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).