Affine bundle

In mathematics, an affine bundle is a fiber bundle whose typical fiber, fibers, trivialization morphisms and transition functions are affine. In mathematics, an affine bundle is a fiber bundle whose typical fiber, fibers, trivialization morphisms and transition functions are affine. Let π ¯ : Y ¯ → X {displaystyle {overline {pi }}:{overline {Y}} o X} be a vector bundle with a typical fiber a vector space F ¯ {displaystyle {overline {F}}} . An affine bundle modelled on a vector bundle π ¯ : Y ¯ → X {displaystyle {overline {pi }}:{overline {Y}} o X} is a fiber bundle π : Y → X {displaystyle pi :Y o X} whose typical fiber F {displaystyle F} is an affine space modelled on F ¯ {displaystyle {overline {F}}} so that the following conditions hold: (i) All the fiber Y x {displaystyle Y_{x}} of Y {displaystyle Y} are affine spaces modelled over the corresponding fibers Y ¯ x {displaystyle {overline {Y}}_{x}} of a vector bundle Y ¯ {displaystyle {overline {Y}}} . (ii) There is an affine bundle atlas of Y → X {displaystyle Y o X} whose local trivializations morphisms and transition functions are affine isomorphisms. Dealing with affine bundles, one uses only affine bundle coordinates ( x μ , y i ) {displaystyle (x^{mu },y^{i})} possessing affine transition functions There are the bundle morphisms where ( y ¯ i ) {displaystyle ({overline {y}}^{i})} are linear bundle coordinates on a vector bundle Y ¯ {displaystyle {overline {Y}}} , possessing linear transition functions y ¯ ′ i = A j i ( x ν ) y ¯ j {displaystyle {overline {y}}'^{i}=A_{j}^{i}(x^{ u }){overline {y}}^{j}} . An affine bundle has a global section, but in contrast with vector bundles, there is no canonical global section of an affine bundle. Let π : Y → X {displaystyle pi :Y o X} be an affine bundle modelled on a vector bundle π ¯ : Y ¯ → X {displaystyle {overline {pi }}:{overline {Y}} o X} . Every global section s {displaystyle s} of an affine bundle Y → X {displaystyle Y o X} yields the bundle morphisms In particular, every vector bundle Y {displaystyle Y} has a natural structure of an affine bundle due to these morphisms where s = 0 {displaystyle s=0} is the canonical zero-valued section of Y {displaystyle Y} . For instance, the tangent bundle T X {displaystyle TX} of a manifold X {displaystyle X} naturally is an affine bundle.