Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 1633.6 Tensor Fields on Smooth Manifolds3.6.1 Tensor BundleA tensor bundle T associated to a smooth n−manifold M is defined as atensor product of tangent and cotangent bundles:T =q⊗T ∗ M ⊗p⊗T M ={ }} {p timesT M ⊗ ... ⊗ T M ⊗{ }} {q timesT ∗ M ⊗ ... ⊗ T ∗ M.Tensor bundles are special case of more general fibre bundles (see section4.1 below).A tensor–field of type (p, q) on a smooth n−manifold M is defined as asmooth section τ : M −→ T of the tensor bundle T . The coefficients of thetensor–field τ are smooth (C k ) functions with p indices up and q indicesdown. The classical position of indices can be explained in modern termsas follows. If (U, φ) is a chart at a point m ∈ M with local coordinates(x 1 , ..., x n ), we have the holonomous frame field∂ x i 1 ⊗ ∂ x i 2 ⊗ ... ⊗ ∂ x ip ⊗ dx j1 ⊗ dx j2 ... ⊗ dx jq ,for i ∈ {1, ..., n} p , j = {1, ..., n} q , over U of this tensor bundle, and for any(p, q)−tensor–field τ we haveτ|U = τ i1...ipj 1...j q∂ x i 1 ⊗ ∂ x i 2 ⊗ ... ⊗ ∂ x ip ⊗ dx j1 ⊗ dx j2 ... ⊗ dx jq .For such tensor–fields the Lie derivative along any vector–field is defined(see section 3.7 below), and it is a derivation (i.e., both linearity and Leibnizrules hold) with respect to the tensor product. Tensor bundle T admitsmany natural transformations (see [Kolar et al. (1993)]). For example, a‘contraction’ like the trace T ∗ M ⊗ T M = L (T M, T M) → M × R, butapplied just to one specified factor of type T ∗ M and another one of typeT M, is a natural transformation. And any ‘permutation of the same kindof factors’ is a natural transformation.The tangent bundle π M : T M → M of a manifold M (introduced above)is a special tensor bundle over M such that, given an atlas {(U α , ϕ α )} ofM, T M has the holonomic atlasΨ = {(U α , ϕ α = T ϕ α )}.The associated linear bundle coordinates are the induced coordinates (ẋ λ )at a point m ∈ M with respect to the holonomic frames {∂ λ } in tangent