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Why does manifold not have a volume form?

asked 2020-11-06 23:57:31 +0200

Richard_L gravatar image

I create a subset of a Riemannian manifold. It is orientable, but it does not have the attribute 'volume_form'. Why? Note that the metric (g) has the attribute 'volume_form'. [Some variables, all real, elided below]

# Define manifold
M = Manifold(3,'R^3',field='real',start_index=1)
U = M.open_subset('U')

# Add chart and metric
Rho.<r12,r13,r23> = U.chart("r12:(0,+oo) r13:(0,+oo) r23:(0,+oo)")
Rho.add_restrictions([r23<r12+r13, r13<r12+r23, r12<r13+r23])
g = U.riemannian_metric('g');
g[:] = G[:].simplify_full()

# Establish affine connection
nabla = g.connection()

# Check that we have an orientation
U.orientation()

# Volume form 
#VolForm = U.volume_form()  # FAILS: 'DifferentiableManifold_with_category' object has no attribute 'volume_form'
VolForm = g.volume_form()  # This succeeds as expected
VolForm.display()

The on-line documentation shows examples of manifolds with attribute 'volume_form', thus my question.

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answered 2020-11-07 07:50:03 +0200

eric_g gravatar image

updated 2020-11-07 16:25:46 +0200

You have to define the manifold M with the keyword structure='pseudo-Riemannian', or structure='Riemannian' or structure='Lorentzian', then it has a volume form; see here.

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Comments

Thank you. I missed that.

Richard_L gravatar imageRichard_L ( 2020-11-09 16:57:02 +0200 )edit

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Asked: 2020-11-06 23:57:31 +0200

Seen: 66 times

Last updated: Nov 07 '20