UDC 513

MATHEMATICS

V. K. IONIN

ON RIEMANNIAN SPACES WITH THE EUCLIDEAN ISOPERIMETRIC INEQUALITY

(Presented by Academician A. D. Aleksandrov on 26 II 1969)

In the present note we consider only twice continuously differentiable Riemannian spaces homeomorphic to Euclidean spaces.

1°. By a body in the \(n\)-dimensional Euclidean space \(E^n\) we shall mean the closure of a bounded connected open domain. In an \(n\)-dimensional Riemannian space \(V^n\) we shall call a body the image of any body from \(E^n\) under a homeomorphic mapping of \(E^n\) onto \(V^n\). We shall say that the Euclidean isoperimetric inequality holds in \(V^n\) if for every body in \(V^n\) there is a body in \(E^n\) with no smaller volume and with no larger area (i.e., \((n-1)\)-dimensional volume) of the boundary. We denote by \(I_n\) the set of all such \(n\)-dimensional (\(n \ge 2\)) Riemannian spaces. It is known (see \((^{1,2})\)) that \(I_2\) consists only of two-dimensional Riemannian spaces with nonpositive Gaussian curvature. For \(n \ge 3\) we have proved the following.

Theorem. If in an \(n\)-dimensional Riemannian space one can introduce a semigeodesic coordinate system \((x^1,\ldots,x^n)\) with fundamental form

\[ ds^2 = g_{ij}dx^idx^j + (dx^n)^2 \qquad (i,j=1,\ldots,n-1) \]

so that each hypersurface \(x^n=\mathrm{const}\) belongs to \(I_{n-1}\), and the determinant

\[ |g_{ij}|=\lambda(x^n)\mu(x^1,\ldots,x^{n-1}), \tag{*} \]

where \(\lambda>0\), \(\mu>0\), and

\[ \left(\frac{1}{\lambda^{\,1/(2n-2)}}\right)'' \ge 0, \tag{**} \]

then \(V^n \in I_n\), i.e., in \(V^n\) the isoperimetric inequality of the Euclidean space \(E^n\) holds.

Condition \((*)\) means that geodesics orthogonal to the hypersurfaces \(x^n=\mathrm{const}\) establish between any such hypersurfaces a homeomorphic mapping under which the ratio of the volume of any \((n-1)\)-dimensional body to the volume of its image is constant.

The geometric meaning of condition \((**)\) is somewhat clarified by means of the following, easily verified assertion. Let some \((n-1)\)-dimensional Riemannian space \(V^{n-1}\) with fundamental form

\[ ds^2 = a_{ij}dx^idx^j \qquad (i,j=1,\ldots,n-1) \]

have nonpositive curvature, i.e., have nonpositive curvature at each point in each two-dimensional direction. In order that the \(n\)-dimensional Riemannian space \(V^n\) with fundamental form

\[ ds^2 = g_{ij}dx^idx^j + (dx^n)^2, \qquad g_{ij} = [\lambda(x^n)]^{1/(n-1)}a_{ij} \]

have nonpositive curvature, it is sufficient that condition \((**)\) be satisfied.

2°. We give some special cases of the theorem. Consider an arbitrary \((n-1)\)-dimensional Riemannian space \(V^{n-1}\in I_{n-1}\). Let

\[ (ds')^2 = g_{ij}dx^idx^j \qquad (i,j=1,\ldots,n-1) \]

its fundamental form in some system of coordinates. In the direct product \(V^n=V^{n-1}\times R\) of the space \(V^n\) with the real line \(R\), introduce a metric by means of the line element

\[ ds^2=a(x^n)g_{ij}dx^i dx^j+(dx^n)^2, \]

where \(a\) is a positive twice continuously differentiable function of one real variable. It follows easily from the theorem that if \((\sqrt a)''\geq 0\), then \(V^n\in I_n\).

Let us note that if \(V^{n-1}\) is Lobachevskii space of curvature \(K\), then, as is not hard to see, \(V^n\) for \(a=\operatorname{ch}^2\sqrt{-K}\,x\) is also Lobachevskii space of curvature \(K\). Thus every Lobachevskii space is a space with the Euclidean isoperimetric inequality.

Consider a Riemannian space \(V^n\) admitting a system of \(n\)-orthogonal hypersurfaces, i.e., a space whose fundamental form in some system of coordinates has the form

\[ ds^2=g_{11}(dx^1)^2+\cdots+g_{nn}(dx^n)^2. \]

With the aid of our theorem it is not difficult to establish that, in order that the space \(V^n\in I_n\), it is sufficient that the following conditions be satisfied:

1) \(g_{nn}=\mathrm{const}\), \(g_{ii}=\alpha_{i\,i+1}(x^{i+1})\cdots \alpha_{in}(x^n)\), where \(\alpha_{ij}\) is a positive twice continuously differentiable function of one real variable;

2) for each \(i=2,\ldots,n\) the inequality

\[ \left[(\alpha_{1i}\cdots \alpha_{i-1\,i})^{1/(2i-2)}\right]''\geq 0 \]

holds.

We shall give precise definitions of the symmetrizations by means of which the theorem is proved.

\(3^\circ\). The symmetrization \(S_1\) assigns to an arbitrary body \(Q\) of the space \(V^n\) a certain body \(S_1Q\) of an \(n\)-dimensional Riemannian space \(W^n\) with fundamental form

\[ ds^2=[\lambda(y^n)]^{1/(n-1)}[(dy^1)^2+\cdots+(dy^{n-1})^2]+(dy^n)^2. \]

Denote by \(\Gamma\) the geodesic \(y^1=\cdots=y^{n-1}=0\). Let the body \(S_1Q\) be determined by the following conditions:

1) \(\Gamma\) is its axis of rotation;

2) for every real \(x\), the area of the section of the body \(Q\) by the hypersurface \(x^n=x\) is equal to the area of the section of the body \(S_1Q\) by the hypersurface \(y^n=x\).

The symmetrization \(S_1\), in the case when the space \(V^n\) is Euclidean, coincides with the well-known Schwarz symmetrization \((^3)\).

\(4^\circ\). The symmetrization \(S_2\) assigns to an arbitrary body \(Q\) of the space \(W^n\) a certain body \(S_2Q\) of Euclidean space \(E^n\). Denote by \(\Gamma_\rho\) the set of points in the space \(W^n\) the distance from each of which to the geodesic \(\Gamma\) is equal to \(\rho\). In \(E^n\) introduce some rectangular coordinate system \((z^1,\ldots,z^n)\). Denote by \(\gamma\) the line \(z^1=\cdots=z^{n-1}=0\), and by \(\gamma_\rho\) the set of points at distance \(\rho\) from \(\gamma\). Construct in \(E^n\) a body \(\bar Q\), determined by the following conditions:

1) \(\bar Q\) is symmetric with respect to the hyperplane \(z^n=0\);

2) for every \(\rho>0\), the area of the section of the body \(Q\) by the hypersurface \(\Gamma_\rho\) is equal to the area of the section of the body \(\bar Q\) by the hypersurface \(\gamma_\rho\).

Replace each section of the body \(\bar Q\) by the hypersurface \(z^n=\mathrm{const}\) by its convex hull. The body obtained in this way will be denoted by \(S_2Q\).

\(5^\circ\). It is known \((^3)\) that Schwarz symmetrization does not decrease the volume of a body and does not increase the area of the boundary of this body. It can be proved (specifically for this purpose the conditions \((*)\) and \((**)\) were imposed on the space \(V^n\)) that the symmetrization \(S_1\) and the superposition \(S_2S_1\) of the symmetrizations \(S_1\) and \(S_2\) possess this same property. Thus, for an arbitrary body \(Q \subset V^n\) we have succeeded in constructing a body \(S_2S_1Q \subset E^n\) with no smaller volume and no greater boundary area, and this means that \(V^n \in I_n\).

CITED LITERATURE

\(^1\) A. D. Aleksandrov, DAN, 47, 239 (1945).
\(^2\) Yu. G. Reshetnyak, Vestn. Leningr. Univ., 19, 58 (1961).
\(^3\) W. Blaschke, Kreis und Kugel, Moscow, 1967.