Tangent bundle

Informally, the tangent bundle of a manifold (in this case the circle) is obtained by considering all tangent spaces (above) and combining them smoothly without intersections (below)

Tangent bundle of a smooth manifold $M$ ${\ displaystyle M}$ $M$ - there is a vector bundle over $M$ ${\ displaystyle M}$ $M$ whose layer is at a point $x\in M$ ${\ displaystyle x \ in M}$ $x \ in M$ is tangent space $T_{x}M$ ${\ displaystyle T_ {x} M}$ $T_ {x} M$ at the point $x$ ${\ displaystyle x}$ $x$ . Tangent bundle is usually denoted by $T M$ ${\ displaystyle TM}$ $TM$ .

Element of total space $T M$ ${\ displaystyle TM}$ $TM$ Is a couple $(x,\;v)$ ${\ displaystyle (x, \; v)}$ $(x, \; v)$ where $x\in M$ ${\ displaystyle x \ in M}$ $x \ in M$ and $v\in T_{x}M$ ${\ displaystyle v \ in T_ {x} M}$ $v \ in T_ {x} M$ . The tangent bundle has a natural topology (not the topology of a disjunctive union) and a smooth structure that turns it into a manifold. Dimension $T M$ ${\ displaystyle TM}$ $TM$ equal to twice the dimension $M$ ${\ displaystyle M}$ $M$ .

Topology and Smooth Structure

If a $M$ ${\ displaystyle M}$ $M$ - $n$ ${\ displaystyle n}$ $n$ -dimensional variety, then it has a map atlas $(U_{\alpha },\;\varphi _{\alpha })$ ${\ displaystyle (U _ {\ alpha}, \; \ varphi _ {\ alpha})}$ $(U_{\alpha },\;\varphi _{\alpha })$ where $U_{\alpha }$ ${\ displaystyle U _ {\ alpha}}$ $U_{\alpha }$ - open subset $M$ ${\ displaystyle M}$ $M$ and

\varphi _{\alpha }\colon U_{\alpha }\to \mathbb {R} ^{n}

{\ displaystyle \ varphi _ {\ alpha} \ colon U _ {\ alpha} \ to \ mathbb {R} ^ {n}}

\varphi _{\alpha }\colon U_{\alpha }\to \mathbb{R} ^{n}

- homeomorphism .

These local coordinates on $U$ ${\ displaystyle U}$ $U$ generate an isomorphism between $T_{x}M$ ${\ displaystyle T_ {x} M}$ $T_{x}M$ and $\mathbb {R} ^{n}$ ${\ displaystyle \ mathbb {R} ^ {n}}$ $\mathbb {R} ^{n}$ for anyone $x\in U$ ${\ displaystyle x \ in U}$ $x\in U$ . You can define a mapping

{\tilde {\varphi }}_{\alpha }\colon \pi ^{-1}(U_{\alpha })\to \mathbb {R} ^{2n}

{\ displaystyle {\ tilde {\ varphi}} _ {\ alpha} \ colon \ pi ^ {- 1} (U _ {\ alpha}) \ to \ mathbb {R} ^ {2n}}

{\tilde \varphi }_{\alpha }\colon \pi ^{{-1}}(U_{\alpha })\to \mathbb{R} ^{{2n}}

as

{\tilde {\varphi }}_{\alpha }(x,\;v^{i}\partial _{i})=(\varphi _{\alpha }(x),\;v^{1},\;\ldots ,\;v^{n}).

{\ displaystyle {\ tilde {\ varphi}} _ {\ alpha} (x, \; v ^ {i} \ partial _ {i}) = (\ varphi _ {\ alpha} (x), \; v ^ {1}, \; \ ldots, \; v ^ {n}).}

{\tilde \varphi }_{\alpha }(x,\;v^{i}\partial _{i})=(\varphi _{\alpha }(x),\;v^{1},\;\ldots ,\;v^{n}).

These mappings are used to determine the topology and smooth structure on $T M$ ${\ displaystyle TM}$ $TM$ .

Subset $A$ ${\ displaystyle A}$ $A$ of $T M$ ${\ displaystyle TM}$ $TM$ open if and only if ${\tilde {\varphi }}_{\alpha }(A\cap \pi ^{-1}(U_{\alpha }))$ ${\ displaystyle {\ tilde {\ varphi}} _ {\ alpha} (A \ cap \ pi ^ {- 1} (U _ {\ alpha}))}$ ${\tilde \varphi }_{\alpha }(A\cap \pi ^{{-1}}(U_{\alpha }))$ - open in $\mathbb {R} ^{2n}$ ${\ displaystyle \ mathbb {R} ^ {2n}}$ $\mathbb{R} ^{{2n}}$ for anyone $\alpha$ ${\ displaystyle \ alpha}$ $\alpha$ . These mappings are homeomorphisms of open subsets $T M$ ${\ displaystyle TM}$ $TM$ and $\mathbb {R} ^{2n}$ ${\ displaystyle \ mathbb {R} ^ {2n}}$ $\mathbb{R} ^{{2n}}$ therefore they form maps of smooth structure on $T M$ ${\ displaystyle TM}$ $TM$ . Transition Functions at Map Intersections $\pi ^{-1}(U_{\alpha }\cap U_{\beta })$ ${\ displaystyle \ pi ^ {- 1} (U _ {\ alpha} \ cap U _ {\ beta})}$ $\pi ^{{-1}}(U_{\alpha }\cap U_{\beta })$ are defined by the Jacobi matrices of the corresponding coordinate transformations; therefore, they are smooth mappings of open subsets $\mathbb {R} ^{2n}$ ${\ displaystyle \ mathbb {R} ^ {2n}}$ $\mathbb{R} ^{{2n}}$ .

Tangent bundle is a special case of a more general construction called vector bundle . Tangent bundle $n$ ${\ displaystyle n}$ $n$ -dimensional diversity $M$ ${\ displaystyle M}$ $M$ can be defined as a vector bundle of rank $n$ ${\ displaystyle n}$ $n$ above $M$ ${\ displaystyle M}$ $M$ whose transition functions are specified by the Jacobian of the corresponding coordinate transformations.

Examples

The simplest example is obtained for $\mathbb {R} ^{n}$ ${\ displaystyle \ mathbb {R} ^ {n}}$ . In this case, the tangent bundle is trivial and isomorphic to the projection $\mathbb {R} ^{2n}\to \mathbb {R} ^{n}$ ${\ displaystyle \ mathbb {R} ^ {2n} \ to \ mathbb {R} ^ {n}}$ .
Unit circle $S^{1}$ ${\ displaystyle S ^ {1}}$ . Its tangent bundle is also trivial and isomorphic. $S^{1}\times \mathbb {R}$ ${\ displaystyle S ^ {1} \ times \ mathbb {R}}$ . Geometrically, it is a cylinder of infinite height (see image above).
A simple example of a nontrivial tangent bundle is obtained on the unit sphere $S^{2},$ ${\ displaystyle S ^ {2},}$ this tangent bundle is nontrivial due to the hedgehog combing theorem .

Unfortunately, only tangent bundles of the real line can be represented $R$ ${\ displaystyle R}$ and unit circle $S^{1}$ ${\ displaystyle S ^ {1}}$ which are both trivial. For two-dimensional manifolds, the tangent bundle is a 4-dimensional manifold, so it is difficult to imagine.

Vector Fields

A vector field is a smooth vector function on a manifold $M$ ${\ displaystyle M}$ whose value at each point is a vector tangent to $M$ ${\ displaystyle M}$ i.e. smooth display

V\colon M\to TM

{\ displaystyle V \ colon M \ to TM}

such that the image $x$ ${\ displaystyle x}$ denoted by $V_{x}$ ${\ displaystyle V_ {x}}$ lies in $T_{x}M$ ${\ displaystyle T_ {x} M}$ - tangent space at a point $x$ ${\ displaystyle x}$ . In the language of locally trivial bundles , such a mapping is called a section . Vector box on $M$ ${\ displaystyle M}$ Is the section of the tangent bundle over $M$ ${\ displaystyle M}$ .

The set of all vector fields over $M$ ${\ displaystyle M}$ is indicated $\Gamma (TM)$ ${\ displaystyle \ Gamma (TM)}$ . Vector fields can be added pointwise:

(V+W)_{x}=V_{x}+W_{x}

{\ displaystyle (V + W) _ {x} = V_ {x} + W_ {x}}

and multiply by smooth functions by $M$ ${\ displaystyle M}$

(fV)_{x}=f(x)V_{x},

{\ displaystyle (fV) _ {x} = f (x) V_ {x},}

getting new vector fields. The set of all vector fields $\Gamma (TM)$ ${\ displaystyle \ Gamma (TM)}$ obtains the structure of the module over the commutative algebra of smooth functions on $M$ ${\ displaystyle M}$ (indicated by $C^{\infty }(M)$ ${\ displaystyle C ^ {\ infty} (M)}$ )

If a $f$ ${\ displaystyle f}$ is a smooth function, then the differentiation operation along the vector field $X$ ${\ displaystyle X}$ gives a new smooth function $X f$ ${\ displaystyle Xf}$ . This differentiation operator has the following properties:

Additivity: $X(f+h)=Xf+Xh$ ${\ displaystyle X (f + h) = Xf + Xh}$ .
Leibniz rule : $X(fh)=(Xf)\cdot h+f\cdot (Xh)$ ${\ displaystyle X (fh) = (Xf) \ cdot h + f \ cdot (Xh)}$ .

A vector field on a manifold can also be defined as an operator possessing the above properties.

Local vector field on $M$ ${\ displaystyle M}$ Is the local section of the tangent bundle. A local vector field is determined only on some open subset $U$ ${\ displaystyle U}$ of $M$ ${\ displaystyle M}$ at the same time at each point of $U$ ${\ displaystyle U}$ defines a vector from the corresponding tangent space. The set of local vector fields on $M$ ${\ displaystyle M}$ forms a structure called a sheaf of real vector spaces over $M$ ${\ displaystyle M}$ .

Canonical Vector Field on TM

On each tangent bundle $T M$ ${\ displaystyle TM}$ canonical vector field can be defined. If a $(x,\;y)$ ${\ displaystyle (x, \; y)}$ - local coordinates on $T M$ ${\ displaystyle TM}$ , then the vector field has the form

V=\left.y^{i}{\frac {\partial }{\partial y^{i}}}\right|_{(x,\;y)}.

{\ displaystyle V = \ left.y ^ {i} {\ frac {\ partial} {\ partial y ^ {i}}} \ right | _ {(x, \; y)}.}

$V$ ${\ displaystyle V}$ is a mapping $V\colon TM\to TTM$ ${\ displaystyle V \ colon TM \ to TTM}$ .

The existence of such a vector field on $T M$ ${\ displaystyle TM}$ can be compared with the existence of a canonical 1-form on a cotangent bundle .

Links

Arnold V.I. Mathematical methods of classical mechanics. - 5th ed., Stereotyped. - M .: URSS editorial, 2003 .-- 416 p. - 1,500 copies - ISBN 5-354-00341-5 .
Vasiliev V.A. Introduction to topology. - M .: FAZIS, 1997 .-- 132 p. - ISBN 5-7036-0036-7 .
John M. Lee. Introduction to Smooth Manifolds. - New York: Springer-Verlag, 2003 .-- ISBN 0-387-95495-3 .
Jurgen Jost. Riemannian Geometry and Geometric Analysis. - Berlin: Springer-Verlag, 2002 .-- ISBN 3-540-42627-2 .
Todd Rowland Tangent Bundle on Wolfram MathWorld .
Tangent Bundle on the PlanetMath website .