double vector bundle

A double vector bundle consists of $(E,\; E^H,\; E^V,\; B)$, where

1. the side bundles $E^H$ and $E^V$ are vector bundles over the base $B$,
2. $E$ is a vector bundle on both side bundles $E^H$ and $E^V$,
3. the projection, the addition, the scalar multiplication and the zero map on E for both vector bundle structures are morphisms.

A double vector bundle morphism $(f\_E,\; f\_H,\; f\_V,\; f\_B)$ consists of maps $f\_E\; :\; E\; \backslash mapsto\; E\text{'}$, $f\_H\; :\; E^H\; \backslash mapsto\; E^H\{\}\text{'}$, $f\_V\; :\; E^V\; \backslash mapsto\; E^V\{\}\text{'}$ and $f\_B\; :\; B\; \backslash mapsto\; B\text{'}$ such that $(f\_E,\; f\_V)$ is a bundle morphism from $(E,\; E^V)$ to $(E\text{'},\; E^V\{\}\text{'})$, $(f\_E,\; f\_H)$ is a bundle morphism from $(E,\; E^H)$ to $(E\text{'},\; E^H\{\}\text{'})$, $(f\_V,\; f\_B)$ is a bundle morphism from $(E^V,\; B)$ to $(E^V\{\}\text{'},\; B\text{'})$ and $(f\_H,\; f\_B)$ is a bundle morphism from $(E^H,\; B)$ to $(E^H\{\}\text{'},\; B\text{'})$.

The 'flip of the double vector bundle $(E,\; E^H,\; E^V,\; B)$ is the double vector bundle $(E,\; E^V,\; E^H,\; B)$.

If $(E,\; M)$ is a vector bundle over a differentiable manifold $M$ then $(TE,\; E,\; TM,\; M)$ is a double vector bundle when considering its secondary vector bundle structure.

If $M$ is a differentiable manifold, then its double tangent bundle $(TTM,\; TM,\; TM,\; M)$ is a double vector bundle.

{{Citation

| last = Mackenzie

| first = K.

| title = Double Lie algebroids and second-order geometry, I

| journal = Advances in Mathematics

| volume = 94

| number = 2

| date = 1992

| pages = 180–239

| doi=10.1016/0001-8708(92)90036-k

| doi-access = free

}}

Category:Differential geometry

Category:Topology

Category:Differential topology