MATHEMATICS

Yu. D. BURAGO

REALIZATION OF A TWO-DIMENSIONAL METRIZED MANIFOLD BY A SURFACE IN \(E^3\)

(Presented by Academician V. I. Smirnov, 4 VII 1960)

1. In the works of J. Nash \((^1)\) and N. Kuiper \((^2)\), an isometric embedding of an \(n\)-dimensional Riemannian space \(R^n\) into \((n+1)\)-dimensional Euclidean space \(E^{n+1}\) was carried out. In these works the metric in \(R^n\) is given by means of a quadratic form with coefficients of class \(C^1\).

In the case of a two-dimensional manifold, in the present paper a realization is constructed for more general metrics, given by a function of a pair of points. From these results it follows, in particular, that the geometry of any manifold of bounded curvature in the sense of A. D. Aleksandrov \((^3)\) can be regarded as the intrinsic geometry of a certain surface in \(E^3\).

1. Let an intrinsic metric \(\rho\) be given in a manifold \(M\), and let there be a sequence of polyhedral metrics \(\rho_n\). The metrics \(\rho_n\) are called proportionally convergent to \(\rho\) if

\[ \frac{\rho_n(X,Y)}{\rho(X,Y)} \to 1 \quad \text{as } n \to \infty \]

uniformly with respect to \(X, Y\).

If the metric \(\rho\) has a tangent cone at each point (in the sense of intrinsic geometry), then, as Yu. G. Reshetnyak proved \((^4)\), \(\rho\) is approximated by proportionally convergent polyhedral metrics.

Denote by \(\Phi\) the class of non-self-intersecting surfaces in \(E^3\) at each point of which there exists a tangent plane. (These surfaces, generally speaking, are not smooth.)

Theorem. Let the metric \(\rho\), given on a two-dimensional orientable manifold \(M\), admit an approximation by proportionally convergent polyhedral metrics \(\rho_n\). Then there exists a surface \(F \in \Phi\) with intrinsic metric \(\rho\).

Corollary 1. Let at each point of a two-dimensional orientable manifold \(M\) with metric \(\rho\) there exist a tangent cone. Then there exists a surface \(F \in \Phi\) with intrinsic metric \(\rho\).

Corollary 2. Every orientable manifold of bounded curvature, having no points with curvature \(2\pi\), with boundary in the form of a curve of bounded variation of rotation or without boundary, is isometric to a certain surface \(F \in \Phi\).

1. We outline the plan of the proof of the theorem. Let \(\rho\) be an arbitrary metric satisfying the condition of the theorem; let \(\rho_n\) be polyhedral metrics proportionally convergent to \(\rho\). With each metric \(\rho_n\) we associate a certain topological mapping \(\varphi_n\) of the manifold \(M\) onto itself.

Lemma 1. The sequence of metrics \(\rho_n\) and mappings \(\varphi_n\) can be chosen so that:

1) \(\rho_{n+1}(\varphi_n(X), \varphi_n(Y)) > \rho_n(X,Y)\);

2) \(\varphi_n \in C^\infty(D_n)\), \(D_n = M \setminus \bigcup A_i\), where \(A_i\) are the vertices of all metrics \(\rho_k\) for \(k \leq n\);

3) \(\rho_{n+1}(X,\varphi_n(X)) \to 0\) uniformly with respect to \(X\);

4)
\[ \frac{\rho_{n+1}\bigl(\varphi_n(X),\varphi_n^{*}(Y)\bigr)}{\rho(X,Y)} \to 1 \]
uniformly with respect to \(X,Y\), and the sequences 3)—4) converge just as rapidly as \(\rho_n(X,Y)\to \rho(X,Y)\).

The mappings \(\varphi_n\) are constructed by means of triangulations of the manifold \(M\). These constructions are based on the following:

Lemma 2. Suppose the total angles about the vertices and the angles between adjacent edges of the boundary of the development \(K\) are bounded below by a number \(\alpha>0\). Then the development \(K\) can be divided into \(n\) plane triangles adjacent along whole sides, in such a way that all angles of the resulting triangles are bounded below by a number \(C(\alpha)>0\), depending only on \(\alpha\).

The proof of Lemma 2 is similar to the proof of Theorem 2 in \((^5)\).

With the aid of Caper’s construction, each metric \(\rho_n\) (beginning with some sufficiently large \(n\)) is embedded in \(E^3\) as a surface \(F_n\in C^\infty(D_n)\). The surface \(F_n\in\Phi\) has a metric close to \(\rho_n\) and is situated near the surface \(F_{n-1}\), which, in view of 1)—2), serves as a “short” embedding for \(\rho_n\) (see \((^2)\)). With a suitable choice of the parameters in Caper’s formulas for the surfaces \(F_n\), the desired surface \(F\), realizing the metric \(\rho\), is the limit of the sequence of surfaces \(F_n\). The proof of this fact is based on properties 2)—4) of the metrics \(\rho_n\).

Leningrad State University
named after A. A. Zhdanov

Received
16 VI 1960

REFERENCES CITED

\(^1\) J. Nash, Sbornik perevodov Matematika, 1, Nos. 2, 3 (1957). \(^2\) N. Caper, ibid., p. 17. \(^3\) A. D. Aleksandrov, DAN, 60, No. 9 (1948). \(^4\) Yu. G. Reshetnyak, Izv. Sibirsk. otd. AN SSSR, 10, 15 (1959). \(^5\) Yu. D. Burago, V. A. Zalgaller, Vestn. LGU, 7, 66 (1960).