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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]


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]


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

Geometry perspective of assignment

Assignment and Laplacian

Optimization of Dirichlet energy


  1. 1Rock (, What is the volume of the sphere in hyperbolic space?, URL (version: 2020-05-01):