MATHEMATICS

A. V. POGORELOV

ON A TRANSFORMATION OF ISOMETRIC SURFACES

(Presented by Academician V. I. Smirnov on 30 IV 1958)

In this note a standard method will be given for assigning to each pair of isometric surfaces of a space of constant curvature (K \ne 0) a pair of isometric surfaces of Euclidean space, and conversely, to each pair of isometric surfaces of Euclidean space—a pair of isometric surfaces of a space of constant curvature. This gives a new approach to the problem of unique determination of convex surfaces in spaces of constant curvature.

Without loss of generality, we shall assume (K = 1) for elliptic space and (K = -1) for Lobachevsky space.

Let (R) be elliptic space. Introduce Weierstrass coordinates (x_i) ((i = 0, 1, 2, 3)) in (R), and associate with each point of the space (R) a pair of points of four-dimensional Euclidean space with Cartesian coordinates (x_i) and (-x_i). These points fill the unit sphere, since the Weierstrass coordinates satisfy the condition

[
x^2 = x_0^2 + x_1^2 + x_2^2 + x_3^2 = 1.
]

Denote by (E_0) the three-dimensional Euclidean space (x_0 = 0).

Theorem 1. Suppose that in elliptic space (R) we have two isometric surfaces (F') and (F''), given by equations in Weierstrass coordinates

[
x = x'(u,v), \qquad x = x''(u,v),
]

where the points of the surfaces corresponding under the isometry have the same coordinates (u, v).

Then the equations in Cartesian coordinates (y_i)

[
y = \frac{x'(u,v) - e_0\bigl(x'(u,v)e_0\bigr)}
{e_0\bigl(x'(u,v) \pm x''(u,v)\bigr)},
]

[
y = \frac{x''(u,v) - e_0\bigl(x''(u,v)e_0\bigr)}
{e_0\bigl(x'(u,v) \pm x''(u,v)\bigr)}
\tag{1}
]

define two isometric surfaces in the Euclidean space (E_0).

In equations (1), (e_0) is the unit vector along the axis (x_0), and the scalar product is expressed by the usual formula.

Let now (R) be Lobachevsky space ((K=-1)). Introduce Weierstrass coordinates (x_i) in (R), and associate with each point of (R) the point of four-dimensional Euclidean space with Cartesian coordinates (x_i). Then (R) is mapped onto the two-sheeted hyperboloid

[
- x_0^2 + x_1^2 + x_2^2 + x_3^2 = -1.
]

Theorem 2. Suppose that in Lobachevsky space (R) we have two isometric surfaces (F') and (F''), given by equations in Weierstrass coordinates

[
x = x'(u,v), \qquad x = x''(u,v).
]

Then the equations in Cartesian coordinates (y_i)

[
y=\frac{x'(u,v)+e_0\bigl(x'(u,v)e_0\bigr)}
{e_0\bigl(x'(u,v)\pm x'(u,v)\bigr)},
]

[
y=\frac{x''(u,v)+e_0\bigl(x''(u,v)e_0\bigr)}
{e_0\bigl(x'(u,v)\pm x''(u,v)\bigr)}
\tag{2}
]

define two isometric surfaces in the Euclidean space (E_0).

In equations (2) the scalar product is taken with respect to the form

[
-x_0^2+x_1^2+x_2^2+x_3^2 .
]

Theorem 3. Let in the Euclidean space (E_0) there be two isometric surfaces (\Phi') and (\Phi''), given in Cartesian coordinates (y_i) by the equations

[
y=y'(u,v),\qquad y=y''(u,v).
]

Then the equations in Weierstrass coordinates (x_i)

[
\begin{aligned}
x&=\rho{2y'(u,v)\pm e_0(1-y''^{2}(u,v)+y''^{2}(u,v))},\
x&=\rho{2y''(u,v)\pm e_0(1-y''^{2}(u,v)+y'^{2}(u,v))}
\end{aligned}
\tag{3}
]

define two isometric surfaces in the elliptic space (R) ((K=1)).

In equations (3) (\rho) is a normalizing factor, determined by the condition (x^2=1), and the scalar product is determined by the usual formula for vectors of four-dimensional space.

Theorem 4. Let in the Euclidean space (E_0) there be two isometric surfaces (\Phi') and (\Phi''), given in Cartesian coordinates (y_i) by the equations

[
y=y'(u,v),\qquad y=y''(u,v).
]

Then the equations in Weierstrass coordinates (x_i)

[
x=\rho{2y'(u,v)\pm e_0(1-y''^{2}(u,v)+y'^{2}(u,v))},
]

[
x=\rho{2y''(u,v)\pm e_0(1-y'^{2}(u,v)+y''^{2}(u,v))}
\tag{4}
]

define two isometric surfaces in Lobachevsky space (R) ((K=-1)).

In equations (4) the scalar multiplication is taken with respect to the quadratic form

[
-x_0^2+x_1^2+x_2^2+x_3^2,
]

and the normalizing factor (\rho) is determined by the condition (x^2=-1).

In all Theorems 1–4, if the given surfaces are simply congruent, i.e. can be brought into coincidence by a motion, then the surfaces obtained are also congruent.

We shall give an example of the use of Theorem 1 for proving the unique determination of closed convex surfaces in elliptic space.

Let (F') and (F'') be closed isometric, equally oriented convex surfaces in elliptic space (R) ((K=1)). Place the surfaces (F') and (F'') so that they do not intersect the plane (x_0=0). Then the Weierstrass coordinates of the points of the surfaces can be normalized by the additional condition (x_0>0). Move the given surfaces (F') and (F''), remaining in the region (x_0>0), into such a position that the point ((1,0,0,0)) lies inside each of the surfaces.

If we now take equations (1) with the plus sign in the denominator, then the corresponding surfaces of the Euclidean space (E_0) will be not only isometric but also convex. And such surfaces, as is known, are congruent. Consequently, the surfaces (F') and (F'') are also congruent.

Received
28 IV 1958