By J.P. Jouanolou

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1). R will be called the curvature tensor of the affine connection V. If V = TM, the identity mapping 9 : TM -p TM can be regarded as a 1-form and dv9 is called the torsion tensor of the connection V. 1. For any w E AT(VM), r > 0, 1. 4W=RAW, - 2. do (R Aw) = R Adow . ) If VM is the trivial bundle M x R, then the space of all r-forms is denoted by AT (M). AT (M) _ {0} for r < 0 and for r > dim M. Denote the direct product lITr>O AT (M) by A* (M). If VM is the trivial bundle M x R with trivial connection D, then do will be denoted simply by d.

Obviously E1 is densely embedded in E and the inclusion mapping is continuous. Let E-1 be the dual space of E1 with the usual operator norm. E-1 is a Banach space at this moment, and since E1 is dense in E one obtains E C E-1 and the inclusion mapping is continuous. For every x E E1, the linear function y H (y, x)1 is understood as an element of E-1. We denote this element by Jx. J : E1 -* E-1 is a continuous linear injection and is also a surjection by the Riesz theorem. By the closed graph theorem, J : E1 -* E-1 is a linear isomorphism.

Vector bundles and affine connections, given in §2, are fundamental concepts in infinite-dimenaional calculus. In §3 we use these concepts to discuss covariant exterior derivatives and Lie derivatives. We assume C°° differentiability unless stated otherwise. 3 in Chapter I, we do not need to consider the delicate difference between various definitions of differentiabilities. In §5, the Frobenius theorem on B-manifolds is given in a form that can be applied to control theory. In §6, Sobolev manifolds are presented as a nonlinear version of Sobolev chains.