A. V. CHAKMAZYAN

EVOLUTE SURFACES OF A TWO-DIMENSIONAL DUALLY NORMALIZED \(D_2\) IN \(E_4\)

(Presented by Academician P. S. Aleksandrov on 8 II 1962)

1. We shall say that a surface \(X_m\) in the projective space \(P_n\) is dually normalized if it is normalized in the sense of A. P. Norden \(\left({}^{1}\right)\) and its normal of the first kind contains the characteristic of the family of hyperplanes tangent to \(X_m\). A number of properties of dually normalized surfaces were obtained in the author’s papers \(\left({}^{2}\right)\). In this note we shall restrict ourselves to the case of a surface of two dimensions \(X_2\), immersed in the Euclidean \(E_4\).

Suppose that \(X_2\) can be completed to a hypersurface in such a way that the characteristics of the family of tangent hyperplanes are perpendicular to the tangent plane of the surface \(X_2\). It is obvious that then the natural normalization of \(X_2\) will at the same time also be dual. The surfaces \(X_2\) admitting a dual normalization form a certain class, which in what follows we shall denote by \(D_2\).

The basic differential equations of surfaces of the class \(D_2\) have the form \(\left({}^{2б}\right)\)

\[ \nabla_j r_i = h_{ij}X + k_{ij}Y,\qquad \mathbf{X}_j=-h_j^{\,l}r_l,\qquad \mathbf{Y}_j=-k_j^{\,l}r_l, \tag{A} \]

where \(h_{ij}=-\partial_j r\,\partial_i X=\mathbf{X}\nabla_j r_i\), \(k_{ij}=-\partial_j r\,\partial_i Y=\mathbf{Y}\nabla_j r_i\), and \(\mathbf{X}\) and \(\mathbf{Y}\) denote, respectively, the normal vector of the tangent hyperplane and the vector of the characteristic line.

1. Let us construct the congruence of normals of a dually normalized surface, going in the direction of the unit vectors

\[ X^* = X\cos\alpha + Y\sin\alpha,\qquad \alpha=\mathrm{const}. \]

\[ \mathbf{R}=\mathbf{r}(u^1,u^2)+\rho X^*(u^1,u^2). \tag{1} \]

Let us pose the question: do focal surfaces of this congruence exist, i.e., such surfaces which are touched by all its rays? In this case there must exist a function \(\rho(u^1,u^2)\) such that the vectors \(\mathbf{R}_1,\mathbf{R}_2\) and \(X^*\) are coplanar. Taking (1) and (A) into account, we obtain

\[ \mathbf{R}_i=\left[\delta_i^s-\rho\left(h_i^s\cos\alpha+k_i^s\sin\alpha\right)\right]\mathbf{r}_s+\rho_iX^*. \]

Taking the latter into account, we must require the existence of numbers \(\mu^1,\mu^2,\gamma\), not all simultaneously equal to zero, such that

\[ \mu^i\mathbf{R}_i+\gamma X^*=0, \]

i.e.

\[ \mu^i\left[\delta_i^s-\rho\left(h_i^s\cos\alpha+k_i^s\sin\alpha\right)\right]\mathbf{r}_s+(\mu^i\rho_i+\gamma)X^*=0. \]

Hence it follows that

\[ \mu^i\left[\delta_i^s-\rho\left(h_i^s\cos\alpha+k_i^s\sin\alpha\right)\right]=0. \]

Thus, the desired function \(\rho(u^1,u^2)\) must be a quantity reciprocal to a root of the equation

\[ \operatorname{Det}\left\|h_i^{\,s}\cos\alpha+k_i^{\,s}\sin\alpha-\omega\delta_i^s\right\|=0. \tag{2} \]

Since the tensors \(h_{ij}, k_{ij}\) have common principal directions \((^{26})\), the roots of equation (2) are equal to

\[ \omega_i=\sigma_i\cos\alpha+\tau_i\sin\alpha, \]

where \(\sigma_i,\tau_i\) are the principal values of the tensors \(h_{ij}, k_{ij}\), respectively.
Thus, on the ray (1) there exist two desired focal points

\[ \underset{1}{\mathbf R} =\mathbf r+ \frac{1}{\sigma_1\cos\alpha+\tau_1\sin\alpha} (\mathbf X\cos\alpha+\mathbf Y\sin\alpha), \]

\[ \underset{2}{\mathbf R} =\mathbf r+ \frac{1}{\sigma_2\cos\alpha+\tau_2\sin\alpha} (\mathbf X\cos\alpha+\mathbf Y\sin\alpha). \tag{3} \]

If in (1) we put \(\alpha=0\) and \(\alpha=\pi/2\), then we obtain the congruences of normals \(D_2\) going in the directions of the unit vectors \(\mathbf X\) and \(\mathbf Y\), respectively.
Thus, on each normal to \(D_2\) there exist two focal points. Let us determine how the focal points are situated in the normal plane to \(D_2\). Introduce in this plane a rectangular coordinate system with origin at the point of the surface \(D_2\) and with axes going in the directions of the unit vectors \(\mathbf X,\mathbf Y\). Then from (3) it follows that the coordinates of the focal points \(F_1(\underset{1}{\mathbf R})\) are equal to

\[ x=\frac{\cos\alpha}{\sigma_1\cos\alpha+\tau_1\sin\alpha}, \qquad y=\frac{\sin\alpha}{\sigma_1\cos\alpha+\tau_1\sin\alpha}. \]

Eliminating the parameter \(\alpha\), we obtain the equation

\[ \sigma_1x+\tau_1y=1. \tag{4} \]

Thus, the focal points \(F_1(\underset{1}{\mathbf R})\) are situated in the normal plane on a straight line not passing through the point of the surface \(D_2\). Similarly, we obtain that the focal points \(F_2(\underset{2}{\mathbf R})\) lie on the straight line

\[ \sigma_2x+\tau_2y=1. \tag{5} \]

We shall call the straight lines (4) and (5) the axes of curvature of the surface \(D_2\). From (4) and (5) we find the geometric meaning of the invariants \(\chi_1=\sqrt{\sigma_1^2+\tau_1^2}\) and \(\chi_2=\sqrt{\sigma_2^2+\tau_2^2}\): their reciprocals \(\frac{1}{\chi_1}\) and \(\frac{1}{\chi_2}\) give the distances from the point of the surface \(D_2\) to the axes of curvature (4) and (5), respectively.

Let us pose the question: do there exist envelopes of the family of normal planes of the surface \(D_2\)?
Consider the points

\[ \mathbf R=\mathbf r(u^1,u^2)+a(u^1,u^2)\mathbf X+b(u^1,u^2)\mathbf Y \]

in the normal plane \(D_2\), and require that at these points it be tangent to some surface, i.e. that the derivatives \(\mathbf R_1,\mathbf R_2\) decompose with respect to \(\mathbf X\) and \(\mathbf Y\). We have

\[ \mathbf R_i=(\delta_i^s-ah_i^{\,s}-bk_i^{\,s})\mathbf r_s+a_i\mathbf X+b_i\mathbf Y, \]

whence we obtain the condition

\[ \delta_i^s-ah_i^{\,s}-bk_i^{\,s}=0. \tag{6} \]

Let \(a^i\) and \(\widetilde a^i\) be common principal directions of the tensors \(h_{ij}\) and \(k_{ij}\). Then from (6) we have

\[ a^i-a\sigma_1a^i-b\tau_1a^i=0,\qquad \widetilde a^i-a\sigma_2\widetilde a^i-b\tau_2\widetilde a^i=0 \]

or

\[ \begin{aligned} a\sigma_1+b\tau_1&=1,\\ a\sigma_2+b\tau_2&=1. \end{aligned} \tag{7} \]

In this case our requirements are satisfied and the envelope exists. Comparing (7) with (4) and (5), we see that \(a(u^1,u^2)\) and \(b(u^1,u^2)\) are the coordinates of the points of intersection of the axes of curvature, which, thus, lie on the envelope of the family of normal planes \(D_2\). Thus, we have proved that the normal planes \(D_2\), embedded in \(E_4\), admit an enveloping surface. We shall call this surface the evolute surface.

We shall now prove that if the normal planes of a two-dimensional surface embedded in \(E_4\) admit an envelope, then this surface is dually normalizable.

Let

\[ \mathbf R=\mathbf R(u^1,u^2) \]

be the parametric equation of the evolute surface. If \(\mathbf m,\mathbf n\) are unit and mutually perpendicular normal vectors of the evolute surface, then its basic derivative equations have the form

\[ \nabla_j\mathbf R_i=a_{ij}\mathbf m+b_{ij}\mathbf n, \tag{8} \]

where \(a_{ij}=-\partial_j\mathbf R\,\partial_i\mathbf m=\mathbf m\nabla_j\mathbf R_i\), \(b_{ij}=-\partial_j\mathbf R\,\partial_i\mathbf n=\mathbf n\nabla_j\mathbf R_i\), and \(\nabla\) is the symbol of covariant differentiation in the intrinsic connection of the evolute surface. If the radius vector of a point of the original surface is represented in the form

\[ \mathbf R^*=\mathbf R(u^1,u^2)+\lambda^k(u^1,u^2)\mathbf R_k, \tag{9} \]

then we must require that the derivatives \(\mathbf R^*_1,\mathbf R^*_2\) be resolved along \(\mathbf m\) and \(\mathbf n\). We have:

\[ \mathbf R_i^*=(\delta_i^m+\nabla_i\lambda^k)\mathbf R_k+\lambda^k a_{ik}\mathbf m+\lambda^k b_{ik}\mathbf n, \]

whence we obtain the condition

\[ g_{ij}+\nabla_i\lambda_j=0; \tag{10} \]

\(\lambda_i\) is a gradient, since \(g_{ij}\) is symmetric.

Forming the integrability condition for (10), we obtain \(R_{\cdot kij}^{\ \ \ l}\lambda_l=0\), or \({}^{(3)}\)

\[ K\varepsilon_{jk}\varepsilon_i^{\ l}\lambda_l=0, \tag{11} \]

where \(K\) is the Gaussian curvature of the intrinsic geometry of the evolute surface. From (11) it follows that \(K=0\), and this means that the intrinsic geometry of the evolute surface is Euclidean. The line element of this surface can be reduced to the form \(ds^2=du^2+dv^2\). Hence, in turn, it follows that

\[ \mathbf R_1^2=1,\qquad \mathbf R_2^2=1,\qquad \mathbf R_1\mathbf R_2=0. \]

From (10) we obtain

\[ \partial_1\lambda_1=-1,\qquad \partial_2\lambda_2=-1,\qquad \partial_1\lambda_2=\partial_2\lambda_1=0; \]

whence it follows that

\[ \lambda_1=-u+a,\qquad \lambda_2=-v+b, \]

where \(a,b\) are constants.

From what has been said it is clear that equation (9) can be rewritten in the form

\[ \mathbf R^*=\mathbf R(u,v)+(a-u)\mathbf R_1+(b-v)\mathbf R_2. \tag{12} \]

It is easy to see that equation (12) represents the surface \(D_2\). Indeed, the vectors \(\mathbf R_1, \mathbf R_2\) are unit vectors, and for them the conditions
\(\partial_i\mathbf R_j=a_{ij}\mathbf m+b_{ij}\mathbf n\) hold, where \(\mathbf m\) and \(\mathbf n\) are normal vectors. Hence

\[ \mathbf R_1\partial_j\mathbf R_2=0,\qquad \mathbf R_2\partial_j\mathbf R_1=0,\qquad j=1,2. \]

This means that the surface is doubly normalized.

Thus, we have proved the following theorems:

Theorem 1. In order that a surface be doubly normalizable, it is necessary and sufficient that its normal planes have an envelope of zero curvature.

Theorem 2. Every \(D_2\) in \(E_4\) cuts orthogonally the tangent planes of some surface of zero curvature.

Thus, the problem of finding \(D_2\) in \(E_4\) is equivalent to the problem of finding \(X_2\) of zero curvature. We shall call surfaces that cut orthogonally the tangent planes of a given surface its evolvent surfaces. Then this result can be formulated as follows:

Every \(D_2\) in \(E_4\) is an evolvent surface of an arbitrary surface of zero curvature.

In conclusion I express my gratitude to A. P. Norden, under whose guidance the present work was carried out.

Yerevan State
University

Received
2 XII 1961

REFERENCES CITED

1. A. P. Norden, Spaces of Affine Connection, 1950.
2. A. V. Chakmazyan, Dokl. Acad. Sci. ArmSSR, a) 28, No. 24, 151 (1959); b) 29, No. 1, 3 (1959); c) 31, 129 (1960); d) 30, No. 4, 187 (1960); e) 33, No. 3 (1961).
3. A. P. Norden, Theory of Surfaces, 1956.