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)[source]#

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