By By (author) Wolfgang Bertram

Differential Geometry, Lie teams and Symmetric areas Over common Base Fields and earrings

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Transition functions and chart representation of higher order tangent maps. For a good understanding of the structure of T k M we will need to know the structure of the transition functions T k ϕij of T k M , and more generally, the structure of higher order tangent maps T k f . For k = 1 , recall that, for a smooth map f : V ⊃ U → W (which represents a smooth map M → N with respect to charts), we have T f (x + εv) = f (x) + εdf (x)v. 6 (iv)), we may calculate d(T f ) and hence T T f . One gets T 2 f (x + ε1 v1 + ε2 v2 + ε1 ε2 v12 ) = f (x) + ε1 df (x)v1 + ε2 df (x)v2 + ε1 ε2 df (x)v12 + d2 f (x)(v1 , v2 ) .

3 by checking the definition of differentiability over K[ε] directly, leads to fairly long and involved calculations. ) 42 DIFFERENTIAL GEMETRY OVER GENERAL BASE FIELDS AND RINGS 7. Scalar extensions. 1. Higher order tangent bundles. For any manifold M , the second order tangent bundle is T 2 M := T (T M ) , and the k -th order tangent bundle is inductively defined by T k M := T (T k−1 M ) . 2. The k -th order tangent bundle T k M is, in a canonical way, a manifold over the ring of iterated dual numbers T k K = K[ε1 ] .

2. If M is a manifold of class C k+1 over K, modelled on V , then T M is, in a canonical way, a manifold of class C k over K[ε], modelled on the scalar extended K[ε] -module V ⊕ εV . If f : M → N is of class C k+1 over K, then T f : T M → T N is of class C k over K[ε] . Thus the tangent functor can be characterized as the “functor of scalar extension” from manifolds over K into manifolds over K[ε] agreeing on open submanifolds of K-modules with the algebraic scalar extension functor. Proof. Change of charts on T M is described by the transition maps T ϕij (x, v) = (ϕij (x), dϕij (x)v) .