Assignment

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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]

(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

  1. 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