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Projective Geometry

In: Differential Geometry of Spray and Finsler Spaces

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  • Zhongmin Shen

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

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.

Suggested Citation

  • Zhongmin Shen, 2001. "Projective Geometry," Springer Books, in: Differential Geometry of Spray and Finsler Spaces, chapter 0, pages 173-195, Springer.
    • Handle: RePEc:spr:sprchp:978-94-015-9727-2_13
    DOI: 10.1007/978-94-015-9727-2_13
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