MATHEMATICS

I. Ya. Bakelman

GENERALIZED CHEBYSHEV NETS AND MANIFOLDS OF BOUNDED CURVATURE

(Presented by Academician V. I. Smirnov on 9 January 1961)

1. In the plane of variables \(\{u, v\}\), consider the square \(D\), \(0 < u < a,\ 0 < v < a\). Let \(\alpha(u)\), \(\beta(v)\) be functions of bounded variation, left-continuous on the interval \([0,a)\), and let \(\mu(M)\) be a completely additive function of Borel sets from the square \(D\), having bounded variation. Denote by \(Q_{u_0,v_0}\) the rectangle \(0 < u < u_0,\ 0 < v < v_0\).

A function \(f(u,v)\) defined in the square \(D\) will be called a function of bounded variation if it admits the representation

\[ f(u,v)=\alpha(u)+\beta(v)+\mu(Q_{u,v}), \]

where the properties of the functions \(\alpha(u)\), \(\beta(v)\), \(\mu(Q_{u,v})\) are described above.

We shall say that a certain function of bounded variation \(f(u,v)\) generates in the square \(D\) a generalized Chebyshev line element

\[ ds^2=du^2+2\cos f(u,v)\,du\,dv+dv^2, \tag{1} \]

if the function \(f(u,v)\) admits the representation

\[ f(u,v)=-A+\alpha(u)+\beta(v)+\mu(Q_{u,v}), \]

where: 1) \(0<A=\alpha(0)=\beta(0)<\pi\); 2) \(0<\alpha(u)<\pi,\ 0<\beta(v)<\pi\) for \(0\le u<a,\ 0\le v<a\); 3) for all \((u,v)\in D\) we have \(0<f(u,v)<\pi\).

If one considers a two-dimensional Riemannian manifold on which a Chebyshev coordinate net has been introduced, then in this net the line element of the manifold has the form (1), where \(f(u,v)\) is at least twice continuously differentiable; \(f(u,v)\) is, obviously, the angle between the coordinate lines at the point of the manifold with coordinates \((u,v)\). The geometric meaning of the quantities \(A\), \(\alpha(u)\), \(\beta(v)\), \(\mu(Q_{u,v})\) is as follows:

\[ A=f(0,0),\qquad \alpha(u)=f(u,0),\qquad \beta=f(0,v), \]

\[ \mu(Q_{u,v})=\int_0^u\int_0^v K(u,v)\,d\sigma, \]

where \(K(u,v)\) is the Gaussian curvature, and \(d\sigma\) is the area element of the manifold under consideration. It is obvious that in this case the functions \(\alpha(u)\), \(\beta(v)\), \(\mu(Q_{u,v})\), \(f(u,v)\) satisfy conditions 1)—3).

2. A curve \(L\) in \(D\) will be called smooth if there exists a parametrization of it \(u=u(t)\), \(v=v(t)\), \(a\le t\le b\), such that \(u(t), v(t)\in C^1\) and everywhere on \([a,b]\), \(u'^2+v'^2\ne0\).

Theorem 1. Let a generalized Chebyshev line element be defined in \(D\). Then along every smooth curve \(L\) the function

\[ \sqrt{u'^2(t)+2\cos f(u(t),v(t))\,u'(t)v'(t)+v'^2(t)} \]

is summable on \([a,b]\).

We define the length of the curve \(L\) with respect to the generalized Chebyshev line element under consideration by the formula

\[ s_f(L)=\int_a^b \sqrt{u'^2+2\cos f(u,v)\,u'v'+v'^2}\,dt . \]

Let \(A\) and \(B\) be two points of the square \(D\). Put

\[ \rho_f(A,B)=\inf s_f(L), \tag{2} \]

where the greatest lower bound is taken over all smooth curves \(L\subset D\) with endpoints at \(A\) and \(B\). The function of a pair of points \(\rho_f(A,B)\) satisfies, as is not difficult to see, the axioms of a metric space. Moreover, this metric is evidently intrinsic.

In what follows we shall say that the generalized Chebyshev line element (1) generates the intrinsic metric (2).*

Thus, on the square \(D\), by means of the metric (2), the generalized Chebyshev line element generates a certain two-dimensional metric manifold, which we shall denote by \(D_f\).

Theorem 2. Let a generalized Chebyshev line element (1) be given in \(D\). Then the manifold \(D_f\) generated by this element is a manifold of bounded curvature in the sense of A. D. Aleksandrov (see (¹)); moreover, the set function \(\mu(M)\) coincides with the intrinsic integral curvature of this manifold.

1. Theorem 2 admits a converse in the following sense:

Theorem 3. Let \(R\) be a manifold of bounded curvature and let \(G\subset R\) be a domain homeomorphic to the square \(D\), with \(\omega^+(G)<\pi/2\), \(\omega^-(G)<\pi/2\) (\(\omega^+\) and \(\omega^-\) are the positive and negative parts of the intrinsic integral curvature of the manifold \(R\)). Then in \(G\) one can introduce a coordinate net with a generalized Chebyshev line element (1), which generates a metric manifold \(G_f\) isometric to the domain \(G\). In this case one may assume that \(A=\pi/2\).

Leningrad State
Pedagogical Institute
named after A. I. Herzen

Received
4 I 1961

CITED LITERATURE

¹ A. D. Aleksandrov, DAN, 60, No. 9 (1948).

* According to A. D. Aleksandrov (¹), the metric \(\rho_f(A,B)\) is called intrinsic if \(\rho_f(A,B)=\inf s(L)\) over all smooth curves \(L\) in \(D\) with endpoints \(A\) and \(B\), where the length of the curve \(s(L)\) is defined as the least upper bound of the sums

\[ \sum_{i=0}^{n-1}\rho_f(X_i,X_{i+1}); \]

\(A=X_0,X_1,\ldots,X_n=B\) is a system of successive points on the curve \(L\).