92 lines
2.5 KiB
Python
92 lines
2.5 KiB
Python
|
r'''
|
||
|
|
unit test describing the hyperbolic half-plane with the Poincare metric. This
|
||
|
|
is a basic model of hyperbolic geometry on the (positive) half-space
|
||
|
|
|
||
|
|
{(x,y) \in R^2 | y > 0}
|
||
|
|
|
||
|
|
with the Riemannian metric
|
||
|
|
|
||
|
|
ds^2 = (dx^2 + dy^2)/y^2
|
||
|
|
|
||
|
|
It has constant negative scalar curvature = -2
|
||
|
|
|
||
|
|
https://en.wikipedia.org/wiki/Poincare_half-plane_model
|
||
|
|
'''
|
||
|
|
from sympy.matrices.dense import diag
|
||
|
|
from sympy.diffgeom import (twoform_to_matrix,
|
||
|
|
metric_to_Christoffel_1st, metric_to_Christoffel_2nd,
|
||
|
|
metric_to_Riemann_components, metric_to_Ricci_components)
|
||
|
|
import sympy.diffgeom.rn
|
||
|
|
from sympy.tensor.array import ImmutableDenseNDimArray
|
||
|
|
|
||
|
|
|
||
|
|
def test_H2():
|
||
|
|
TP = sympy.diffgeom.TensorProduct
|
||
|
|
R2 = sympy.diffgeom.rn.R2
|
||
|
|
y = R2.y
|
||
|
|
dy = R2.dy
|
||
|
|
dx = R2.dx
|
||
|
|
g = (TP(dx, dx) + TP(dy, dy))*y**(-2)
|
||
|
|
automat = twoform_to_matrix(g)
|
||
|
|
mat = diag(y**(-2), y**(-2))
|
||
|
|
assert mat == automat
|
||
|
|
|
||
|
|
gamma1 = metric_to_Christoffel_1st(g)
|
||
|
|
assert gamma1[0, 0, 0] == 0
|
||
|
|
assert gamma1[0, 0, 1] == -y**(-3)
|
||
|
|
assert gamma1[0, 1, 0] == -y**(-3)
|
||
|
|
assert gamma1[0, 1, 1] == 0
|
||
|
|
|
||
|
|
assert gamma1[1, 1, 1] == -y**(-3)
|
||
|
|
assert gamma1[1, 1, 0] == 0
|
||
|
|
assert gamma1[1, 0, 1] == 0
|
||
|
|
assert gamma1[1, 0, 0] == y**(-3)
|
||
|
|
|
||
|
|
gamma2 = metric_to_Christoffel_2nd(g)
|
||
|
|
assert gamma2[0, 0, 0] == 0
|
||
|
|
assert gamma2[0, 0, 1] == -y**(-1)
|
||
|
|
assert gamma2[0, 1, 0] == -y**(-1)
|
||
|
|
assert gamma2[0, 1, 1] == 0
|
||
|
|
|
||
|
|
assert gamma2[1, 1, 1] == -y**(-1)
|
||
|
|
assert gamma2[1, 1, 0] == 0
|
||
|
|
assert gamma2[1, 0, 1] == 0
|
||
|
|
assert gamma2[1, 0, 0] == y**(-1)
|
||
|
|
|
||
|
|
Rm = metric_to_Riemann_components(g)
|
||
|
|
assert Rm[0, 0, 0, 0] == 0
|
||
|
|
assert Rm[0, 0, 0, 1] == 0
|
||
|
|
assert Rm[0, 0, 1, 0] == 0
|
||
|
|
assert Rm[0, 0, 1, 1] == 0
|
||
|
|
|
||
|
|
assert Rm[0, 1, 0, 0] == 0
|
||
|
|
assert Rm[0, 1, 0, 1] == -y**(-2)
|
||
|
|
assert Rm[0, 1, 1, 0] == y**(-2)
|
||
|
|
assert Rm[0, 1, 1, 1] == 0
|
||
|
|
|
||
|
|
assert Rm[1, 0, 0, 0] == 0
|
||
|
|
assert Rm[1, 0, 0, 1] == y**(-2)
|
||
|
|
assert Rm[1, 0, 1, 0] == -y**(-2)
|
||
|
|
assert Rm[1, 0, 1, 1] == 0
|
||
|
|
|
||
|
|
assert Rm[1, 1, 0, 0] == 0
|
||
|
|
assert Rm[1, 1, 0, 1] == 0
|
||
|
|
assert Rm[1, 1, 1, 0] == 0
|
||
|
|
assert Rm[1, 1, 1, 1] == 0
|
||
|
|
|
||
|
|
Ric = metric_to_Ricci_components(g)
|
||
|
|
assert Ric[0, 0] == -y**(-2)
|
||
|
|
assert Ric[0, 1] == 0
|
||
|
|
assert Ric[1, 0] == 0
|
||
|
|
assert Ric[0, 0] == -y**(-2)
|
||
|
|
|
||
|
|
assert Ric == ImmutableDenseNDimArray([-y**(-2), 0, 0, -y**(-2)], (2, 2))
|
||
|
|
|
||
|
|
## scalar curvature is -2
|
||
|
|
#TODO - it would be nice to have index contraction built-in
|
||
|
|
R = (Ric[0, 0] + Ric[1, 1])*y**2
|
||
|
|
assert R == -2
|
||
|
|
|
||
|
|
## Gauss curvature is -1
|
||
|
|
assert R/2 == -1
|