UDC 513.873

MATHEMATICS

S. Z. SHEFEL’

ISOMETRIC IMMERSIONS OF CLASS $C^1$

(Presented by Academician A. D. Aleksandrov, March 27, 1970)

An immersion in Euclidean space $\mathbf R^3$ of a two-dimensional Riemannian manifold by a surface $F$ of class $C^1$ or even $C^{1,\alpha}$ for small $\alpha$, as follows from works $(^{1-3})$, may be such that for the surface $F$ the relations between its intrinsic and extrinsic geometry that are characteristic of surfaces of class $C^2$ are violated. Thus, for example, a closed surface with positive intrinsic curvature need not be convex, a surface with negative curvature need not be saddle-shaped, and a surface isometric to the plane need not be a cylinder.

In this connection it makes sense to consider isometric immersions in $\mathbf R^3$ of class $C^1$ subject to certain additional restrictions.

Such restrictions include the condition that the extrinsic curvature of the surface be bounded $(^4)$ and the condition that the surface belong to the class $C^{1,\alpha}$ for $\alpha > {}^2/{}_3$ $(^5)$.

In the present note, the condition of $\mathfrak A$-regularity is imposed on isometric immersions by a surface of class $C^1$.

Definition. An isometric immersion of a metric of class $K$ by a surface $F$ in $\mathbf R^3$ is called regular in the class $K$ with respect to a group of affine transformations $\mathfrak A$ of the space $\mathbf R^3$ ($\mathfrak A$-regular) if, under any affine transformation of $\mathbf R^3$, the surface $F$ is transformed into a surface that is an isometric immersion of some metric of the same class $K$.

Immersions in $\mathbf R^n$ regular with respect to the group of transformations of $\mathbf R^n$ were considered in $(^6)$.

We denote by $A, A^+, A^-, A^0$, respectively, the classes of metrics of bounded curvature, nonnegative, nonpositive curvature, and zero curvature (the class of locally Euclidean metrics).

Theorems 1–3 establish structural properties of $\mathfrak A$-regular immersions of metrics of the classes $A^+, A^-, A^0$. Theorems 4–6 establish connections between the conditions of bounded extrinsic curvature, $\mathfrak A$-regularity, and membership of the surface in the class $C^{1,\alpha}$ for $\alpha > {}^2/{}_3$. At the same time one should additionally note that the immersion of a metric of any of the classes $A, A^+, A^-, A^0$ by a surface of bounded extrinsic curvature is, obviously, $\mathfrak A$-regular. Theorem 6 is a generalization of an analogous theorem of Yu. F. Borisov $(^5)$.

Theorem 1. If a surface $F \in C^1$ is an $\mathfrak A$-regular immersion in $\mathbf R^3$ of a metric of class $A^+$, then for every point $M$ of it at least one of the following assertions is true:

$(\alpha)$. Some neighborhood of the point $M$ on the surface $F$ is a convex surface (in particular, this neighborhood may be a plane domain).

$(\beta)$. Through the point $M$ there passes a rectilinear segment $\pi(M)$ lying on the surface $F$, its ends are situated on the boundary of $F$, and the tangent plane to $F$ along the segment $\pi(M)$ is stationary. If the point $M$ has no plane neighborhood on the surface $F$, then such a segment is unique and none of its points has a plane neighborhood on $F$.

Theorem 1′. If a surface \(F\) satisfies the conditions of Theorem 1, then every cap cut from the surface \(F\) by a plane is a convex surface.

Theorem 2. If a surface \(F\) is an \(\mathfrak A\)-regular immersion in \(\mathbb R^3\) of a metric of class \(A^{-}\), then \(F\) is a saddle surface.

Theorem 3. If a surface \(F\) in \(\mathbb R^3\) of class \(C^1\) is an \(\mathfrak A\)-regular immersion of a metric of class \(A^0\), then for any of its points \(M\) at least one of the following assertions is true:

\((\alpha)\). Some neighborhood of the point \(M\) on the surface \(F\) is a planar domain.

\((\beta)\). Through the point \(M\) there passes a rectilinear segment \(\pi(M)\), lying on the surface \(F\), with endpoints on the boundary of \(F\); the tangent plane to \(F\) along \(\pi(M)\) is stationary. If the point \(M\) has no planar neighborhood on the surface \(F\), then such a segment is unique and none of its points has a planar neighborhood on \(F\).

Theorem 4. If a surface \(F\) of class \(C^{1,\alpha}\) \((\alpha>1/2)\) is an \(\mathfrak A\)-regular immersion in \(\mathbb R^3\) of a metric of one of the classes \(A^{+}, A^{-}, A^0\), then \(F\) is a surface of bounded extrinsic curvature.

Theorem 5. If a surface \(F\) of class \(C^{1,\alpha}\) \((\alpha>2/3)\) is an immersion in \(\mathbb R^3\) of a metric of one of the classes \(A, A^{+}, A^{-}, A^0\), then this immersion \(F\) is an \(\mathfrak A\)-regular immersion.

From Theorems 1–5 it follows that

Theorem 6. If a surface \(F \in C^{1,\alpha}\) \((\alpha>2/3)\) is an immersion in \(\mathbb R^3\) of a metric of one of the classes \(A^{+}, A^{-}, A^0\), then it is a surface of bounded extrinsic curvature and for it the assertion of the corresponding structural Theorem 1, 2, or 3 is satisfied.

The proof of Theorem 1 is based on the following lemmas.

Lemma 1. If a surface \(F\) satisfies the conditions of Theorem 1, is given by the equation \(z=f(x,y)\), \((x,y)\in K\), where \(K\) is a closed domain in the \(x,y\)-plane, and if at the interior points of \(K\) \(f(x,y)>0\), while on the boundary of \(K\) \(f(x,y)=0\), then \(F\) is a convex surface.

The central point in the proof of Lemma 1 is the following assertion: some neighborhood of the “vertex” of any cap cut by a plane from the surface \(F\) is a convex surface.

Lemma 2. If a surface \(F\in C^1\) is saddle and is an isometric immersion of a metric of class \(A^0\), then for it the assertions of Theorem 2 are fulfilled.

Proof. Introduce on the surface \(F\) a new topology induced by the Gauss map of \(F\) onto the sphere. Divide \(F\) into two sets \(F^0\) and \(F'=F\setminus F^0\), where \(F^0\) is the set of points of \(F\) interior with respect to the topology introduced above. If \(F^0\) is nonempty, then the extrinsic curvature of \(F^0\) coincides with the intrinsic curvature, which is proved by approximating the surface \(F^0\) by a \(C^1\)-sequence of saddle polyhedra. Hence, by a theorem of A. V. Pogorelov \((^4)\), it follows that for the points of \(F^0\) the assertions of Theorem 2 are fulfilled.

The proof of Theorem 3 is given in \((^6)\).

The proof of Theorem 4 is carried out analogously to the proof of Lemma 2. We note only that from the condition \(F\in C^{1,\alpha}\) \((\alpha>1/2)\) it follows that the image of the set \(F'\) on the sphere has measure zero.

The proof of Theorem 5 is based on the work \((^7)\). Let \(F\in C^{1,\alpha}\) \((\alpha>2/3)\), and let the intrinsic metric of \(F\) belong to class \(A\). Define on \(F\) a set function \(\sigma\), completely additive on the ring of Borel sets, which on each domain \(G\subset F\) with rectifiable boundary \(L\) coincides with the quantity \(\Delta(L')\), where \(L'\) is the spherical image of \(L\), and \(\Delta\) is the oriented area of \(L'\).

Let \(\alpha\) be an affine transformation of \(\mathbb R^3\), and let \(G\) be a domain on \(F\) bounded by a smooth curve.

If the variation \(|\sigma(F)| < \infty\), then the formula holds

\[ \sigma(\alpha(G)) = \int_G J(x')\,\sigma(dG), \]

where \(x'\) is the spherical image of the point \(x \in G\), and \(J(x')\) is the Jacobian of the diffeomorphism \(\beta\) of the unit sphere \(S\) onto itself, defined by the equality \(x' = \beta(\alpha(x'))\), for all \(x' \in S\). It follows from (7) that in order for the surface \(F \in C^1\), \((\alpha > 1/2)\), to be an isometric immersion of a metric of class \(A\), it is necessary and sufficient that the variation \(|\sigma(F)| < \infty\). In this case the intrinsic curvature \(\omega(F) \equiv \sigma(F)\).

The author expresses his gratitude to Yu. F. Borisov for a number of useful discussions.

REFERENCES

\(^{1}\) J. Nash, Collected Translations. Mathematics, 1, 2 (1957).
\(^{2}\) N. H. Kuiper, ibid.
\(^{3}\) Yu. F. Borisov, Uspekhi Mat. Nauk, 15, no. 3 (93) (1960).
\(^{4}\) A. V. Pogorelov, Intrinsic Geometry of Convex Surfaces, 1969.
\(^{5}\) Yu. F. Borisov, Vestn. Leningrad Univ., no. 19 (1960).
\(^{6}\) S. Z. Shefel’, Siberian Mathematical Journal, 20, 2 (1970).
\(^{7}\) Yu. F. Borisov, Vestn. Leningrad Univ., no. 7, no. 19 (1958); no. 1, no. 13 (1959).