Tangent Structures to a Manifold| Tangent Structures to a Manifold |
Tangent Vector at a Point > s.a. vector;
vector calculus; vector field.
$ Def: There are various possible definitions, including:
(1) A derivation on the algebra of germs of differentiable functions at x ∈ X;
(2) An equivalence class of triples (x, φ, V),
with (x, φ', V') ~ (x, φ, V)
if V' = D(φ' \(\circ\)
φ−1)|x
V (i.e., V transforms like a vector);
(3) An equivalence class of curves, tangent to each other at x.
> Online resources:
see MathWorld page;
Wikipedia page on tangent vector
and tangent space.
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, \(\mathbb R\)).
* 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, \(\mathbb R\)).
@ References: Yano & Ishihara 73;
Morandi et al PRP(90);
Hindeleh 09 [of Lie groups].
> Online resources:
see Wikipedia page.
Related Concepts > s.a. Jet and Jet Bundle;
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.
* Push-forward map: Given a map f :
M → N between differentiable manifolds, the pushforward f '
or f* is a map between vector fields.
* Tangent map: Given a map f :
M → N between differentiable manifolds, the tangent map
Tf is a map between vectors (elements of TM and TN).
Cotangent Structures > s.a. differential forms.
$ Cotangent vector: A cotangent vector
at a point p ∈ M is a dual vector, i.e., a map ω:
TpM → \(\mathbb R\) from vectors
to the reals.
$ Cotangent bundle: The set T*M of
all cotangent vectors at all points of an n-dimensional manifold M, with
a differentiable fiber bundle structure.
Generalizations
> s.a. Topological Tangent Bundle.
@ Second-order tangent structures:
Dodson & Galanis JGP(04) [infinite-dimensional manifolds].
@ Related topics: in Boroojerdian IJTP(13)-a1211
[\(\mathbb Z\)2-graded tangent bundle].
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send feedback and suggestions to bombelli at olemiss.edu – modified 14 jan 2016