Smooth bundle

A smooth bundle is a locally trivial bundle with smooth transition functions.

Definition

Let be $Y$ ${\ displaystyle Y}$ $Y$ and $X$ ${\ displaystyle X}$ $X$ - smooth manifolds . Epimorphism of varieties $\pi \colon Y\to X$ ${\ displaystyle \ pi \ colon Y \ to X}$ $\pi \colon Y\to X$ called smooth delamination if exist: open cover $(U_{i})$ ${\ displaystyle (U_ {i})}$ $(U_{i})$ varieties $X$ ${\ displaystyle X}$ $X$ variety $V$ ${\ displaystyle V}$ $V$ and the family of diffeomorphisms $\varphi _{i}\colon \pi ^{-1}(U_{i})\to U_{i}\times V$ ${\ displaystyle \ varphi _ {i} \ colon \ pi ^ {- 1} (U_ {i}) \ to U_ {i} \ times V}$ $\varphi _{i}\colon \pi ^{-1}(U_{i})\to U_{i}\times V$ related smooth transition functions $\rho _{ij}=\varphi _{i}\varphi _{j}^{-1}$ ${\ displaystyle \ rho _ {ij} = \ varphi _ {i} \ varphi _ {j} ^ {- 1}}$ $\rho _{ij}=\varphi _{i}\varphi _{j}^{-1}$ on $U_{i}\cap U_{j}\times V$ ${\ displaystyle U_ {i} \ cap U_ {j} \ times V}$ $U_{i}\cap U_{j}\times V$ .

A smooth bundle is a locally trivial bundle with a bundle space. $Y$ ${\ displaystyle Y}$ $Y$ base $X$ ${\ displaystyle X}$ $X$ typical layer $V$ ${\ displaystyle V}$ $V$ and satin layering $(U_{i},\;\varphi _{i},\;\rho _{ij})$ ${\ displaystyle (U_ {i}, \; \ varphi _ {i}, \; \ rho _ {ij})}$ $(U_{i},\;\varphi _{i},\;\rho _{ij})$ . Closed subvariety $\pi ^{-1}(x)\subset Y$ ${\ displaystyle \ pi ^ {- 1} (x) \ subset Y}$ $\pi ^{-1}(x)\subset Y$ called a typical layer of a smooth bundle at a point $x\in X$ ${\ displaystyle x \ in X}$ $x\in X$ .

Examples

Vector bundle , in particular the tangent bundle
Main bundle

Properties

Space bundle $Y$ ${\ displaystyle Y}$ $Y$ endowed with coordinate atlas $(x^{\mu },\;y^{a})$ ${\ displaystyle (x ^ {\ mu}, \; y ^ {a})}$ $(x^{\mu },\;y^{a})$ where $(y^{a})$ ${\ displaystyle (y ^ {a})}$ $(y^{a})$ - coordinates on $V$ ${\ displaystyle V}$ $V$ and $(x^{\mu })$ ${\ displaystyle (x ^ {\ mu})}$ $(x^{\mu })$ - coordinates on $X$ ${\ displaystyle X}$ $X$ whose transition functions are independent of coordinates $(y^{a})$ ${\ displaystyle (y ^ {a})}$ $(y^{a})$ .
For every point $x\in X$ ${\ displaystyle x \ in X}$ $x\in X$ there is an open neighborhood $U$ ${\ displaystyle U}$ $U$ and attachment $s\colon U\to Y$ ${\ displaystyle s \ colon U \ to Y}$ $s\colon U\to Y$ such that $\pi \circ s=\mathrm {Id} \,(U)$ ${\ displaystyle \ pi \ circ s = \ mathrm {Id} \, (U)}$ $\pi \circ s=\mathrm {Id} \,(U)$ . This map is called a (local) section of a smooth bundle.

Variations and generalizations

Foliation
Superbundle
Graduated stratification
Banach and Hilbert bundles

Literature

Greub, W., Halperin S., Vanstone R. Connections, curvature and cohomology, vol. I — III. - N.-Y .: Academic Press, 1972-1976.
Kobayashi Sh., Nomizu K. Basics of differential geometry. - M .: Science, 1981. - T. 1. - 344 p.
Sardanashvili G. A. Modern methods of field theory. 1. Geometry and classical fields. - M .: URSS, 1996. - 224 p. - ISBN 5-88417-087-4 . .
Sardanashvily, G. , Fiber bundles, jet manifolds and Lagrangian theory. Lectures for theoreticians, arXiv: 0908.1886