25. Developable Surfaces. Minimal Surfaces

25. Developable Surfaces. Minimal Surfaces I n this section we shall comment briefly on two well known types of regular surfaces S in the Euclidean metric space E of three dimensions. One of these is the developable surface which is characterized geometrically by the condition that it can be rolled, without stretching or tearing, upon a plane. This implies that the developable surface is intrinsically flat, i.e. that its curvature tensor vanishes (see Remark 1 in Sect. 13). I t can be shown in fact that a surface S i s a developable surface if, and only i f , its Gaussian curvature K i s equal to zero. Special developable surfaces are ( a ) the plane, ( p ) the cone and ( y ) the cylinder which can be considered as a cone whose vertex is a t infinity. More generally it can be shown that a developable surface is a tangent developable, i.e. the locus of the tangents of a regular curve C in the Euclidean metric space E ; the curve C is called the edge of regression of the tangent developable. Another type of surface to which we wish to call special attention is the m i n i m a l surface which may be defined as a surface whose mean curvature L? is equal to zero a t each point. A t a n arbitrary point P of a minimal surface S , not a plane, the Gaussian curvature K must be negative. I n fact we must have K~ K~ = 0 over a minimal surface S from the second equation (21.14) where K~ and K~ are the principal curvatures of S ; hence neither K~ or K~ can be equal to zero since otherwise we would have K~ = K~ = 0 and it would follow that the coefficients b,, of the second fundamental form vanish over S (see Remark in Sect. 21); but the vanishing of the coefficients b,, means that S is a plane (see Remark in Sect. 18) contrary to hypothesis. Hence K = - K; < 0 from the first equation (21.14).

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25. DEVELOPABLE SURFACES.

MINIMAL SURFACES

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Minimal surfaces arise in the existence theoretic problem of finding a connected surface of minimum area bounded by a given simple closed curve in the space E . This problem, known as the problem of Plateau, has attracted the attention of a number of distinguished mathematicians. It was finally solved by J. Douglas, Solution of the problem of Plateau, Trans. Am. Math. SOC.33 (1931), pp. 263-321. For a detailed treatment of the geometry of the above and other surfaces of special type the reader is referred to the standard texts on differential geometry. General References L. P. EISENHART, Riemannian Geometry, Princeton University Press, Princeton, New Jersey, 1926. T. LEVI-CIVITA,The Absolute Differential Calculus, Blackie and Son, London and Glasgow, 1927. 0. VEBLEN, Invariants of Quadratic Differential Forms, Cambridge Tracts in Mathematics and Mathematical Physics, No. 24, Cambridge University Press, London and New York, 1927.

0. VEBLENand J . H. C. WHITEHEAD, T h e Foundations of Dijferenfial Geometry, Cambridge Tracts in Mathematics and Mathematical Physics, No. 29, Cambridge University Press, London and New York, 1932. T. Y. THOMAS,T h e Differential Invariants of Generalized Spaces, University Press, London and New York, 1934.

Cambridge

E. P. LANE, Metric Differential Geometry of Curves and Surfaces, University of Chicago Press, Chicago, Illinois, 1940. A. J . MCCONNELL,Applications of the Absolute Differential Calculus, Blackie and Son, London and Glasgow, 1943.