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By Eisenhart L. P.

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Extra resources for Affine Geometries of Paths Possessing an Invariant Integral

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Gy counterpart of the notion of a quadratic extension (in the context of symmetric triples). 7. Let N be an affine symmetric space. A special affine fibration over N is a surjective affine map q W M ! N , where M is a pseudo-Riemannian symmetric space and the fibres of q are flat, coisotropic, and connected. l; Âl /. l; Âl ; 0/. Then the bundle projection p W T N ! N is the simplest example of a special affine fibration over N . l; Â l /. We now want to construct special affine fibrations that correspond to quite general quadratic extensions in the same way as p W T N !

G. Then J acts on M , and we would like to set q to be the projection onto the orbit space N D J nM . That the orbits are flat and coisotropic is a simple consequence of the properties of i? The main problem is to show that the orbit space is a manifold (the orbits have to be closed, in particular). It is this point, where we need Condition (a) or (b). Without these conditions, it is not difficult to construct examples with non-closed J -orbits. For them the resulting geometric structure will be a foliation only, not a fibration.

L; Âl ; a/=Gl;Âl ;a for a 2 Anl;Â l . Take a D a1; . If Finally, let us determine HQ n 5, then in the generic case . X // 2 Rn 3 and . l; Âl ; a/=Gl;Âl ;a Š Zl;0 =Z2 . Hence, dH D 2. In the non-generic case Gl;Âl ;a becomes larger and we get dH D 0. For n D 3; 4 6, then in the generic case we have dH D 0. Now take a D a2; . If n n 4 n 4 . X// 2 R and . 2/. Hence, dH D 3. In the non-generic case we get dH D 1. For n D 5 we have dH D 1 and for n D 4 we get dH D 0. 1 Examples of geometric structures.

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