Trac #31781: Improve handling of metrics on pseudo-Riemannian submani…

…folds

In Sage 9.3.rc5, we have
{{{
sage: S2 = manifolds.Sphere(2)
sage: g = S2.metric()
sage: g
Riemannian metric g on the 2-sphere S^2 of radius 1 smoothly embedded
 in the Euclidean space E^3
}}}
So far so good, but then
{{{
sage: g.display()
...
ValueError: no basis could be found for computing the components
 in the Coordinate frame (A, (d/dtheta,d/dphi))
}}}
{{{
sage: g.ricci_scalar()
...
ValueError: no basis could be found for computing the components
 in the Coordinate frame (A, (d/dtheta,d/dphi))
}}}

URL: https://trac.sagemath.org/31781
Reported by: egourgoulhon
Ticket author(s): Eric Gourgoulhon
Reviewer(s): Michael Jung

src/sage/manifolds/differentiable/degenerate.py

@@ -31,8 +31,8 @@ class DegenerateManifold(DifferentiableManifold):
 
     - ``n`` -- positive integer; dimension of the manifold
     - ``name`` -- string; name (symbol) given to the manifold
-    - ``metric_name`` -- (default: ``'g'``) string; name (symbol) given to the
-      metric
+    - ``metric_name`` -- (default: ``None``) string; name (symbol) given to the
+      metric; if ``None``, ``'g'`` is used
     - ``signature`` -- (default: ``None``) signature `S` of the metric as a
       tuple: `S = (n_+, n_-, n_0)`, where `n_+` (resp. `n_-`, resp. `n_0`) is the
       number of positive terms (resp. negative terms, resp. zero tems) in any
@@ -98,12 +98,12 @@ class DegenerateManifold(DifferentiableManifold):
     - [DB1996]_
     - [DS2010]_
     """
-    def __init__(self, n, name, metric_name='g', signature=None, base_manifold=None,
-                 diff_degree=infinity, latex_name=None,
+    def __init__(self, n, name, metric_name=None, signature=None,
+                 base_manifold=None, diff_degree=infinity, latex_name=None,
                  metric_latex_name=None, start_index=0, category=None,
                  unique_tag=None):
         r"""
-        Construct a pseudo-Riemannian manifold.
+        Construct a degenerate manifold.
 
         TESTS::
 
@@ -130,7 +130,9 @@ def __init__(self, n, name, metric_name='g', signature=None, base_manifold=None,
                                         category=category)
         self._metric = None # to be initialized by metric()
         self._metric_signature = signature
-        if not isinstance(metric_name, str):
+        if metric_name is None:
+            metric_name = 'g'
+        elif not isinstance(metric_name, str):
             raise TypeError("{} is not a string".format(metric_name))
         self._metric_name = metric_name
         if metric_latex_name is None:

src/sage/manifolds/differentiable/degenerate_submanifold.py

@@ -182,30 +182,24 @@ class DegenerateSubmanifold(DegenerateManifold, DifferentiableSubmanifold):
 
     - ``n`` -- positive integer; dimension of the manifold
     - ``name`` -- string; name (symbol) given to the manifold
-    - ``field`` -- field `K` on which the manifold is
-      defined; allowed values are
-
-      - ``'real'`` or an object of type ``RealField`` (e.g., ``RR``) for
-        a manifold over `\RR`
-      - ``'complex'`` or an object of type ``ComplexField`` (e.g., ``CC``)
-        for a manifold over `\CC`
-      - an object in the category of topological fields (see
-        :class:`~sage.categories.fields.Fields` and
-        :class:`~sage.categories.topological_spaces.TopologicalSpaces`)
-        for other types of manifolds
-
+    - ``ambient`` -- (default: ``None``) pseudo-Riemannian manifold `M` in
+      which the submanifold is embedded (or immersed). If ``None``, it is set
+      to ``self``
+    - ``metric_name`` -- (default: ``None``) string; name (symbol) given to the
+      metric; if ``None``, ``'g'`` is used
     - ``signature`` -- (default: ``None``) signature `S` of the metric as a
       tuple: `S = (n_+, n_-, n_0)`, where `n_+` (resp. `n_-`, resp. `n_0`) is the
       number of positive terms (resp. negative terms, resp. zero tems) in any
       diagonal writing of the metric components; if ``signature`` is not
       provided, `S` is set to `(ndim-1, 0, 1)`, being `ndim` the manifold's dimension
-    - ``ambient`` -- (default: ``None``) manifold of destination
-      of the immersion. If ``None``, set to ``self``
     - ``base_manifold`` -- (default: ``None``) if not ``None``, must be a
       topological manifold; the created object is then an open subset of
       ``base_manifold``
+    - ``diff_degree`` -- (default: ``infinity``) degree of differentiability
     - ``latex_name`` -- (default: ``None``) string; LaTeX symbol to
       denote the manifold; if none are provided, it is set to ``name``
+    - ``metric_latex_name`` -- (default: ``None``) string; LaTeX symbol to
+      denote the metric; if none is provided, it is set to ``metric_name``
     - ``start_index`` -- (default: 0) integer; lower value of the range of
       indices used for "indexed objects" on the manifold, e.g., coordinates
       in a chart
@@ -227,12 +221,12 @@ class DegenerateSubmanifold(DegenerateManifold, DifferentiableSubmanifold):
         :mod:`~sage.manifolds.differentiable.differentiable_submanifold`
 
     """
-    def __init__(self, n, name, ambient=None, metric_name='g', signature=None,
+    def __init__(self, n, name, ambient=None, metric_name=None, signature=None,
                  base_manifold=None, diff_degree=infinity, latex_name=None,
                  metric_latex_name=None, start_index=0, category=None,
                  unique_tag=None):
         r"""
-        Construct a pseudo-Riemannian submanifold.
+        Construct a degenerate submanifold.
 
         EXAMPLES:
 
@@ -245,7 +239,6 @@ def __init__(self, n, name, ambient=None, metric_name='g', signature=None,
             differentiable manifold M
 
         """
-
         DegenerateManifold.__init__(self, n, name=name,
                                           metric_name=metric_name,
                                           signature=signature,

src/sage/manifolds/differentiable/examples/sphere.py

@@ -52,14 +52,14 @@
 As a submanifold of a Riemannian manifold, namely the Euclidean space,
 the 2-sphere admits an induced metric::
 
-    sage: h = S2_r.induced_metric()
-    sage: h.display()
-    gamma = r^2 dtheta⊗dtheta + r^2*sin(theta)^2 dphi⊗dphi
+    sage: g = S2_r.induced_metric()
+    sage: g.display()
+    g = r^2 dtheta⊗dtheta + r^2*sin(theta)^2 dphi⊗dphi
 
 The induced metric is also known as the *first fundamental form* (see
 :meth:`~sage.manifolds.differentiable.pseudo_riemannian_submanifold.PseudoRiemannianSubmanifold.first_fundamental_form`)::
 
-    sage: h is S2_r.first_fundamental_form()
+    sage: g is S2_r.first_fundamental_form()
     True
 
 The *second fundamental form* encodes the extrinsic curvature of the
@@ -157,12 +157,12 @@
 vector field pointing outwards. Henceforth, the manifold admits
 a volume form::
 
-    sage: h = S1c.induced_metric()
-    sage: h.display()
-    gamma = dchi⊗dchi
-    sage: eps = h.volume_form()
+    sage: g = S1c.induced_metric()
+    sage: g.display()
+    g = dchi⊗dchi
+    sage: eps = g.volume_form()
     sage: eps.display()
-    eps_gamma = -dchi
+    eps_g = -dchi
 
 """
 
@@ -380,7 +380,8 @@ def __init__(self, n, radius=1, ambient_space=None, center=None, name=None,
         PseudoRiemannianSubmanifold.__init__(self, n, name,
                                              ambient=ambient_space,
                                              signature=n, latex_name=latex_name,
-                                             start_index=1, category=category)
+                                             metric_name='g', start_index=1,
+                                             category=category)
         # set attributes
         self._radius = radius
         self._center = center

src/sage/manifolds/differentiable/pseudo_riemannian.py

@@ -297,8 +297,8 @@ class PseudoRiemannianManifold(DifferentiableManifold):
 
     - ``n`` -- positive integer; dimension of the manifold
     - ``name`` -- string; name (symbol) given to the manifold
-    - ``metric_name`` -- (default: ``'g'``) string; name (symbol) given to the
-      metric
+    - ``metric_name`` -- (default: ``None``) string; name (symbol) given to the
+      metric; if ``None``, ``'g'`` is used
     - ``signature`` -- (default: ``None``) signature `S` of the metric as a
       single integer: `S = n_+ - n_-`, where `n_+` (resp. `n_-`) is the
       number of positive terms (resp. number of negative terms) in any
@@ -412,8 +412,8 @@ class PseudoRiemannianManifold(DifferentiableManifold):
         -2
 
     """
-    def __init__(self, n, name, metric_name='g', signature=None, base_manifold=None,
-                 diff_degree=infinity, latex_name=None,
+    def __init__(self, n, name, metric_name=None, signature=None,
+                 base_manifold=None, diff_degree=infinity, latex_name=None,
                  metric_latex_name=None, start_index=0, category=None,
                  unique_tag=None):
         r"""
@@ -450,7 +450,9 @@ def __init__(self, n, name, metric_name='g', signature=None, base_manifold=None,
                                         category=category)
         self._metric = None # to be initialized by metric()
         self._metric_signature = signature
-        if not isinstance(metric_name, str):
+        if metric_name is None:
+            metric_name = 'g'
+        elif not isinstance(metric_name, str):
             raise TypeError("{} is not a string".format(metric_name))
         self._metric_name = metric_name
         if metric_latex_name is None:

src/sage/manifolds/differentiable/pseudo_riemannian_submanifold.py

@@ -221,8 +221,8 @@ class PseudoRiemannianSubmanifold(PseudoRiemannianManifold,
     - ``ambient`` -- (default: ``None``) pseudo-Riemannian manifold `M` in
       which the submanifold is embedded (or immersed). If ``None``, it is set
       to ``self``
-    - ``metric_name`` -- (default: ``'g'``) string; name (symbol) given to the
-      submanifold's metric
+    - ``metric_name`` -- (default: ``None``) string; name (symbol) given to the
+      metric; if ``None``, ``'gamma'`` is used
     - ``signature`` -- (default: ``None``) signature `S` of the metric as a
       single integer: `S = n_+ - n_-`, where `n_+` (resp. `n_-`) is the
       number of positive terms (resp. number of negative terms) in any
@@ -236,7 +236,8 @@ class PseudoRiemannianSubmanifold(PseudoRiemannianManifold,
     - ``latex_name`` -- (default: ``None``) string; LaTeX symbol to
       denote the submanifold; if none is provided, it is set to ``name``
     - ``metric_latex_name`` -- (default: ``None``) string; LaTeX symbol to
-      denote the metric; if none is provided, it is set to ``metric_name``
+      denote the metric; if none is provided, it is set to ``metric_name`` if
+      the latter is not ``None`` and to ``r'\gamma'`` otherwise
     - ``start_index`` -- (default: 0) integer; lower value of the range of
       indices used for "indexed objects" on the submanifold, e.g. coordinates
       in a chart
@@ -300,10 +301,10 @@ class PseudoRiemannianSubmanifold(PseudoRiemannianManifold,
         :mod:`~sage.manifolds.differentiable.differentiable_submanifold`
 
    """
-    def __init__(self, n, name, ambient=None, metric_name='g', signature=None,
-                 base_manifold=None, diff_degree=infinity, latex_name=None,
-                 metric_latex_name=None, start_index=0, category=None,
-                 unique_tag=None):
+    def __init__(self, n, name, ambient=None, metric_name=None,
+                 signature=None, base_manifold=None, diff_degree=infinity,
+                 latex_name=None, metric_latex_name=None, start_index=0,
+                 category=None, unique_tag=None):
         r"""
         Construct a pseudo-Riemannian submanifold.
 
@@ -337,6 +338,9 @@ def __init__(self, n, name, ambient=None, metric_name='g', signature=None,
              5-dimensional pseudo-Riemannian manifold M
 
         """
+        if metric_name is None:
+            metric_name = 'gamma'
+            metric_latex_name = r'\gamma'
         PseudoRiemannianManifold.__init__(self, n, name=name,
                                           metric_name=metric_name,
                                           signature=signature,
@@ -513,7 +517,7 @@ def first_fundamental_form(self):
         A sphere embedded in Euclidean space::
 
             sage: M.<x,y,z> = EuclideanSpace()
-            sage: N = Manifold(2, 'N', ambient=M, structure="Riemannian")
+            sage: N = Manifold(2, 'N', ambient=M, structure='Riemannian')
             sage: C.<th,ph> = N.chart(r'th:(0,pi):\theta ph:(-pi,pi):\phi')
             sage: r = var('r', domain='real')
             sage: assume(r>0)
@@ -535,16 +539,105 @@ def first_fundamental_form(self):
             [          r^2             0]
             [            0 r^2*sin(th)^2]
 
+        By default, the first fundamental form is named ``gamma``, but this
+        can be customized by means of the argument ``metric_name`` when
+        declaring the submanifold::
+
+            sage: P = Manifold(1, 'P', ambient=M, structure='Riemannian',
+            ....:              metric_name='g')
+            sage: CP.<t> = P.chart()
+            sage: F = P.diff_map(M, [t, 2*t, 3*t])
+            sage: P.set_embedding(F)
+            sage: P.induced_metric()
+            Riemannian metric g on the 1-dimensional Riemannian submanifold P
+             embedded in the Euclidean space E^3
+            sage: P.induced_metric().display()
+            g = 14 dt⊗dt
+
         """
         if self._first_fundamental_form is None:
-            self._first_fundamental_form = self.metric()
+            self._first_fundamental_form = super().metric()
             self._first_fundamental_form.set(
                                self._immersion.pullback(self.ambient_metric()))
-            self._first_fundamental_form.set_name("gamma", r"\gamma")
         return self._first_fundamental_form
 
     induced_metric = first_fundamental_form
 
+    def metric(self, name=None, signature=None, latex_name=None,
+               dest_map=None):
+        r"""
+        Return the induced metric (first fundamental form) or define a new
+        metric tensor on the submanifold.
+
+        A new (uninitialzed) metric is returned only if the argument ``name``
+        is provided and differs from the metric name declared at the
+        construction of the submanifold; otherwise, the first fundamental
+        form is returned.
+
+        INPUT:
+
+        - ``name`` -- (default: ``None``) name given to the metric; if ``name``
+          is ``None`` or equals the metric name declared when constructing
+          the submanifold, the first fundamental form is returned (see
+          :meth:`first_fundamental_form`)
+        - ``signature`` -- (default: ``None``; ignored if ``name`` is ``None``)
+          signature `S` of the metric as a single integer: `S = n_+ - n_-`,
+          where `n_+` (resp. `n_-`) is the number of positive terms (resp.
+          number of negative terms) in any diagonal writing of the metric
+          components; if ``signature`` is not provided, `S` is set to the
+          submanifold's dimension (Riemannian signature)
+        - ``latex_name`` -- (default: ``None``; ignored if ``name`` is ``None``)
+          LaTeX symbol to denote the metric; if ``None``, it is formed from
+          ``name``
+        - ``dest_map`` -- (default: ``None``; ignored if ``name`` is ``None``)
+          instance of
+          class :class:`~sage.manifolds.differentiable.diff_map.DiffMap`
+          representing the destination map `\Phi:\ U \rightarrow M`, where `U`
+          is the current submanifold; if ``None``, the identity map is assumed
+          (case of a metric tensor field *on* `U`)
+
+        OUTPUT:
+
+        - instance of
+          :class:`~sage.manifolds.differentiable.metric.PseudoRiemannianMetric`
+
+        EXAMPLES:
+
+        Induced metric on a straight line of the Euclidean plane::
+
+            sage: M.<x,y> = EuclideanSpace()
+            sage: N = Manifold(1, 'N', ambient=M, structure='Riemannian')
+            sage: CN.<t> = N.chart()
+            sage: F = N.diff_map(M, [t, 2*t])
+            sage: N.set_embedding(F)
+            sage: N.metric()
+            Riemannian metric gamma on the 1-dimensional Riemannian
+             submanifold N embedded in the Euclidean plane E^2
+            sage: N.metric().display()
+            gamma = 5 dt⊗dt
+
+        Setting the argument ``name`` to that declared while constructing
+        the submanifold (default = ``'gamma'``) yields the same result::
+
+            sage: N.metric(name='gamma') is N.metric()
+            True
+
+        while using a different name allows one to define a new metric on the
+        submanifold::
+
+            sage: h = N.metric(name='h'); h
+            Riemannian metric h on the 1-dimensional Riemannian submanifold N
+             embedded in the Euclidean plane E^2
+            sage: h[0, 0] = 1  # initialization
+            sage: h.display()
+            h = dt⊗dt
+
+        """
+        if name is None or name == self._metric_name:
+            return self.first_fundamental_form()
+        return super().metric(name=name, signature=signature,
+                              latex_name=latex_name, dest_map=dest_map)
+
     @cached_method
     def difft(self):
         r"""
@@ -1162,7 +1255,7 @@ def second_fundamental_form(self):
                              + " hypersurfaces")
         if self._second_fundamental_form is None:
             resu = self.vector_field_module().tensor((0, 2), name='K',
-                                                     sym=[(0, 1)])
+                                                      sym=[(0, 1)])
             if self._dim_foliation != 0:
                 inverse_subs = {v: k for k, v in self._subs[0].items()}
                 asff = self.ambient_second_fundamental_form()

src/sage/manifolds/manifold.py

@@ -3008,27 +3008,15 @@ def Manifold(dim, name, latex_name=None, field='real', structure='smooth',
                                       start_index=start_index,
                                       unique_tag=unique_tag())
     elif structure in ['pseudo-Riemannian', 'Riemannian', 'Lorentzian','degenerate_metric']:
-        if 'diff_degree' in extra_kwds:
-            diff_degree = extra_kwds['diff_degree']
-        else:
-            diff_degree = infinity
-        if 'metric_name' in extra_kwds:
-            metric_name = extra_kwds['metric_name']
-        else:
-            metric_name = 'g'
-        if 'metric_latex_name' in extra_kwds:
-            metric_latex_name = extra_kwds['metric_latex_name']
-        else:
-            metric_latex_name = None
+        diff_degree = extra_kwds.get('diff_degree', infinity)
+        metric_name = extra_kwds.get('metric_name', None)
+        metric_latex_name = extra_kwds.get('metric_latex_name', None)
         if structure == 'pseudo-Riemannian':
-            if 'signature' in extra_kwds:
-                signature = extra_kwds['signature']
-            else:
-                signature = None
+            signature = extra_kwds.get('signature', None)
         elif structure == 'Riemannian':
             signature = dim
         elif structure == 'degenerate_metric':
-            signature = (0,dim-1,1)
+            signature = (0, dim-1, 1)
         elif structure == 'Lorentzian':
             if 'signature' in extra_kwds:
                 signat = extra_kwds['signature']