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On the projective geometry of paths

Published online by Cambridge University Press:  20 January 2009

J. Haantjes
J. Haantjes
Affiliation:
University of Edinburgh.
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An affine connection in an n-dimensional manifold Xn defines a system of paths, but conversely a connection is not defined uniquely by a system of paths. It was shown by H. Weyl that any two affine connections whose components are related by an equation of the form

where is the unit affinor, give the same system of paths. In the geometry of a system of paths, a particular parameter on the paths, called the projective normal parameter, plays an important part. This parameter, which is invariant under a transformation of connection (1), was introduced by J. H. .C. Whitehead. It can be defined by means of a Schwarzian differential equation and it is determined up to linear fractional transformations. In § 1 this method is briefly discussed.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 5 , Issue 2 , June 1937 , pp. 103 - 115
Copyright
Copyright © Edinburgh Mathematical Society 1937

References

page 103 note 1 Weyl, H., “Zur Infinitesimalgeometrie: Einordnung der projektiven und der konformen Auffassung,” Göttinger Nachrichten (1921), pp. 99–112.Google Scholar

page 103 note In this paper the term “affinor” is used instead of “tensor.”

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page 109 note 1 The equations are obtainable in this form by putting t = u 1/u o in x x = x x (t) and multiplying by an arbitrary homogeneous function of degree 1.

page 111 note 1 The sign means that the equation holds with respect to the coordinate system or systems used in the. equation itself; it need not hold with respect to another system.

page 112 note 1 A. P. D., p. 11.Google Scholar