Differential Geometry of Manifolds

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On the structure forms of a projective structure



A projective structure on a smooth manifold is a maximal atlas such that all its transition maps are the fractional linear transformations. Our aim is to interpret this notion in terms of the higher order frame bundles and their structure forms. It is shown that the projective structure gener­ates the sequence of differential geometric structures. The construction is following:

Step 1. For a smooth manifold the so-called quotient frame bundle as­sociated to the 2nd order frame bundle on the manifold is constructed.

Step 2. Given projective structure on the manifold, the mappings from the quotient frame bundle to the higher order frame bundles are con­structed. These mappings are the differential geometric structures.

Step 3. The pullbacks of the structure forms of the frame bundles via the mappings are considered. These are called structure forms of the pro­jective structure. The expressions of their exterior differentials in terms of the forms themselves are found. These expressions coincide with the structure equations of a projective space.


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