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Descriptive geometry of paths. (Japanese) Zbl 0063.06013

Tensor 5, 81-86 (1942).
In [Tensor 1, 36–39 (1938)] the author studied the projective geometry of generalized paths in which any path is determined by three points and the differential equations of paths are given in the form \(x'''^i + G^i(x,x',x'')=0\) which is invariant under any projective transformation \(t^-=(\alpha t+\beta)/(\gamma t+\delta)\) of the parameter \(t\) of the paths. This paper deals with the descriptive geometry of these generalized paths, that is, the invariant theory of the paths under any transformation of \(t\). The equations of the paths become \(x'''^i+G^i(x,x',x'')+G(x,x',x'')x'^i\) after a transformation of \(t\), where the function \(G\) depends on the transformation but \(G^i\) does not. Hence one must discuss the invariant theory of \(G^i\) under any transformation of coordinate system and \(G^i\to G^i+Gx'^i\). The parameters of connection are obtained which correspond to Thomas’s projective parameters and from \(\Pi^i_{jk}\) there are derived the parameters \(L^i_{jk}\) which are transformed by any transformation of coordinate system just as those of an affine connection. Other invariants are also calculated.

MSC:

53A99 Classical differential geometry
51N05 Descriptive geometry