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Keywords:
second order connection; non-holonomic 2-jet; absolute differentiation; bundle of smooth maps; connection on a functional bundle
Summary:
We study the second order connections in the sense of C. Ehresmann. On a fibered manifold \$Y\$, such a connection is a section from \$Y\$ into the second non-holonomic jet prolongation of \$Y\$. Our main aim is to extend the classical theory to the functional bundle of all smooth maps between the fibers over the same base point of two fibered manifolds over the same base. This requires several new geometric results about the second order connections on \$Y\$, which are deduced in the first part of the paper.
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