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MATHEMATICS
M. A. AKIVIS
ON THE STRUCTURE OF MULTIDIMENSIONAL SURFACES CARRYING A NET OF LINES OF CURVATURE
(Presented by Academician P. S. Novikov, 19 XI 1962)
- Let \(V_m\) be an \(m\)-dimensional nonisotropic surface of the \(n\)-dimensional conformal space \(C_n\) \((m<n)\). As was shown in \((^1)\), with the first-order neighborhood of a point \(A\) of this surface there is associated a quadratic form
\[ \varphi^0=g_{ij}\omega^i\omega^j, \]
which determines an angular metric on \(V_m\), and with its second-order neighborhood—an invariant pencil of quadratic forms
\[ \varphi^\alpha=a^\alpha_{ij}\omega^i\omega^j. \]
We shall call \(V_m\) a surface carrying a net of lines of curvature if at each of its points the quadratic forms \(\varphi^0\) and \(\varphi^\alpha\) can be simultaneously reduced to canonical form. Directions issuing from the point \(A\), conjugate to one another with respect to the forms \(\varphi^0\) and \(\varphi^\alpha\), will be called principal directions, and the lines enveloping them—the lines of curvature of the surface \(V_m\). The principal directions are mutually orthogonal.
A hypersurface \(V_{n-1}\) of the space \(C_n\) always carries a net of lines of curvature, while for \(m<n-1\) surfaces \(V_m\) carrying a net of lines of curvature form a special class of \(m\)-dimensional surfaces. In the present paper the structure of such surfaces is studied.
Under the Darboux transfer, a nonisotropic surface \(V_m\) of the space \(C_n\) is mapped onto a tangentially nondegenerate surface \(U_m\) lying on a second-order hypersurface \(Q_n\) of the projective space \(P_{n+1}\). Moreover, if the surface \(V_m\) carries a net of lines of curvature, then the corresponding surface \(U_m\) carries a net of conjugate lines in the sense of our paper \((^2)\). Therefore, in studying surfaces \(V_m\) of the space \(C_n\) carrying a net of lines of curvature, the results of \((^2)\) can be used.
- Attach to the surface \(V_m\) a conformal frame whose point \(A_0\) coincides with the point \(A\) of the surface; the hyperspheres \(A_i\) \((i=1,\ldots,m)\) are orthogonal to it at the point \(A\); the hyperspheres \(A_\alpha\) \((\alpha=m+1,\ldots,n)\) are tangent to it at this point; and the point \(A_{n+1}\) serves as the second point of intersection of the hyperspheres \(A_i\) and \(A_\alpha\). Then
\[ dA_\xi=\omega^\eta_\xi A_\eta \quad (\xi,\eta=0,1,\ldots,n+1), \]
where the forms \(\omega^\eta_\xi\) are related by a number of relations (see \((^1)\)), one of which has the form
\[ \omega^{n+1}_i=-g_{ij}\omega^j. \]
Here \(g_{ij}=(A_iA_j)\) is the tensor generating the quadratic form \(\varphi^0\).
The surface \(V_m\) is determined by the system of Pfaff equations
\[ \omega^\alpha_0=0, \tag{1} \]
and the forms \(\omega^i=\omega^i_0\) will be linearly independent on the surface \(V_m\). Differential prolongation of the system of equations (1) leads to a new system
of equations
\[ \omega_i^\alpha=\lambda_{ij}^\alpha \omega^j . \]
The coefficients \(\lambda_{ij}^\alpha\) entering here make it possible to construct a pencil of tensors
\[ a_{ij}^\alpha=\lambda_{ij}^\alpha-\frac{1}{m}\lambda_{kl}^\alpha g^{kl}g_{ij}, \]
by means of which an invariant pencil of quadratic forms \(\varphi^\alpha\) is introduced.
If we choose the hyperspheres \(A_i\) of the frame so that they are orthogonal to the corresponding lines of curvature of the surface \(V_m\), then its quadratic forms \(\varphi^0\) and \(\varphi^\alpha\) take the form
\[ \varphi^0=(\omega^1)^2+\ldots+(\omega^m)^2, \]
\[ \varphi^\alpha=a_1^\alpha(\omega^1)^2+\ldots+a_m^\alpha(\omega^m)^2; \]
the \(i\)-th family of lines of curvature is determined on the surface \(V_m\) by the system of equations:
\[ \omega^j=0 \qquad \text{for } j\ne i. \]
Now \(g_{ij}=\delta_{ij}\), where \(\delta_{ij}\) is the Kronecker symbol, \(\omega_i^j+\omega_j^i=0\), and
\[ \omega_i^\alpha=\lambda_i^\alpha\omega^i,\qquad \omega_i^j=\sum_k l_{ik}^j\omega^k \quad (i\ne j), \tag{2} \]
where, here and below, there is no summation over the indices \(i,j,k\) unless this is indicated by the sign \(\sum\).
- The second-order neighborhood of the point \(A_0\) of the surface \(V_m\) is determined by the expression
\[ d^2 A_0 \equiv \sum_i(\lambda_i^\alpha A_\alpha-A_{n+1})(\omega^i)^2 \quad (\operatorname{mod} A_0,A_i). \]
Let us consider the hyperspheres
\[ B_i=\lambda_i^\alpha A_\alpha-A_{n+1}. \]
The totality of hyperspheres passing through the point \(A_0\) and orthogonal to the hyperspheres \(A_i\) and \(B_i\) determines the first osculating sphere of the surface \(V_m\). If by \(m_1\) we denote the number of independent hyperspheres among the \(B_i\) \((m_1\le m)\), then the dimension of this osculating sphere will be equal to \(m+m_1-1\). The number \(m_1\) is equal to the number of linearly independent forms among the quadratic forms \(\varphi^0,\varphi^\alpha\), and the number \(m_1-1\) is equal to the number of linearly independent forms among the quadratic forms \(\varphi^\alpha\). We shall denote the first osculating sphere of the surface \(V_m\) by \(C_{m+m_1-1}\).
- The following theorems relate the structure of the surface \(V_m\) of the space \(C_n\) to the structure of its system of hyperspheres \(B_i\).
Theorem 1a. If the number \(m_1\) of linearly independent hyperspheres \(B_i\) of the surface \(V_m\) is less than \(m\), and any subsystem of the system of hyperspheres \(B_i\) consisting of \(m-1\) hyperspheres has rank \(m_1\), then the surface \(V_m\) lies entirely in its first osculating sphere \(C_{m+m_1-1}\).
Theorem 1b. If all hyperspheres \(B_i\) of the surface \(V_m\) are linearly independent, then its first osculating sphere has dimension \(2m-1\), and the second osculating sphere may have dimension \(2m+m_2-1\), where \(0\le m_2\le m\).
Theorem 1c. Let \(m_1<m\), and suppose that among the hyperspheres \(B_i\) there are \(p\) hyperspheres \(B_1,\ldots,B_p\) such that, when any one of them is omitted, the system of hyperspheres \(B_i\) does not decrease in rank, and \(q=m-p\) hyperspheres \(B_{p+1},\ldots,B_m\) such that, when any one of them is omitted, the system of hyperspheres \(B_i\) decreases in rank by one. Then this surface, on the one hand, decomposes into a \(q\)-parameter family of surfaces \(V_p\), each of which belongs to its own \((p+p_1-1)\)-dimensional osculating sphere (here \(p_1=m_1-q\)), and on the other
from the other side, into a \(p\)-parameter family of surfaces \(V_q\), the first osculating sphere of which has dimension \(2q-1\), and the second osculating sphere dimension \(2q+m_2-1\), where \(0\leq m_2\leq q\). The surface \(V_m\) itself then has the first osculating sphere of dimension \(m+m_1-1\) and the second osculating sphere of dimension \(m+m_1+m_2-1\).
Theorems 1a and 1b are special cases of Theorem 1c, corresponding to \(p=m\) and \(p=0\).
- We shall call a net of lines of curvature on a surface \(V_m\) of the space \(C_n\) completely holonomic if each of the Pfaff equations
\[ \omega^i=0, \]
which define the submanifolds of the net, is completely integrable. If, conversely, none of these equations and no subsystem of them consisting of fewer than \(m-1\) equations is completely integrable, then we shall call the net of lines of curvature of the surface \(V_m\) irreducible. The condition for complete holonomicity of the net of lines of curvature of the surface \(V_m\) is that the coefficients \(l^j_{ik}\) in the expansion (2) of the forms \(\omega_i^j\) vanish for \(k\ne i,\ k\ne j\).
The following theorems relate the question of holonomicity of the net of lines of curvature on a surface \(V_m\) to the structure of its system of hyperspheres \(B_i\).
Theorem 2a. If no three hyperspheres \(B_i\) of the surface \(V_m\) belong to one pencil, then the net of lines of curvature of this surface is completely holonomic.
Theorem 2b. Only surfaces \(V_m\) of classes 1 and 0 (i.e., surfaces \(V_m\) wholly belonging to an \((m+1)\)-dimensional sphere, and \(m\)-dimensional spheres) can have an irreducible net of lines of curvature.
Theorem 2c. A surface \(V_m\) carrying a net of lines of curvature can decompose into a \(q\)-parameter family of \(p\)-dimensional surfaces \(V_p\) \((p+q=m)\), carrying an irreducible net of lines of curvature, in two cases: 1) when, among the system of hyperspheres \(B_i\) of the surface \(V_m\), \(p\) hyperspheres belong to one pencil, but no two of them coincide; the class of the surfaces \(V_p\) in this case is equal to one; 2) when, among the system of hyperspheres \(B_i\), \(p\) hyperspheres coincide with one another; the surfaces \(V_p\) in this case will be \(p\)-dimensional spheres.
Theorem 2d. If on a surface \(V_m\), carrying a net of lines of curvature, among the system of hyperspheres \(B_i\) \(p\) hyperspheres coincide with one another, then this surface is the envelope of an \((m-p)\)-parameter family of \(m\)-dimensional spheres.
- Surfaces \(V_m\) carrying a holonomic net of lines of curvature are determined by the system of Pfaff equations (1), (2), in which now \(l^j_{ik}=0\) for \(k\ne i,\ k\ne j\). Investigation of this system of equations leads to the following existence theorem:
Theorem 3. Surfaces \(V_m\) carrying a holonomic net of lines of curvature, on which the systems of hyperspheres \(B_i\) satisfy the conditions of Theorems 1a and 1b, exist, and the arbitrariness of their existence is equal to \(\dfrac{m(m-1)}{2}\) functions of two arguments.
The proof of the existence theorem for surfaces \(V_m\) carrying a holonomic net of lines of curvature, on which the system of hyperspheres \(B_i\) satisfies the conditions of Theorem 1c for \(p\ne m,\ p\ne 0\), is complicated by a number of technical difficulties.
Moscow Institute
of Steel and Alloys
Received
19 XI 1962
REFERENCES
- M. A. Akivis, Matem. sborn., 53 (95), 1, 53 (1961).
- M. A. Akivis, DAN, 139, No. 6, 1279 (1961).