Abstract: | LetE
n, k
be a pseudoeuclidean space with linear elementdx
1
2
+ +dx
n–k
2
–dx
n–k
+1/2
– –dx
n
2
. The area of a smooth two-dimensional surface inE
n, k
is defined by
, whereE, F, andG are the coefficients of the first fundamental form of the surface andD is the region of variation of the parametersu andv. The following theorem is proved: LetL be a piecewise smooth closed curve inE
n, k
(1 k n–1). Then there exists a two-dimensional piecewise smooth surface of arbitrarily small area bounded by the curveL. 3 figures.Translated from Ukrainskií Geometricheskií Sbornik, No. 30, 1987, pp. 18–22. |