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Summary
The geodesic graph of Riemannian spaces all geodesics of which are orbits of 1-parameter isometry groups was constructed by
J. Szenthe in 1976 and it became a basic tool for studying such spaces, called g.o.\ spaces. This infinitesimal structure
corresponds to the reductive complement
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EN 1993-1.8, Design of steel structures , Part 1.8, Connections, (in Hungarian), 2005. Iványi M. Ductility of steel structures, the model of interactive hinge, Pollack Periodica
Abstract
We prove that a diffeomorphism of a manifold with an Ehresmann connection is an automorphism of the Ehresmann connection, if and only if, it is a totally geodesic map (i.e., sends the geodesies, considered as parametrized curves, to geodesies) and preserves the strong torsion of the Ehresmann connection. This result generalizes and to some extent strengthens the classical theorem on the automorphisms of a D-manifold (manifold with covariant derivative).
Combining several results on related (or conjugate) connections, defined on banachable fibre bundles, we set up a machinery, which permits to study various transformations of linear connections. Global and local methods are applied throughout. As an application, we get an extension of the classical affine transformations to the context of infinite-dimensional vector bundles. Another application shows that, realising the ordinary linear differential equations (in Banach spaces) as connections, we get the usual transformations of (equivalent) equations. Thus, some classical results on differential equations, such as the Theorem of Floquet, can have a “geometric” interpretation.
, L. J. , Ulrich , D. ( 2003 ) Strong, reliable and precise synaptic connections between thalamic relay cells and neurones of the nucleus reticularis in juvenile rats . J. Physiol. 546 , 801 – 811 . 12
