An Account of the Theory of Crystallographic Groups by Louis Auslander

By Louis Auslander

Lawsuits of the yank Mathematical Society
Vol. sixteen, No. 6 (Dec., 1965), pp. 1230-1236
Published by way of: American Mathematical Society
DOI: 10.2307/2035904
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Page count number: 7

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4) where fb, i = 1, 2 are the Christoffel symbols of the surface X. 3). ,2 , Xx 1 X! 1 and Xx 2 x2 to be linearly independent in the ambient space. , 2 (x 1 , xg). , 1 (x 1 , xg) and Xx:. (xt. xg). ngt=mt planes given by Ti(p, x 1 ) =< p- X(xt. x~), N;(x~, xg) >= 0 p E R 3 or R\ 54 I. Transformations of surfaces and applications where the N;, i = 1 or i = 1, 2 ( according to whether the surface is immersed in lf3 or R_4 ) are linearly independent vector fields normal to the surface X spanning the normal space of X.

Proof. Let X : U C R2 --+ M 3 (J<) be a local parametrization of the surface M C M 3 • Assume M has constant mean curvature H = c > 0 ( c > 1 if K = -1 ). 16) = -1. Since, K does not vanish, it follows from equation (3. 15) where a= 1/c if K = 0, tan a= 1/c if K = 1, tanh a= 1/c if K = -1, 2 . 28 I. Transformations of surfaces and applications we conclude that X' has constant mean curvature -c. 6) and the fact that M has no umbilic points. 14), we observe that c2 + K > 0 and c > 0 imply K > 0. In particular, K > 2 when K = 1.

Moreover, one can invert those transformations. 1) until eventually it is transformed into one which has a vanishing Laplace invariant. This equation is integrated and then using the inverse transformation one obtains a solution for the given initial differential equation. There is a geometrical construction for surfaces [Ch3], which corresponds to the Laplace method described above. Before going into the geometry, we would like to make the following observations: 1. Given a. 1), we have seen that if h = 0, we obtain linear first-order equations first for Z:c 2 + az and then for z, which we can easily solve.

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