Trac #29581: New Algorithm for Characteristic Classes

The current algorithm for characteristic forms is comparably slow. The
worst case scenario showed computations times about 1h in dimension
four.

With this ticket, we want to replace the current algorithm and implement
characteristic cohomology classes as sub-ring of the de Rham cohomology
ring (cf. #31691).

The idea is that characteristic (cohomology) classes are generated by
Chern/Pontryagin/Euler classes. The generators can be computed by an
Faddeev-LeVerrier-like algorithm (cf. #30681).

URL: https://trac.sagemath.org/29581
Reported by: gh-mjungmath
Ticket author(s): Michael Jung
Reviewer(s): Travis Scrimshaw, Eric Gourgoulhon

src/doc/en/reference/manifolds/diff_vector_bundle.rst

@@ -8,4 +8,4 @@ Differentiable Vector Bundles
 
    sage/manifolds/differentiable/bundle_connection
 
-   sage/manifolds/differentiable/characteristic_class
+   sage/manifolds/differentiable/characteristic_cohomology_class

src/doc/en/reference/references/index.rst

@@ -1280,6 +1280,10 @@ REFERENCES:
              Coinvariants Quasi-Symétriques*, Electronic Journal of
              Combinatorics Vol 12(1) (2005) N16.
 
+.. [Che1944] \S. Chern, *A simple intrinsic proof of the Gauss-Bonnet formula
+             for closed Riemannian manifolds*, Ann. of Math. (2) 45 (1944),
+             747–752.
+
 .. [CQ2019] \A. Cassella and C. Quadrelli.
             *Right-angled Artin groups and enhanced Koszul properties*.
             Preprint, :arxiv:`1907.03824`, (2019).

src/sage/manifolds/differentiable/bundle_connection.py

@@ -266,7 +266,7 @@ def __init__(self, vbundle, name, latex_name=None):
                             "vector bundle")
         Mutability.__init__(self)
         self._vbundle = vbundle
-        self._base_space = vbundle.base_space()
+        self._domain = vbundle.base_space()
         self._name = name
         if latex_name is None:
             self._latex_name = self._name
@@ -394,7 +394,7 @@ def __eq__(self, other):
             return True
         if not isinstance(other, BundleConnection):
             return False
-        if other._base_space != self._base_space:
+        if other._domain != self._domain:
             return False
         if self._connection_forms == {}:
             return False
@@ -566,7 +566,7 @@ def connection_forms(self, frame=None):
 
         """
         if frame is None:
-            smodule = self._vbundle.section_module(domain=self._base_space)
+            smodule = self._vbundle.section_module(domain=self._domain)
             frame = smodule.default_frame()
             if frame is None:
                 raise ValueError("a frame must be provided")
@@ -832,15 +832,15 @@ def add_connection_form(self, i, j, frame=None):
         """
         self._require_mutable()
         if frame is None:
-            smodule = self._vbundle.section_module(domain=self._base_space)
+            smodule = self._vbundle.section_module(domain=self._domain)
             frame = smodule.default_frame()
             if frame is None:
                 raise ValueError("a frame must be provided")
         # Are the components already known?
         if frame not in self._connection_forms:
             if frame not in self._vbundle._frames:
                 raise ValueError("the {} is not".format(frame) +
-                                 " a frame on the {}".format(self._base_space))
+                                 " a frame on the {}".format(self._domain))
             self._connection_forms[frame] = self._new_forms(frame)
         self._del_derived()  # deletes the derived quantities
         return self._connection_forms[frame][(i, j)]
@@ -967,7 +967,7 @@ def del_other_forms(self, frame=None):
 
         """
         if frame is None:
-            smodule = self._vbundle.section_module(domain=self._base_space)
+            smodule = self._vbundle.section_module(domain=self._domain)
             frame = smodule.default_frame()
             if frame is None:
                 raise ValueError("a frame must be provided")
@@ -1023,7 +1023,7 @@ def curvature_form(self, i, j, frame=None):
 
         """
         if frame is None:
-            smodule = self._vbundle.section_module(domain=self._base_space)
+            smodule = self._vbundle.section_module(domain=self._domain)
             frame = smodule.default_frame()
             if frame is None:
                 raise ValueError("a frame must be provided")
@@ -1127,13 +1127,13 @@ def __getitem__(self, args):
         # extract frame from first index:
         vb = self._vbundle
         if isinstance(args, (int, Integer, slice)):
-            smodule = vb.section_module(domain=self._base_space)
+            smodule = vb.section_module(domain=self._domain)
             frame = smodule.default_frame()
         elif not isinstance(args[0], (int, Integer, slice)):
             frame = args[0]
             args = args[1:]
         else:
-            smodule = vb.section_module(domain=self._base_space)
+            smodule = vb.section_module(domain=self._domain)
             frame = smodule.default_frame()
         # indexing:
         if isinstance(args, slice):
@@ -1200,13 +1200,13 @@ def __setitem__(self, args, value):
         # extract frame from first index:
         vb = self._vbundle
         if isinstance(args, (int, Integer, slice)):
-            smodule = vb.section_module(domain=self._base_space)
+            smodule = vb.section_module(domain=self._domain)
             frame = smodule.default_frame()
         elif not isinstance(args[0], (int, Integer, slice)):
             frame = args[0]
             args = args[1:]
         else:
-            smodule = vb.section_module(domain=self._base_space)
+            smodule = vb.section_module(domain=self._domain)
             frame = smodule.default_frame()
         # determine indices:
         if isinstance(args, slice):
@@ -1350,7 +1350,7 @@ def display(self, frame=None, vector_frame=None, chart=None,
         """
         vb = self._vbundle
         if frame is None:
-            smodule = vb.section_module(domain=self._base_space)
+            smodule = vb.section_module(domain=self._domain)
             frame = smodule.default_frame()
             if frame is None:
                 raise ValueError("a local frame must be provided")