Abstract
Projective differential geometry was initiated in the 1920s, especially by Elie Cartan and Tracey Thomas. Nowadays, the subject is not so well-known. These notes aim to remedy this deficit and present several reasons why this should be done at this time. The deeper underlying reason is that projective differential geometry provides the most basic application of what has come to be known as the ‘Bernstein-Gelfand-Gelfand machinery’. As such, it is completely parallel to conformal differential geometry. On the other hand, there are direct applications within Riemannian differential geometry. We shall soon see, for example, a good geometric reason why the symmetries of the Riemann curvature tensor constitute an irreducible representation of SL(n,ℝ) (rather than SO(n) as one might naively expect). Projective differential geometry also provides the simplest setting in which overdetermined systems of partial differential equations naturally arise.
This article is based on two introductory lectures given at the 2006 Summer Program at the Institute for Mathematics and its Applications at the University of Minnesota. The author would like to thank the IMA for hospitality during this time and the referee of this article for helpful suggestions. The author is supported by the Australian Research Council.
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Eastwood, M. (2008). Notes on Projective Differential Geometry. In: Eastwood, M., Miller, W. (eds) Symmetries and Overdetermined Systems of Partial Differential Equations. The IMA Volumes in Mathematics and its Applications, vol 144. Springer, New York, NY. https://doi.org/10.1007/978-0-387-73831-4_3
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