Existence of elastic deformations with prescribed principal strains and triply orthogonal systems. (English) Zbl 0544.53012

Two geometric problems are studied. The first is prescribing the principal strains of an elastic deformation. An elastic deformation may be viewed as a diffeomorphism between two Riemannian manifolds; the metrics are usually the standard flat one. The principal strains of the deformation are the eigenvalues of the metric in the range manifold viewed as a symmetric matrix with respect to the metric in the domain. Given distinct positive smooth functions on the domain, we prove the local existence of a smooth elastic deformation with these functions as principal strains. We also prove the local existence of a triply orthogonal system on a Riemannian 3-manifold. This is equivalent to the existence of local coordinates with respect to which the metric is diagonal. Each problem has two distinct approaches to it, depending on whether one solves for an appropriately defined diffeomorphism or for an appropriate moving frame. One way works and the other one doesn’t, but for the two problems the situations are opposite. By solving for the diffeomorphism in the first and the moving frame in the second, the problems reduce to nonlinear hyperbolic systems, which are locally solvable. Both are striking examples of natural problems in Riemannian geometry which reduce to nonelliptic partial differential equations.


53B20 Local Riemannian geometry
35L70 Second-order nonlinear hyperbolic equations
74B20 Nonlinear elasticity
58C15 Implicit function theorems; global Newton methods on manifolds
58D25 Equations in function spaces; evolution equations
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