# Assignment

**Assignment** is a mapping from the Riemannian manifold in discussion to a number field, i.e. it assign each point on the Riemannian to a number. Assignment play a central role to take analysis perspective into account in the research context.

## Assignment along gradient lines

Following flow equation in the local gradient-contour coordinate system, we can get

[math]\displaystyle{ a = \pm \frac{\mu}{\lambda} \sinh(c \pm \lambda s) }[/math] |
(1) |

Notice that if we let [math]\displaystyle{ c = 0 }[/math], i.e., the above formula gives us the relationship how an assignment grow when the point departs from a zero point.

And we also know, the area of a circle with radius [math]\displaystyle{ r }[/math] in hyperbolic surface with sectional curvature [math]\displaystyle{ \kappa }[/math] is given by^{[1]}

[math]\displaystyle{ 2\pi \int_0^r \left(\frac{\sinh(\sqrt{\kappa} t)}{\sqrt{\kappa}}\right) \, \mathrm{d}t }[/math] |
(2) |

This formula lead to a very interesting geometry perspective which we describe below.

## Geometry perspective of assignment

## Assignment and Laplacian

## Optimization of Dirichlet energy

## References

- ↑ 1Rock (https://math.stackexchange.com/users/208645/1rock), What is the volume of the sphere in hyperbolic space?, URL (version: 2020-05-01): https://math.stackexchange.com/q/3652822