def curvature_at_identity(
self, tangent_vec_a, tangent_vec_b, tangent_vec_c):
r"""Compute the curvature at identity.
For three tangent vectors at identity :math: `x,y,z`,
the curvature is defined by
:math: `R(x, y)z = \nabla_{[x,y]}z
- \nabla_x\nabla_y z + - \nabla_y\nabla_x z`.
Parameters
----------
tangent_vec_a : array-like, shape=[..., n, n]
Tangent vector at identity.
tangent_vec_b : array-like, shape=[..., n, n]
Tangent vector at identity.
tangent_vec_c : array-like, shape=[..., n, n]
Tangent vector at identity.
Returns
-------
curvature : array-like, shape=[..., n, n]
Tangent vector at identity.
"""
bracket = GeneralLinear.bracket(tangent_vec_a, tangent_vec_b)
bracket_term = self.connection_at_identity(bracket, tangent_vec_c)
left_term = self.connection_at_identity(
tangent_vec_a, self.connection_at_identity(
tangent_vec_b, tangent_vec_c))
right_term = self.connection_at_identity(
tangent_vec_b, self.connection_at_identity(
tangent_vec_a, tangent_vec_c))
return bracket_term - left_term + right_term
def connection_at_identity(self, tangent_vec_a, tangent_vec_b):
r"""Compute the Levi-Civita connection at identity.
For two tangent vectors at identity :math: `x,y`, one can associate
left (respectively right) invariant vector fields :math: `\tilde{x},
\tilde{y}`. Then the vector :math: `(\nabla_\tilde{x}(\tilde{x}))_{
Id}` is computed using the lie bracket and the dual adjoint map. This
is a bilinear map that characterizes the connection [Gallier]_.
Parameters
----------
tangent_vec_a : array-like, shape=[..., n, n]
Tangent vector at identity.
tangent_vec_b : array-like, shape=[..., n, n]
Tangent vector at identity.
Returns
-------
nabla : array-like, shape=[..., n, n]
Tangent vector at identity.
References
----------
.. [Gallier] Gallier, Jean, and Jocelyn Quaintance. Differential
Geometry and Lie Groups: A Computational Perspective.
Geonger International Publishing, 2020.
https://doi.org/10.1007/978-3-030-46040-2.
"""
sign = 1. if self.left_or_right == 'left' else -1.
return sign / 2 * (GeneralLinear.bracket(tangent_vec_a, tangent_vec_b)
- self.dual_adjoint(tangent_vec_a, tangent_vec_b)
- self.dual_adjoint(tangent_vec_b, tangent_vec_a))
def structure_constant(self, tangent_vec_a, tangent_vec_b, tangent_vec_c):
r"""Compute the structure constant of the metric.
For three tangent vectors :math: `x, y, z` at identity,
compute :math: `<[x,y], z>`.
Parameters
----------
tangent_vec_a : array-like, shape=[..., n, n]
Tangent vector at identity.
tangent_vec_b : array-like, shape=[..., n, n]
Tangent vector at identity.
tangent_vec_c : array-like, shape=[..., n, n]
Tangent vector at identity.
Returns
-------
structure_constant : array-like, shape=[...,]
"""
bracket = GeneralLinear.bracket(tangent_vec_a, tangent_vec_b)
return self.inner_product_at_identity(bracket, tangent_vec_c)
def connection_at_identity(self, tangent_vec_a, tangent_vec_b):
r"""Compute the Levi-Civita connection at identity.
For two tangent vectors at identity :math: `x,y`, one can associate
left (respectively right) invariant vector fields :math: `\tilde{x},
\tilde{y}`. Then the vector :math: `(\nabla_\tilde{x}(\tilde{x}))_{
Id}` is computed using the lie bracket and the dual adjoint map. This
is a bilinear map that characterizes the connection.
Parameters
----------
tangent_vec_a : array-like, shape=[..., dim]
Tangent vector at identity.
tangent_vec_b : array-like, shape=[..., dim]
Tangent vector at identity.
Returns
-------
nabla : array-like, shape=[..., dim]
Tangent vector at identity.
"""
return 1. / 2 * (GeneralLinear.bracket(tangent_vec_a, tangent_vec_b) -
self.dual_adjoint(tangent_vec_a, tangent_vec_b) -
self.dual_adjoint(tangent_vec_b, tangent_vec_a))