Projective Geometry

Zhongmin Shen ()
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Zhongmin Shen: Indiana University-Purdue University at Indianapolis, Department of Mathematical Sciences

Chapter Chapter 12 in Differential Geometry of Spray and Finsler Spaces, 2001, pp 173-195 from Springer

Abstract: Abstract Two sprays G and $$ \tilde G$$ on a manifold are said to be pointwise projectively related if they have the same geodesics as point sets. For any geodesic c(t) of G, there is an orientation-preserving reparameterization t = t(s) such that c(s) := c(t(s)) is a geodesic of $$ \tilde G$$ , and vice versa. In this chapter, we will show that two sprays G and $$ \tilde G$$ on a manifold are pointwise projectively related if and only if there is a scalar function P on T M \ {0} such that 1 $$ \tilde G = G - 2P\;Y.$$ Then we prove the Rapcsák theorem on projectively related Finsler metrics. This remarkable theorem plays an important role in the projective geometry of Finsler spaces. See [Th3] for a systematic survey on the early development in this field.

Keywords: Inverse Problem; Constant Curvature; Projective Geometry; Einstein Metrics; Riemann Curvature (search for similar items in EconPapers)
Date: 2001
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-94-015-9727-2_13

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DOI: 10.1007/978-94-015-9727-2_13

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