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tangent space vector mapping

asked 2019-05-20 10:47:33 -0500

rrogers gravatar image

Very simple question.
I am going through the SM_tutorial and branched off into a sidestream; trying to understand and put things in a context that I already know.
The tutorial defines a function f() on 3 space and defines the associated tangent_space. I have a couple of questions
1) How do I take a vector in the tangent_space
Say: v = Tp.an_element(); print(v)
"Tangent vector at Point p on the 3-dimensional differentiable manifold M"
or vxx = Tp((-2,1,5), name='vxx')
and apply it to f() (or f(p) although the TM is only defined at p so far)?
If I define a vector in the base space it works::
v = U.vector_field(name='v')
s = v(f)
I know in standard texts the mapping of vectors in TM_p to the base is defined, but couldn't find it in the Sage Manifold documentation.

2) Taking : v = Tp.an_element(); print(v)
I get something that looks like a vector (with a different ancestry) but has a value
v.display()
∂/∂x+2∂/∂y+3∂/∂z
Where did this value come from? In one of the documentation it (sort of) implies it's an example; is this true?
Why isn't it left undefined?
Ray

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Comments

Regarding (2), I couldn't reproduce this. In particular, when I create a manifold (and define some coordinates on it) and then run

p = theManifold.point(name='p')

TpM = theManifold.tangent_space(p)

v = TpM.an_element()

v.dipslay(),

I get the error message

ValueError: no basis could be found for computing the components in the None

my_screen_name gravatar imagemy_screen_name ( 2019-05-21 06:34:53 -0500 )edit

@my_screen_name: I cannot reproduce your issue. Can you provide the full code, starting from the definition of the manifold?

eric_g gravatar imageeric_g ( 2019-05-21 15:22:08 -0500 )edit

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answered 2019-05-21 15:17:27 -0500

eric_g gravatar image

updated 2019-05-21 15:30:52 -0500

Question 1: v(f), where v is a tangent vector at a given point p on a manifold and f a scalar field on that manifold, should work as is. I've opened the ticket #27856 to fix this. Thanks for your report. A workaround is

v(f.differential().at(p))

Question 2: for any Sage parent (here the tangent space Tp), the method an_element() returns an arbitrary element of it. The end user has no control on which element is returned. For tangent spaces on a manifold of dimension $n$, the element is hard-coded to be the vector of components $(1, 2, \ldots, n)$ in the default basis. Another example is the method an_element of the manifold:

sage: M.an_element()
Point on the 3-dimensional differentiable manifold M
sage: M.an_element().coordinates()
(0, 0, 0)

Actually, the methods an_element are mostly used for test suites in Sage, like

sage: TestSuite(Tp).run(verbose=True)
running ._test_additive_associativity() . . . pass
running ._test_an_element() . . . pass
running ._test_cardinality() . . . pass
running ._test_category() . . . pass
running ._test_elements() . . .
  Running the test suite of self.an_element()
  running ._test_category() . . . pass
  running ._test_eq() . . . pass
  running ._test_new() . . . pass
  running ._test_nonzero_equal() . . . pass
  running ._test_not_implemented_methods() . . . pass
  running ._test_pickling() . . . pass
  pass
running ._test_elements_eq_reflexive() . . . pass
running ._test_elements_eq_symmetric() . . . pass
running ._test_elements_eq_transitive() . . . pass
running ._test_elements_neq() . . . pass
running ._test_eq() . . . pass
running ._test_new() . . . pass
running ._test_not_implemented_methods() . . . pass
running ._test_pickling() . . . pass
running ._test_some_elements() . . . pass
running ._test_zero() . . . pass

They probably should not be exposed in the tutorial. This is more confusing than useful, as you pointed out.

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Great! and thanks :) That in fact corresponds to usage/pullback of T_p vector I remember. I have looked around on the web in the last day and there seem's to be some obstacles to just yanking in down in general. One place has a greater than 1-page proof that it can sometimes be done. Yet another preconception too examine (:

rrogers gravatar imagerrogers ( 2019-05-21 18:57:30 -0500 )edit

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Asked: 2019-05-20 10:47:33 -0500

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Last updated: May 21 '19