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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
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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

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