Vector Bundle Fiber Elements - Manifolds

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# Vector Bundle Fiber Elements

The class `VectorBundleFiberElement` implements vectors in the fiber of a vector bundle.

AUTHORS:

- Michael Jung (2019): initial version

class sage.manifolds.vector_bundle_fiber_element.VectorBundleFiberElement(parent, name=None, latex_name=None)

  Bases: `FiniteRankFreeModuleElement`

  Vector in a fiber of a vector bundle at the given point.

  INPUT:

  - `parent` – `VectorBundleFiber`; the fiber to which the vector belongs

  - `name` – (default: `None`) string; symbol given to the vector

  - `latex_name` – (default: `None`) string; LaTeX symbol to denote the vector; if `None`, `name` will be used

  EXAMPLES:

  A vector \(v\) in a fiber of a rank 2 vector bundle:

  Sage

      sage: M = Manifold(2, 'M', structure='top')
      sage: X.<x,y> = M.chart()
      sage: p = M((1,-1), name='p')
      sage: E = M.vector_bundle(2, 'E')
      sage: e = E.local_frame('e')
      sage: Ep = E.fiber(p)
      sage: v = Ep((-2,1), name='v'); v
      Vector v in the fiber of E at Point p on the 2-dimensional topological
       manifold M
      sage: v.display()
      v = -2 e_0 + e_1
      sage: v.parent()
      Fiber of E at Point p on the 2-dimensional topological manifold M
      sage: v in Ep
      True

  Python

      >>> from sage.all import *
      >>> M = Manifold(Integer(2), 'M', structure='top')
      >>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
      >>> p = M((Integer(1),-Integer(1)), name='p')
      >>> E = M.vector_bundle(Integer(2), 'E')
      >>> e = E.local_frame('e')
      >>> Ep = E.fiber(p)
      >>> v = Ep((-Integer(2),Integer(1)), name='v'); v
      Vector v in the fiber of E at Point p on the 2-dimensional topological
       manifold M
      >>> v.display()
      v = -2 e_0 + e_1
      >>> v.parent()
      Fiber of E at Point p on the 2-dimensional topological manifold M
      >>> v in Ep
      True

  See also

  `FiniteRankFreeModuleElement` for more documentation.