Tangent
Structures to a Manifold| Tangent
Structures to a Manifold |
Tangent Bundle
$ Def: The set TM of
all tangent vectors at all points of an n-dimensional
manifold M, with a differentiable fiber bundle structure.
* Fibers: The tangent
spaces TpM at
each p 
M;
Structure group: GL(n, R).
* Coordinates: Given coordinates {xi}
on M, natural coordinates
on TM are {xi,
/xi}.
* Relationships: It is
an associated bundle to the frame bundle FM of
a manifold M, with structure group GL(n, R).
@ References: Yano & Ishiwara 73; Morandi et al PRP(90).
Tangent Vector at a Point > s.a. vector; vector
calculus;
vector field.
$ Def: There are several possible ones:
(1) A derivation on the algebra of germs of differentiable functions at x X;
(2) An equivalence class of triples (x, 
, V),
with (x, ', V')
(x, , V)
if V' = D(' 
–1)|x V (i.e., V transforms
like a vector);
(3) An equivalence class of curves, tangent to each other at x.
Related Concepts > s.a. tensor; tensor
field.
* Distribution: A distribution S of
dimension r on M is
an assignment, to each p M,
of an r-dimensional subspace Sp 
TpM;
Involutive distribution: A distribution S such that for all X,
Y S,
[X, Y] S.
Generalization > s.a. Topological
Tangent Bundle.
@ Second-order: Dodson & Galanis JGP(04)
[infinite-dim manifolds].
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25 may 2008