analytic Riemannian manifold, then the section set $C$ $p$ {\ displaystyle \ operatorname {C} _ {p}} admits locally finite triangulation to open analytic simplexes.
- Without analyticity $M$ ${\ displaystyle M}$ lots of $\operatorname {C} _{p}$ ${\ displaystyle \ operatorname {C} _ {p}}$ may even be non-triangulated .
The distance from a point to its partition set is equal to the radius of injectivity of this point.

Literature

Burago Yu.D., Zalgaller V.A. Introduction to Riemannian geometry. - St. Petersburg: Nauka, 1994 .-- 318 p.

Lots of sections

Content

Examples

Properties

See also

Literature

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