Ivancevic_Applied-Diff-Geom

Glossary of Frequently Used SymbolsxiiiSmooth Manifolds, Fibre Bundles and Jet SpacesUnless otherwise specified, all manifolds M, N, ... are assumed C k −smooth,real, finite–dimensional, Hausdorff, paracompact, connected and withoutboundary, 1 while all maps are assumed C k −smooth. We use the symbols⊗, ∨, ∧ and ⊕ for the tensor, symmetrized and exterior products, as well asthe Whitney sum 2 , respectively, while ⌋ denotes the interior product (contraction)of (multi)vectors and p−forms, and ↩→ denotes a manifold imbedding(i.e., both a submanifold and a topological subspace of the codomainmanifold). The symbols ∂B A denote partial derivatives with respect to coordinatespossessing multi–indices B A (e.g., ∂ α = ∂/∂x α );T M – tangent bundle of the manifold M;π M : T M → M – natural projection;T ∗ M – cotangent bundle of the manifold M;π : Y → X – fibre bundle;(E, π, M) – vector bundle with total space E, base M and projection π;(Y, π, X, V ) – fibre bundle with total space Y , base X, projection π andstandard fibre V ;J k (M, N) – space of k−jets of smooth functions between manifolds M andN;J k (X, Y ) – k−-jet space of a fibre bundle Y → X; in particular, inmechanics we have a 1–jet space J 1 (R, Q), with 1–jet coordinate mapsjt 1 s : t ↦→ (t, x i , ẋ i ), as well as a 2–jet space J 2 (R, Q), with 2–jet coordinatemaps jt 2 s : t ↦→ (t, x i , ẋ i , ẍ i );jxs k – k−jets of sections s i : X → Y of a fibre bundle Y → X;We use the following kinds of manifold maps: immersion, imbedding, submersion,and projection. A map f : M → M ′ is called the immersion ifthe tangent map T f at every point x ∈ M is an injection (i.e., ‘1–1’ map).When f is both an immersion and an injection, its image is said to be asubmanifold of M ′ . A submanifold which also is a topological subspace iscalled imbedded submanifold. A map f : M → M ′ is called submersion ifthe tangent map T f at every point x ∈ M is a surjection (i.e., ‘onto’ map).If f is both a submersion and a surjection, it is called projection or fibrebundle.1 The only 1D manifolds obeying these conditions are the real line R and the circleS 1 .2 Whitney sum ⊕ is an analog of the direct (Cartesian) product for vector bundles.Given two vector bundles Y and Y ′ over the same base X, their Cartesian product is avector bundle over X × X. The diagonal map induces a vector bundle over X called theWhitney sum of these vector bundles and denoted by Y ⊕ Y ′ .