def __init__(self, metric, name, latex_name=None, init_coef=True):
r"""
Construct a Levi-Civita connection.
TESTS:
Levi-Civita connection of the hyperbolic plane::
sage: M = Manifold(2, 'M')
sage: X.<r,ph> = M.chart(r'r:(0,+oo) ph:(0,2*pi)')
sage: g = M.metric('g')
sage: g[0,0], g[1,1] = 1/(1+r^2), r^2
sage: from sage.manifolds.differentiable.levi_civita_connection \
....: import LeviCivitaConnection
sage: nab = LeviCivitaConnection(g, 'nabla', latex_name=r'\nabla')
sage: nab
Levi-Civita connection nabla associated with the Riemannian metric
g on the 2-dimensional differentiable manifold M
sage: TestSuite(nab).run()
"""
AffineConnection.__init__(self, metric.domain(), name, latex_name)
self._metric = metric
# Initialization of the derived quantities:
LeviCivitaConnection._init_derived(self)
if init_coef:
# Initialization of the Christoffel symbols in the top charts on
# the domain (i.e. disregarding the subcharts)
for chart in self._domain.top_charts():
self.coef(chart._frame)
def __init__(self, metric, name, latex_name=None, init_coef=True):
r"""
Construct a Levi-Civita connection.
TESTS:
Levi-Civita connection of the hyperbolic plane::
sage: M = Manifold(2, 'M')
sage: X.<r,ph> = M.chart(r'r:(0,+oo) ph:(0,2*pi)')
sage: g = M.metric('g')
sage: g[0,0], g[1,1] = 1/(1+r^2), r^2
sage: from sage.manifolds.differentiable.levi_civita_connection \
....: import LeviCivitaConnection
sage: nab = LeviCivitaConnection(g, 'nabla', latex_name=r'\nabla')
sage: nab
Levi-Civita connection nabla associated with the Riemannian metric
g on the 2-dimensional differentiable manifold M
sage: TestSuite(nab).run()
"""
AffineConnection.__init__(self, metric.domain(), name, latex_name)
self._metric = metric
# Initialization of the derived quantities:
LeviCivitaConnection._init_derived(self)
if init_coef:
# Initialization of the Christoffel symbols in the top charts on
# the domain (i.e. disregarding the subcharts)
for chart in self._domain.top_charts():
self.coef(chart._frame)
def _del_derived(self):
r"""
Delete the derived quantities.
TEST::
sage: M = Manifold(5, 'M')
sage: g = M.metric('g')
sage: nab = g.connection()
sage: nab._del_derived()
"""
AffineConnection._del_derived(self)
def _del_derived(self):
r"""
Delete the derived quantities.
TESTS::
sage: M = Manifold(5, 'M')
sage: g = M.metric('g')
sage: nab = g.connection()
sage: nab._del_derived()
"""
AffineConnection._del_derived(self)
def _init_derived(self):
r"""
Initialize the derived quantities.
TESTS::
sage: M = Manifold(5, 'M')
sage: g = M.metric('g')
sage: nab = g.connection()
sage: nab._init_derived()
"""
AffineConnection._init_derived(self)
def riemann(self, name=None, latex_name=None):
r"""
Return the Riemann curvature tensor of the connection.
This method redefines
:meth:`sage.manifolds.differentiable.affine_connection.AffineConnection.riemann`
to set some name and the latex_name to the output.
The Riemann curvature tensor is the tensor field `R` of type (1,3)
defined by
.. MATH::
R(\omega, w, u, v) = \left\langle \omega, \nabla_u \nabla_v w
- \nabla_v \nabla_u w - \nabla_{[u, v]} w \right\rangle
for any 1-form `\omega` and any vector fields `u`, `v` and `w`.
INPUT:
- ``name`` -- (default: ``None``) name given to the Riemann tensor;
if none, it is set to "Riem(g)", where "g" is the metric's name
- ``latex_name`` -- (default: ``None``) LaTeX symbol to denote the
Riemann tensor; if none, it is set to "\\mathrm{Riem}(g)", where "g"
is the metric's name
OUTPUT:
- the Riemann curvature tensor `R`, as an instance of
:class:`~sage.manifolds.differentiable.tensorfield.TensorField`
EXAMPLES:
Riemann tensor of the Levi-Civita connection associated with the
metric of the hyperbolic plane (Poincare disk model)::
sage: M = Manifold(2, 'M', start_index=1)
sage: X.<x,y> = M.chart('x:(-1,1) y:(-1,1)') # Cartesian coord. on the Poincare disk
sage: X.add_restrictions(x^2+y^2<1)
sage: g = M.metric('g')
sage: g[1,1], g[2,2] = 4/(1-x^2-y^2)^2, 4/(1-x^2-y^2)^2
sage: nab = g.connection()
sage: riem = nab.riemann(); riem
Tensor field Riem(g) of type (1,3) on the 2-dimensional
differentiable manifold M
sage: riem.display_comp()
Riem(g)^x_yxy = -4/(x^4 + y^4 + 2*(x^2 - 1)*y^2 - 2*x^2 + 1)
Riem(g)^x_yyx = 4/(x^4 + y^4 + 2*(x^2 - 1)*y^2 - 2*x^2 + 1)
Riem(g)^y_xxy = 4/(x^4 + y^4 + 2*(x^2 - 1)*y^2 - 2*x^2 + 1)
Riem(g)^y_xyx = -4/(x^4 + y^4 + 2*(x^2 - 1)*y^2 - 2*x^2 + 1)
"""
if self._riemann is None:
AffineConnection.riemann(self)
if name is None:
self._riemann._name = "Riem(" + self._metric._name + ")"
else:
self._riemann._name = name
if latex_name is None:
self._riemann._latex_name = r"\mathrm{Riem}\left(" + \
self._metric._latex_name + r"\right)"
else:
self._riemann._latex_name = latex_name
for rst in self._riemann._restrictions.values():
rst._name = self._riemann._name
rst._latex_name = self._riemann._latex_name
return self._riemann
def coef(self, frame=None):
r"""
Return the connection coefficients relative to the given frame.
`n` being the manifold's dimension, the connection coefficients
relative to the vector frame `(e_i)` are the `n^3` scalar fields
`\Gamma^k_{\ \, ij}` defined by
.. MATH::
\nabla_{e_j} e_i = \Gamma^k_{\ \, ij} e_k
If the connection coefficients are not known already, they are computed
* as Christoffel symbols if the frame `(e_i)` is a coordinate frame
* from the above formula otherwise
INPUT:
- ``frame`` -- (default: ``None``) vector frame relative to which the
connection coefficients are required; if none is provided, the
domain's default frame is assumed
OUTPUT:
- connection coefficients relative to the frame ``frame``, as an
instance of the class :class:`~sage.tensor.modules.comp.Components`
with 3 indices ordered as `(k,i,j)`; for Christoffel symbols,
an instance of the subclass
:class:`~sage.tensor.modules.comp.CompWithSym` is returned.
EXAMPLES:
Christoffel symbols of the Levi-Civita connection associated to
the Euclidean metric on `\RR^3` expressed in spherical coordinates::
sage: M = Manifold(3, 'R^3', start_index=1)
sage: c_spher.<r,th,ph> = M.chart(r'r:(0,+oo) th:(0,pi):\theta ph:(0,2*pi):\phi')
sage: g = M.metric('g')
sage: g[1,1], g[2,2], g[3,3] = 1, r^2 , (r*sin(th))^2
sage: g.display()
g = dr*dr + r^2 dth*dth + r^2*sin(th)^2 dph*dph
sage: nab = g.connection()
sage: gam = nab.coef() ; gam
3-indices components w.r.t. Coordinate frame (R^3, (d/dr,d/dth,d/dph)),
with symmetry on the index positions (1, 2)
sage: gam[:]
[[[0, 0, 0], [0, -r, 0], [0, 0, -r*sin(th)^2]],
[[0, 1/r, 0], [1/r, 0, 0], [0, 0, -cos(th)*sin(th)]],
[[0, 0, 1/r], [0, 0, cos(th)/sin(th)], [1/r, cos(th)/sin(th), 0]]]
sage: # The only non-zero Christoffel symbols:
sage: gam[1,2,2], gam[1,3,3]
(-r, -r*sin(th)^2)
sage: gam[2,1,2], gam[2,3,3]
(1/r, -cos(th)*sin(th))
sage: gam[3,1,3], gam[3,2,3]
(1/r, cos(th)/sin(th))
Connection coefficients of the same connection with respect to the
orthonormal frame associated to spherical coordinates::
sage: ch_basis = M.automorphism_field()
sage: ch_basis[1,1], ch_basis[2,2], ch_basis[3,3] = 1, 1/r, 1/(r*sin(th))
sage: e = c_spher.frame().new_frame(ch_basis, 'e')
sage: gam_e = nab.coef(e) ; gam_e
3-indices components w.r.t. Vector frame (R^3, (e_1,e_2,e_3))
sage: gam_e[:]
[[[0, 0, 0], [0, -1/r, 0], [0, 0, -1/r]],
[[0, 1/r, 0], [0, 0, 0], [0, 0, -cos(th)/(r*sin(th))]],
[[0, 0, 1/r], [0, 0, cos(th)/(r*sin(th))], [0, 0, 0]]]
sage: # The only non-zero connection coefficients:
sage: gam_e[1,2,2], gam_e[2,1,2]
(-1/r, 1/r)
sage: gam_e[1,3,3], gam_e[3,1,3]
(-1/r, 1/r)
sage: gam_e[2,3,3], gam_e[3,2,3]
(-cos(th)/(r*sin(th)), cos(th)/(r*sin(th)))
"""
from sage.manifolds.differentiable.vectorframe import CoordFrame
if frame is None:
frame = self._domain._def_frame
if frame not in self._coefficients:
# the coefficients must be computed
#
# Check whether frame is a subframe of a frame in which the
# coefficients are already known:
for oframe in self._coefficients:
if frame in oframe._subframes:
self._coefficients[frame] = self._new_coef(frame)
comp_store = self._coefficients[frame]._comp
ocomp_store = self._coefficients[oframe]._comp
for ind, value in ocomp_store.items():
comp_store[ind] = value.restrict(frame._domain)
break
else:
# If not, the coefficients must be computed from scratch:
manif = self._domain
if isinstance(frame, CoordFrame):
# Christoffel symbols
chart = frame._chart
gam = self._new_coef(frame)
gg = self._metric.comp(frame)
ginv = self._metric.inverse().comp(frame)
if Parallelism().get('tensor') != 1:
# parallel computation
nproc = Parallelism().get('tensor')
lol = lambda lst, sz: [
lst[i:i + sz] for i in range(0, len(lst), sz)
]
ind_list = []
for ind in gam.non_redundant_index_generator():
i, j, k = ind
ind_list.append((i, j, k))
ind_step = max(1, int(len(ind_list) / nproc / 2))
local_list = lol(ind_list, ind_step)
# definition of the list of input parameters
listParalInput = []
for ind_part in local_list:
listParalInput.append(
(ind_part, chart, ginv, gg, manif))
# definition of the parallel function
@parallel(p_iter='multiprocessing', ncpus=nproc)
def make_Connect(local_list_ijk, chart, ginv, gg,
manif):
partial = []
for i, j, k in local_list_ijk:
rsum = 0
for s in manif.irange():
if ginv[i, s, chart] != 0:
rsum += ginv[i, s, chart] * (
gg[s, k, chart].diff(j) +
gg[j, s, chart].diff(k) -
gg[j, k, chart].diff(s))
partial.append([i, j, k, rsum / 2])
return partial
# Computation and Assignation of values
for ii, val in make_Connect(listParalInput):
for jj in val:
gam[jj[0], jj[1], jj[2], ii[0][1]] = jj[3]
else:
# sequential
for ind in gam.non_redundant_index_generator():
i, j, k = ind
# The computation is performed at the CoordFunction level:
rsum = 0
for s in manif.irange():
rsum += ginv[i, s, chart] * (
gg[s, k, chart].diff(j) +
gg[j, s, chart].diff(k) -
gg[j, k, chart].diff(s))
gam[i, j, k, chart] = rsum / 2
# Assignation of results
self._coefficients[frame] = gam
else:
# Computation from the formula defining the connection coef.
return AffineConnection.coef(self, frame)
return self._coefficients[frame]
def riemann(self, name=None, latex_name=None):
r"""
Return the Riemann curvature tensor of the connection.
This method redefines
:meth:`sage.manifolds.differentiable.affine_connection.AffineConnection.riemann`
to set some name and the latex_name to the output.
The Riemann curvature tensor is the tensor field `R` of type (1,3)
defined by
.. MATH::
R(\omega, w, u, v) = \left\langle \omega, \nabla_u \nabla_v w
- \nabla_v \nabla_u w - \nabla_{[u, v]} w \right\rangle
for any 1-form `\omega` and any vector fields `u`, `v` and `w`.
INPUT:
- ``name`` -- (default: ``None``) name given to the Riemann tensor;
if none, it is set to "Riem(g)", where "g" is the metric's name
- ``latex_name`` -- (default: ``None``) LaTeX symbol to denote the
Riemann tensor; if none, it is set to "\\mathrm{Riem}(g)", where "g"
is the metric's name
OUTPUT:
- the Riemann curvature tensor `R`, as an instance of
:class:`~sage.manifolds.differentiable.tensorfield.TensorField`
EXAMPLES:
Riemann tensor of the Levi-Civita connection associated with the
metric of the hyperbolic plane (Poincaré disk model)::
sage: M = Manifold(2, 'M', start_index=1)
sage: X.<x,y> = M.chart('x:(-1,1) y:(-1,1)') # Cartesian coord. on the Poincaré disk
sage: X.add_restrictions(x^2+y^2<1)
sage: g = M.metric('g')
sage: g[1,1], g[2,2] = 4/(1-x^2-y^2)^2, 4/(1-x^2-y^2)^2
sage: nab = g.connection()
sage: riem = nab.riemann(); riem
Tensor field Riem(g) of type (1,3) on the 2-dimensional
differentiable manifold M
sage: riem.display_comp()
Riem(g)^x_yxy = -4/(x^4 + y^4 + 2*(x^2 - 1)*y^2 - 2*x^2 + 1)
Riem(g)^x_yyx = 4/(x^4 + y^4 + 2*(x^2 - 1)*y^2 - 2*x^2 + 1)
Riem(g)^y_xxy = 4/(x^4 + y^4 + 2*(x^2 - 1)*y^2 - 2*x^2 + 1)
Riem(g)^y_xyx = -4/(x^4 + y^4 + 2*(x^2 - 1)*y^2 - 2*x^2 + 1)
"""
if self._riemann is None:
AffineConnection.riemann(self)
if name is None:
self._riemann._name = "Riem(" + self._metric._name + ")"
else:
self._riemann._name = name
if latex_name is None:
self._riemann._latex_name = r"\mathrm{Riem}\left(" + \
self._metric._latex_name + r"\right)"
else:
self._riemann._latex_name = latex_name
for rst in self._riemann._restrictions.values():
rst._name = self._riemann._name
rst._latex_name = self._riemann._latex_name
return self._riemann
def coef(self, frame=None):
r"""
Return the connection coefficients relative to the given frame.
`n` being the manifold's dimension, the connection coefficients
relative to the vector frame `(e_i)` are the `n^3` scalar fields
`\Gamma^k_{\ \, ij}` defined by
.. MATH::
\nabla_{e_j} e_i = \Gamma^k_{\ \, ij} e_k
If the connection coefficients are not known already, they are computed
* as Christoffel symbols if the frame `(e_i)` is a coordinate frame
* from the above formula otherwise
INPUT:
- ``frame`` -- (default: ``None``) vector frame relative to which the
connection coefficients are required; if none is provided, the
domain's default frame is assumed
OUTPUT:
- connection coefficients relative to the frame ``frame``, as an
instance of the class :class:`~sage.tensor.modules.comp.Components`
with 3 indices ordered as `(k,i,j)`; for Christoffel symbols,
an instance of the subclass
:class:`~sage.tensor.modules.comp.CompWithSym` is returned.
EXAMPLES:
Christoffel symbols of the Levi-Civita connection associated to
the Euclidean metric on `\RR^3` expressed in spherical coordinates::
sage: M = Manifold(3, 'R^3', start_index=1)
sage: c_spher.<r,th,ph> = M.chart(r'r:(0,+oo) th:(0,pi):\theta ph:(0,2*pi):\phi')
sage: g = M.metric('g')
sage: g[1,1], g[2,2], g[3,3] = 1, r^2 , (r*sin(th))^2
sage: g.display()
g = dr*dr + r^2 dth*dth + r^2*sin(th)^2 dph*dph
sage: nab = g.connection()
sage: gam = nab.coef() ; gam
3-indices components w.r.t. Coordinate frame (R^3, (d/dr,d/dth,d/dph)),
with symmetry on the index positions (1, 2)
sage: gam[:]
[[[0, 0, 0], [0, -r, 0], [0, 0, -r*sin(th)^2]],
[[0, 1/r, 0], [1/r, 0, 0], [0, 0, -cos(th)*sin(th)]],
[[0, 0, 1/r], [0, 0, cos(th)/sin(th)], [1/r, cos(th)/sin(th), 0]]]
sage: # The only non-zero Christoffel symbols:
sage: gam[1,2,2], gam[1,3,3]
(-r, -r*sin(th)^2)
sage: gam[2,1,2], gam[2,3,3]
(1/r, -cos(th)*sin(th))
sage: gam[3,1,3], gam[3,2,3]
(1/r, cos(th)/sin(th))
Connection coefficients of the same connection with respect to the
orthonormal frame associated to spherical coordinates::
sage: ch_basis = M.automorphism_field()
sage: ch_basis[1,1], ch_basis[2,2], ch_basis[3,3] = 1, 1/r, 1/(r*sin(th))
sage: e = c_spher.frame().new_frame(ch_basis, 'e')
sage: gam_e = nab.coef(e) ; gam_e
3-indices components w.r.t. Vector frame (R^3, (e_1,e_2,e_3))
sage: gam_e[:]
[[[0, 0, 0], [0, -1/r, 0], [0, 0, -1/r]],
[[0, 1/r, 0], [0, 0, 0], [0, 0, -cos(th)/(r*sin(th))]],
[[0, 0, 1/r], [0, 0, cos(th)/(r*sin(th))], [0, 0, 0]]]
sage: # The only non-zero connection coefficients:
sage: gam_e[1,2,2], gam_e[2,1,2]
(-1/r, 1/r)
sage: gam_e[1,3,3], gam_e[3,1,3]
(-1/r, 1/r)
sage: gam_e[2,3,3], gam_e[3,2,3]
(-cos(th)/(r*sin(th)), cos(th)/(r*sin(th)))
"""
from sage.manifolds.differentiable.vectorframe import CoordFrame
if frame is None:
frame = self._domain._def_frame
if frame not in self._coefficients:
# the coefficients must be computed
#
# Check whether frame is a subframe of a frame in which the
# coefficients are already known:
for oframe in self._coefficients:
if frame in oframe._subframes:
self._coefficients[frame] = self._new_coef(frame)
comp_store = self._coefficients[frame]._comp
ocomp_store = self._coefficients[oframe]._comp
for ind, value in ocomp_store.items():
comp_store[ind] = value.restrict(frame._domain)
break
else:
# If not, the coefficients must be computed from scratch:
manif = self._domain
if isinstance(frame, CoordFrame):
# Christoffel symbols
chart = frame._chart
gam = self._new_coef(frame)
gg = self._metric.comp(frame)
ginv = self._metric.inverse().comp(frame)
if Parallelism().get('tensor') != 1:
# parallel computation
nproc = Parallelism().get('tensor')
lol = lambda lst, sz: [lst[i:i+sz] for i in
range(0, len(lst), sz)]
ind_list = []
for ind in gam.non_redundant_index_generator():
i, j, k = ind
ind_list.append((i,j,k))
ind_step = max(1,int(len(ind_list)/nproc/2))
local_list = lol(ind_list,ind_step)
# definition of the list of input parameters
listParalInput = []
for ind_part in local_list:
listParalInput.append((ind_part,chart,ginv,gg,manif))
# definition of the parallel function
@parallel(p_iter='multiprocessing',ncpus=nproc)
def make_Connect(local_list_ijk,chart,ginv,gg,manif):
partial = []
for i,j,k in local_list_ijk:
rsum = 0
for s in manif.irange():
if ginv[i,s, chart]!=0:
rsum += ginv[i,s, chart] * (
gg[s,k, chart].diff(j)
+ gg[j,s, chart].diff(k)
- gg[j,k, chart].diff(s) )
partial.append([i,j,k,rsum / 2])
return partial
# Computation and Assignation of values
for ii, val in make_Connect(listParalInput):
for jj in val:
gam[jj[0],jj[1],jj[2],ii[0][1]] = jj[3]
else:
# sequential
for ind in gam.non_redundant_index_generator():
i, j, k = ind
# The computation is performed at the ChartFunction level:
rsum = 0
for s in manif.irange():
rsum += ginv[i,s, chart] * (
gg[s,k, chart].diff(j)
+ gg[j,s, chart].diff(k)
- gg[j,k, chart].diff(s) )
gam[i,j,k, chart] = rsum / 2
# Assignation of results
self._coefficients[frame] = gam
else:
# Computation from the formula defining the connection coef.
return AffineConnection.coef(self, frame)
return self._coefficients[frame]