Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-17T19:57:24.531Z Has data issue: false hasContentIssue false
  • English
  • Français

On Linear Partial Differential Equations of the Second Order Having Geodesic Solutions

Published online by Cambridge University Press:  20 November 2018

G. F. D. Duff
G. F. D. Duff*
Affiliation:
University of Toronto
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The coefficients of the second derivatives in an elliptic or hyperbolic differential equation of the second order determine a Riemannian metric on the space of the independent variables. A Riemannian space has been called harmonic if the Laplace equation corresponding to it has a solution which is a function only of geodesic distance in that space. Harmonic spaces have been studied in some detail (1; 3). In this note are examined the circumstances under which a non-parabolic second order linear equation may have a “geodesic” solution of the type described. It will be shown that the equation must be self-adjoint, that the Riemann space corresponding to the equation must be harmonic and that the coefficient of the dependent variable must be a constant. Conversely, if these conditions are satisfied, the equation has two geodesic solutions, one of which is an elementary solution. In the case of elliptic equations, the second solution is connected with a certain mean value property which is valid in harmonic spaces.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 6 , 1954 , pp. 73 - 79
Copyright
Copyright © Canadian Mathematical Society 1954

References

1. Copson, E. T. and Ruse, H. S., Harmonic Riemannian spaces, Proc. Roy. Soc. Edinburgh, 60 (1940), 117–133.Google Scholar
2. Courant, R. and Hilbert, D., Methoden der mathematischen Physik, vol. 2 (Berlin, 1937).Google Scholar
3. Lichnerowicz, A. and Walker, A. G., C.R. Acad. Sci., Paris, 221 (1945), 394–396.Google Scholar
4. Thomas, T. Y., Differential invariants of generalized spaces (Cambridge, 1934).Google Scholar
5. Thomas, T. Y. and Titt, E. W., On the elementary solution of the linear second order equation, J. Math, pures appl. (9), 17 (1939), 217–248.Google Scholar
6. Willmore, T. J., Mean value theorems in harmonic Riemannian spaces, J. London Math. Soc, 25 (1950), 54–57.Google Scholar