How to create/use matrices valued in differential forms?

asked 2019-05-25 17:16:00 +0100

sum8tion
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updated 2019-08-29 18:36:03 +0100

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First, for context, I am working with holomorphic Hermitian vector bundles, and I need to compute the connection and curvature matrices, and compute some representatives for Chern classes and the Chern form.

Ultimately I want to have a matrix which is valued in differential forms, and when I multiply matrices, I want the component-wise multiplication to be the wedge product of forms.

When I naively try to build a matrix of forms 'by hand', or when I pre-define my matrix space to be valued in the module of differential forms, I get an error telling me that the space of differential forms is a not a ring.

For example,

import sys
import mpmath
sys.modules['sympy.mpmath'] = mpmath
M = Manifold(2, 'M', field='complex')
U = M.open_subset('U')
c_xy.<x,y> = U.chart()
e_xy = c_xy.frame()
a = M.diff_form(2, name='a')
a.parent()
eU = c_xy.frame()
a.degree()
a[eU,0,1] = x*y^2 + 2*x
a.display(eU)
A = matrix([[a, a], [a, a]])

prints out

Module Omega^2(M) of 2-forms on the 2-dimensional complex manifold M
2
a = (x*y^2 + 2*x) dx/\dy

and raises a type error:

TypeError: base_ring (=Module Omega^2(M) of 2-forms on the 2-dimensional complex manifold M) must be a ring

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Comments

maybe you can give your code, so someone might be more easily able to help you.

your post is in duplicate, maybe you can delete one of them?(but in fact I do not know if it is possible on the sagemath site !

ortollj ( 2019-05-25 19:14:06 +0100 )edit

just for information: there is a button 101 which allows you to display code more readable.

ortollj ( 2019-05-26 08:06:33 +0100 )edit

It seems that someone fixed it for me. How do I do it myself?

sum8tion ( 2019-05-30 16:25:09 +0100 )edit

To display blocks of code or error messages, skip a line above and below, and do one of the following (all give the same result):

indent all code lines with 4 spaces
select all code lines and click the "code" button (the icon with '101 010')
select all code lines and hit ctrl-K

For instance, typing

If we define `f` by

    def f(x, y, z):
        return x * y * z

then `f(2, 3, 5)` returns `30` but `f(2*3*5)` gives:

    TypeError: f() takes exactly 3 arguments (1 given)

produces:

If we define f by
def f(x, y, z):
    return x * y * z
then f(2, 3, 5) returns 30 but f(2*3*5) gives:
TypeError: f() takes exactly 3 arguments (1 given)

slelievre ( 2019-06-01 15:40:56 +0100 )edit

add a comment

sage: aa = M.mixed_form(comp=[0, 0, a], name='aa') sage: aa.parent() Graded algebra Omega^*(M) of mixed differential forms on the 2-dimensional complex manifold M sage: aa.display(eU) aa = [0] + [0] + [(x*y^2 + 2*x) dx/\dy]

sage: A = matrix([[aa, aa], [aa, aa]]) sage: A [Mixed differential form aa on the 2-dimensional complex manifold M Mixed differential form aa on the 2-dimensional complex manifold M] [Mixed differential form aa on the 2-dimensional complex manifold M Mixed differential form aa on the 2-dimensional complex manifold M] sage: A.base_ring() Graded algebra Omega^*(M) of mixed differential forms on the 2-dimensional complex manifold M sage: A[0,1] Mixed differential form aa on the 2-dimensional complex manifold M sage: det(A) Mixed differential form aa/\aa-aa/\aa on the 2-dimensional complex manifold M sage: det(A).display(eU) aa/\aa-aa/\aa = [0] + [0] + [0]

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How to create/use matrices valued in differential forms?

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