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Tangent bundle
Informally, the tangent bundle of a manifold (which in this case is a circle) is obtained by considering all the tangent spaces (top), and joining them together in a smooth and non-overlapping manner (bottom).
A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Formally, in differential geometry, the tangent bundle of a differentiable manifold
M
{\displaystyle M}
is a manifold
T
M
{\displaystyle TM}
which assembles all the tangent vectors in
M
{\displaystyle M}
. As a set, it is given by the disjoint union of the tangent spaces of
M
{\displaystyle M}
. That is,
T M
=
⨆
x
∈
M
T
x
M
=
⋃
x
∈
M
{
x
}
×
T
x
M
=
⋃
x
∈
M
{
(
x
,
y
)
∣
y
∈
T
x
M
}
=
{
(
x
,
y
)
∣
x
∈
M
,
y
∈
T
x
M
}
{\displaystyle {\begin{aligned}TM&=\bigsqcup _{x\in M}T_{x}M\\&=\bigcup _{x\in M}\left\{x\right\}\times T_{x}M\\&=\bigcup _{x\in M}\left\{(x,y)\mid y\in T_{x}M\right\}\\&=\left\{(x,y)\mid x\in M,\,y\in T_{x}M\right\}\end{aligned}}}
where
T
x
M
{\displaystyle T_{x}M}
denotes the tangent space to
M
{\displaystyle M}
at the point
x
{\displaystyle x}
. So, an element of
T
M
{\displaystyle TM}
can be thought of as a pair
(
x
,
v
)
{\displaystyle (x,v)}
, where
x
{\displaystyle x}
is a point in
M
{\displaystyle M}
and
v
{\displaystyle v}
is a tangent vector to
M
{\displaystyle M}
at
x
{\displaystyle x}
.
There is a natural projection
π : T M ↠ M
{\displaystyle \pi :TM\twoheadrightarrow M}
defined by
π
(
x
,
v
)
=
x
{\displaystyle \pi (x,v)=x}
. This projection maps each element of the tangent space
T
x
M
{\displaystyle T_{x}M}
to the single point
x
{\displaystyle x}
.
The tangent bundle comes equipped with a natural topology (described in a section below). With this topology, the tangent bundle to a manifold is the prototypical example of a vector bundle (which is a fiber bundle whose fibers are vector spaces). A section of
T
M
{\displaystyle TM}
is a vector field on
M
{\displaystyle M}
, and the dual bundle to
T
M
{\displaystyle TM}
is the cotangent bundle, which is the disjoint union of the cotangent spaces of
M
{\displaystyle M}
. By definition, a manifold
M
{\displaystyle M}
is parallelizable if and only if the tangent bundle is trivial. By definition, a manifold
M
{\displaystyle M}
is framed if and only if the tangent bundle
T
M
{\displaystyle TM}
is stably trivial, meaning that for some trivial bundle
E
{\displaystyle E}
the Whitney sum
T
M
⊕
E
{\displaystyle TM\oplus E}
is trivial. For example, the n-dimensional sphere Sn is framed for all n, but parallelizable only for n = 1, 3, 7 (by results of Bott-Milnor and Kervaire).
One of the main roles of the tangent bundle is to provide a domain and range for the derivative of a smooth function. Namely, if
f
:
M
→
N
{\displaystyle f:M\rightarrow N}
is a smooth function, with
M
{\displaystyle M}
and
N
{\displaystyle N}
smooth manifolds, its derivative is a smooth function
D
f
:
T
M
→
T
N
{\displaystyle Df:TM\rightarrow TN}
.
The tangent bundle comes equipped with a natural topology (not the disjoint union topology) and smooth structure so as to make it into a manifold in its own right. The dimension of
T
M
{\displaystyle TM}
is twice the dimension of
M
{\displaystyle M}
.
Each tangent space of an n-dimensional manifold is an n-dimensional vector space. If
U
{\displaystyle U}
is an open contractible subset of
M
{\displaystyle M}
, then there is a diffeomorphism
T
U
→
U
×
R
n
{\displaystyle TU\to U\times \mathbb {R} ^{n}}
which restricts to a linear isomorphism from each tangent space
T
x
U
{\displaystyle T_{x}U}
to
{
x
}
×
R
n
{\displaystyle \{x\}\times \mathbb {R} ^{n}}
. As a manifold, however,
T
M
{\displaystyle TM}
is not always diffeomorphic to the product manifold
M
×
R
n
{\displaystyle M\times \mathbb {R} ^{n}}
. When it is of the form
M
×
R
n
{\displaystyle M\times \mathbb {R} ^{n}}
, then the tangent bundle is said to be trivial. Trivial tangent bundles usually occur for manifolds equipped with a 'compatible group structure'; for instance, in the case where the manifold is a Lie group. The tangent bundle of the unit circle is trivial because it is a Lie group (under multiplication and its natural differential structure). It is not true however that all spaces with trivial tangent bundles are Lie groups; manifolds which have a trivial tangent bundle are called parallelizable. Just as manifolds are locally modeled on Euclidean space, tangent bundles are locally modeled on
U
×
R
n
{\displaystyle U\times \mathbb {R} ^{n}}
, where
U
{\displaystyle U}
is an open subset of Euclidean space.
If M is a smooth n-dimensional manifold, then it comes equipped with an atlas of charts
(
U
α
,
ϕ
α
)
{\displaystyle (U_{\alpha },\phi _{\alpha })}
, where
U
α
{\displaystyle U_{\alpha }}
is an open set in
M
{\displaystyle M}
and
ϕ
α
:
U
α
→
R
n
{\displaystyle \phi _{\alpha }:U_{\alpha }\to \mathbb {R} ^{n}}
is a diffeomorphism. These local coordinates on
U
α
{\displaystyle U_{\alpha }}
give rise to an isomorphism
T
x
M
→
R
n
{\displaystyle T_{x}M\rightarrow \mathbb {R} ^{n}}
for all
x
∈
U
α
{\displaystyle x\in U_{\alpha }}
. We may then define a map
ϕ ~
α
:
π
−
1
(
U
α
)
→
R
2
n
{\displaystyle {\widetilde {\phi }}_{\alpha }:\pi ^{-1}\left(U_{\alpha }\right)\to \mathbb {R} ^{2n}}
by
ϕ ~
α
(
x
,
v
i
∂
i
)
=
(
ϕ
α
(
x
)
,
v
1
,
⋯
,
v
n
)
{\displaystyle {\widetilde {\phi }}_{\alpha }\left(x,v^{i}\partial _{i}\right)=\left(\phi _{\alpha }(x),v^{1},\cdots ,v^{n}\right)}
We use these maps to define the topology and smooth structure on
T
M
{\displaystyle TM}
. A subset
A
{\displaystyle A}
of
T
M
{\displaystyle TM}
is open if and only if
ϕ ~
α
(
A
∩
π
−
1
(
U
α
)
)
{\displaystyle {\widetilde {\phi }}_{\alpha }\left(A\cap \pi ^{-1}\left(U_{\alpha }\right)\right)}
is open in
R
2
n
{\displaystyle \mathbb {R} ^{2n}}
for each
α
.
{\displaystyle \alpha .}
These maps are homeomorphisms between open subsets of
T
M
{\displaystyle TM}
and
R
2
n
{\displaystyle \mathbb {R} ^{2n}}
and therefore serve as charts for the smooth structure on
T
M
{\displaystyle TM}
. The transition functions on chart overlaps
π
−
1
(
U
α
∩
U
β
)
{\displaystyle \pi ^{-1}\left(U_{\alpha }\cap U_{\beta }\right)}
are induced by the Jacobian matrices of the associated coordinate transformation and are therefore smooth maps between open subsets of
R
2
n
{\displaystyle \mathbb {R} ^{2n}}
.
The tangent bundle is an example of a more general construction called a vector bundle (which is itself a specific kind of fiber bundle). Explicitly, the tangent bundle to an
n
{\displaystyle n}
-dimensional manifold
M
{\displaystyle M}
may be defined as a rank
n
{\displaystyle n}
vector bundle over
M
{\displaystyle M}
whose transition functions are given by the Jacobian of the associated coordinate transformations.
The simplest example is that of
R
n
{\displaystyle \mathbb {R} ^{n}}
. In this case the tangent bundle is trivial: each
T
x
R
n
{\displaystyle T_{x}\mathbf {\mathbb {R} } ^{n}}
is canonically isomorphic to
T
0
R
n
{\displaystyle T_{\mathbf {0} }\mathbb {R} ^{n}}
via the map
R
n
→
R
n
{\displaystyle \mathbb {R} ^{n}\to \mathbb {R} ^{n}}
which subtracts
x
{\displaystyle x}
, giving a diffeomorphism
T
R
n
→
R
n
×
R
n
{\displaystyle T\mathbb {R} ^{n}\to \mathbb {R} ^{n}\times \mathbb {R} ^{n}}
.
Another simple example is the unit circle,
S
1
{\displaystyle S^{1}}
(see picture above). The tangent bundle of the circle is also trivial and isomorphic to
S
1
×
R
{\displaystyle S^{1}\times \mathbb {R} }
. Geometrically, this is a cylinder of infinite height.
The only tangent bundles that can be readily visualized are those of the real line
R
{\displaystyle \mathbb {R} }
and the unit circle
S
1
{\displaystyle S^{1}}
, both of which are trivial. For 2-dimensional manifolds the tangent bundle is 4-dimensional and hence difficult to visualize.
A simple example of a nontrivial tangent bundle is that of the unit sphere
S
2
{\displaystyle S^{2}}
: this tangent bundle is nontrivial as a consequence of the hairy ball theorem. Therefore, the sphere is not parallelizable.
A smooth assignment of a tangent vector to each point of a manifold is called a vector field. Specifically, a vector field on a manifold
M
{\displaystyle M}
is a smooth map
V : M → T M
{\displaystyle V\colon M\to TM}
such that
V
(
x
)
=
(
x
,
V
x
)
{\displaystyle V(x)=(x,V_{x})}
with
V
x
∈
T
x
M
{\displaystyle V_{x}\in T_{x}M}
for every
x
∈
M
{\displaystyle x\in M}
. In the language of fiber bundles, such a map is called a section. A vector field on
M
{\displaystyle M}
is therefore a section of the tangent bundle of
M
{\displaystyle M}
.
The set of all vector fields on
M
{\displaystyle M}
is denoted by
Γ
(
T
M
)
{\displaystyle \Gamma (TM)}
. Vector fields can be added together pointwise
( V + W
)
x
=
V
x
+
W
x
{\displaystyle (V+W)_{x}=V_{x}+W_{x}}
and multiplied by smooth functions on M
( f V
)
x
=
f
(
x
)
V
x
{\displaystyle (fV)_{x}=f(x)V_{x}}
to get other vector fields. The set of all vector fields
Γ
(
T
M
)
{\displaystyle \Gamma (TM)}
then takes on the structure of a module over the commutative algebra of smooth functions on M, denoted
C
∞
(
M
)
{\displaystyle C^{\infty }(M)}
.
A local vector field on
M
{\displaystyle M}
is a local section of the tangent bundle. That is, a local vector field is defined only on some open set
U
⊂
M
{\displaystyle U\subset M}
and assigns to each point of
U
{\displaystyle U}
a vector in the associated tangent space. The set of local vector fields on
M
{\displaystyle M}
forms a structure known as a sheaf of real vector spaces on
M
{\displaystyle M}
.
The above construction applies equally well to the cotangent bundle – the differential 1-forms on
M
{\displaystyle M}
are precisely the sections of the cotangent bundle
ω
∈
Γ
(
T
∗
M
)
{\displaystyle \omega \in \Gamma (T^{*}M)}
,
ω
:
M
→
T
∗
M
{\displaystyle \omega :M\to T^{*}M}
that associate to each point
x
∈
M
{\displaystyle x\in M}
a 1-covector
ω
x
∈
T
x
∗
M
{\displaystyle \omega _{x}\in T_{x}^{*}M}
, which map tangent vectors to real numbers:
ω
x
:
T
x
M
→
R
{\displaystyle \omega _{x}:T_{x}M\to \mathbb {R} }
. Equivalently, a differential 1-form
ω
∈
Γ
(
T
∗
M
)
{\displaystyle \omega \in \Gamma (T^{*}M)}
maps a smooth vector field
X
∈
Γ
(
T
M
)
{\displaystyle X\in \Gamma (TM)}
to a smooth function
ω
(
X
)
∈
C
∞
(
M
)
{\displaystyle \omega (X)\in C^{\infty }(M)}
.
Since the tangent bundle
T
M
{\displaystyle TM}
is itself a smooth manifold, the second-order tangent bundle can be defined via repeated application of the tangent bundle construction:
T
2
M
=
T
(
T
M
)
.
{\displaystyle T^{2}M=T(TM).\,}
In general, the
k
{\displaystyle k}
th order tangent bundle
T
k
M
{\displaystyle T^{k}M}
can be defined recursively as
T
(
T
k
−
1
M
)
{\displaystyle T\left(T^{k-1}M\right)}
.
A smooth map
f
:
M
→
N
{\displaystyle f:M\rightarrow N}
has an induced derivative, for which the tangent bundle is the appropriate domain and range
D
f
:
T
M
→
T
N
{\displaystyle Df:TM\rightarrow TN}
. Similarly, higher-order tangent bundles provide the domain and range for higher-order derivatives
D
k
f
:
T
k
M
→
T
k
N
{\displaystyle D^{k}f:T^{k}M\to T^{k}N}
.
A distinct but related construction are the jet bundles on a manifold, which are bundles consisting of jets.
On every tangent bundle
T
M
{\displaystyle TM}
, considered as a manifold itself, one can define a canonical vector field
V
:
T
M
→
T
2
M
{\displaystyle V:TM\rightarrow T^{2}M}
as the diagonal map on the tangent space at each point. This is possible because the tangent space of a vector space W is naturally a product,
T
W
≅
W
×
W
,
{\displaystyle TW\cong W\times W,}
since the vector space itself is flat, and thus has a natural diagonal map
W
→
T
W
{\displaystyle W\to TW}
given by
w
↦
(
w
,
w
)
{\displaystyle w\mapsto (w,w)}
under this product structure. Applying this product structure to the tangent space at each point and globalizing yields the canonical vector field. Informally, although the manifold
M
{\displaystyle M}
is curved, each tangent space at a point
x
{\displaystyle x}
,
T
x
M
≈
R
n
{\displaystyle T_{x}M\approx \mathbb {R} ^{n}}
, is flat, so the tangent bundle manifold
T
M
{\displaystyle TM}
is locally a product of a curved
M
{\displaystyle M}
and a flat
R
n
.
{\displaystyle \mathbb {R} ^{n}.}
Thus the tangent bundle of the tangent bundle is locally (using
≈
{\displaystyle \approx }
for "choice of coordinates" and
≅
{\displaystyle \cong }
for "natural identification"):
T ( T M ) ≈ T ( M ×
R
n
)
≅
T
M
×
T
(
R
n
)
≅
T
M
×
(
R
n
×
R
n
)
{\displaystyle T(TM)\approx T(M\times \mathbb {R} ^{n})\cong TM\times T(\mathbb {R} ^{n})\cong TM\times (\mathbb {R} ^{n}\times \mathbb {R} ^{n})}
and the map
T
T
M
→
T
M
{\displaystyle TTM\to TM}
is the projection onto the first coordinates:
( T M → M ) × (
R
n
×
R
n
→
R
n
)
.
{\displaystyle (TM\to M)\times (\mathbb {R} ^{n}\times \mathbb {R} ^{n}\to \mathbb {R} ^{n}).}
Splitting the first map via the zero section and the second map by the diagonal yields the canonical vector field.
If
(
x
,
v
)
{\displaystyle (x,v)}
are local coordinates for
T
M
{\displaystyle TM}
, the vector field has the expression
V =
∑
i
v
i
∂
∂
v
i
|
(
x
,
v
)
.
{\displaystyle V=\sum _{i}\left.v^{i}{\frac {\partial }{\partial v^{i}}}\right|_{(x,v)}.}
More concisely,
(
x
,
v
)
↦
(
x
,
v
,
0
,
v
)
{\displaystyle (x,v)\mapsto (x,v,0,v)}
– the first pair of coordinates do not change because it is the section of a bundle and these are just the point in the base space: the last pair of coordinates are the section itself. This expression for the vector field depends only on
v
{\displaystyle v}
, not on
x
{\displaystyle x}
, as only the tangent directions can be naturally identified.
Alternatively, consider the scalar multiplication function:
{
R
×
T
M
→
T
M
(
t
,
v
)
⟼
t
v
{\displaystyle {\begin{cases}\mathbb {R} \times TM\to TM\\(t,v)\longmapsto tv\end{cases}}}
The derivative of this function with respect to the variable
R
{\displaystyle \mathbb {R} }
at time
t
=
1
{\displaystyle t=1}
is a function
V
:
T
M
→
T
2
M
{\displaystyle V:TM\rightarrow T^{2}M}
, which is an alternative description of the canonical vector field.
The existence of such a vector field on
T
M
{\displaystyle TM}
is analogous to the canonical one-form on the cotangent bundle. Sometimes
V
{\displaystyle V}
is also called the Liouville vector field, or radial vector field. Using
V
{\displaystyle V}
one can characterize the tangent bundle. Essentially,
V
{\displaystyle V}
can be characterized using 4 axioms, and if a manifold has a vector field satisfying these axioms, then the manifold is a tangent bundle and the vector field is the canonical vector field on it. See for example, De León et al.
There are various ways to lift objects on
M
{\displaystyle M}
into objects on
T
M
{\displaystyle TM}
. For example, if
γ
{\displaystyle \gamma }
is a curve in
M
{\displaystyle M}
, then
γ
′
{\displaystyle \gamma '}
(the tangent of
γ
{\displaystyle \gamma }
) is a curve in
T
M
{\displaystyle TM}
. In contrast, without further assumptions on
M
{\displaystyle M}
(say, a Riemannian metric), there is no similar lift into the cotangent bundle.
The vertical lift of a function
f
:
M
→
R
{\displaystyle f:M\rightarrow \mathbb {R} }
is the function
f
∨
:
T
M
→
R
{\displaystyle f^{\vee }:TM\rightarrow \mathbb {R} }
defined by
f
∨
=
f
∘
π
{\displaystyle f^{\vee }=f\circ \pi }
, where
π
:
T
M
→
M
{\displaystyle \pi :TM\rightarrow M}
is the canonical projection.
- Pushforward (differential)
- Unit tangent bundle
- Cotangent bundle
- Frame bundle
- Musical isomorphism
- Holomorphic tangent bundle
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