Abstract
Full Text
MATHEMATICS
V. S. MALAKHOVSKII
INVARIANT CONSTRUCTION OF THE DIFFERENTIAL GEOMETRY OF A MANIFOLD OF PLANE ALGEBRAIC ELEMENTS
(Presented by Academician I. N. Vekua, 11 V 1963)
In this paper an \(m\)-dimensional manifold \(V_m^k\) is studied, whose element is a nondegenerate algebraic hypersurface of even order \(k=2p\) of a hyperplane of the \(n\)-dimensional projective space \(P_n\) (a plane algebraic element of order \(k\)). A fundamental object has been obtained which determines the manifold up to its position in the space. A number of objects invariantly associated with the manifold are found, and their geometric characterization is clarified.
- If the vertices \(A_\alpha\) \((\alpha,\beta,\sigma_q,\ldots=1,2,\ldots,n)\) of a moving frame are placed in the hyperplane of the algebraic element, and the vertex \(A_{n+1}\) outside it, then the equations of the plane algebraic element are written in the form
\[ a_{x_1\ldots \alpha_k}^{1} x^{\alpha_1}\cdots x^{\alpha_k}=0,\qquad x^{n+1}=0, \tag{1} \]
where the hyperdeterminant \(a\) of the symmetric spatial matrix \((a_{\alpha_1\ldots\alpha_k})\) may always be assumed equal to unity \((^1)\). The system of differential equations of invariance \((^2)\) of the algebraic element (1) has the form
\[ \omega_\alpha=0,\qquad \nabla a_{\alpha_1\ldots\alpha_k}+\frac{k}{n}a_{\alpha_1\ldots\alpha_k}\omega_\beta^\beta=0, \tag{2} \]
where \(\nabla\) is the symbol of covariant differentiation, \(\omega_\alpha=\omega_\alpha^{n+1}\), and \(\omega_{\alpha'}^{\beta'}\) \((\alpha',\beta',\ldots=1,\ldots,n+1)\) are the components of the derivative formulas \(d\bar A_{\alpha'}=\omega_{\alpha'}^{\beta'}\bar A_{\beta'}\) of the moving frame. System (2) determines a \((C_{n+k-1}^k+n-1)\)-dimensional space of plane algebraic elements of order \(k\). If the dimension \(m\) of the manifold \(V_m^k\) exceeds the number \(h\) of parameters on which the hyperplanes containing the algebraic elements in it depend, i.e. \(m=h+p\), then the manifold \(V_m^k\) is considered on \(\infty^h\) submanifolds with a fixed hyperplane. Therefore we shall restrict ourselves to consideration of the case \(m=h\). The closed system of differential equations
\[ \Delta a_{\check i}^{j}=a_{\check i}^{j}\omega_j,\qquad \nabla a_{\alpha_1\ldots\alpha_k}+\frac{k}{n}a_{\alpha_1\ldots\alpha_k}\omega_\beta^\beta =b_{\alpha_1\ldots\alpha_k}^{j}\omega_j, \]
\[ [\Delta a_{\check i}^{j}\omega_j]=0,\qquad [\Delta b_{\alpha_1\ldots\alpha_k}^{j}\omega_j]=0 \tag{3} \]
determines a manifold \(V_m^k\) of this type with an arbitrariness \(C_{n+k-1}^k+n-m-1\) functions of \(m\) arguments. Here \(i,j,\ldots=1,\ldots,m\);
\[ \check i,\check j,\ldots=m+1,\ldots,n;\qquad \Delta a_{\check i}^{j}=\nabla a_{\check i}^{j}+a_{\check i}^{i}a_j^{i}\omega_{\check i}^{j}-\omega_{\check i}^{j}, \]
\[ \Delta b_{\alpha_1\ldots\alpha_k}^{j} =\nabla b_{\alpha_1\ldots\alpha_k}^{j} +b_{\alpha_1\ldots\alpha_k}^{j} \left(\frac{k}{n}\omega_\beta^\beta-\omega_{n+1}^{n+1}\right) +a_j^i b_{\alpha_1\ldots\alpha_k}^{j}\omega_i^j+ \]
\[ +\frac{k}{n}a_{\alpha_1\ldots\alpha_i}\left(\omega_{n+1}^{j}+a_{\check i}^{j}\omega_{n+1}^{\check i}\right) -\{a_{\beta\alpha_2\ldots\alpha_k}(\delta_{\alpha_1}^{j}+\delta_{\alpha_1}^{\check i}a_{\check i}^{j})+\cdots \]
\[ \cdots+a_{\alpha_1\ldots\alpha_{k-1}\beta}(\delta_{\alpha_k}^{j}+\delta_{\alpha_k}^{\check i}a_{\check i}^{j})\}\omega_{n+1}^{\beta}, \]
where the quantities \(b_{\alpha_1\ldots\alpha_k}^{i}\), \(\Delta b_{\alpha_1\ldots\alpha_k}^{i}\) are symmetric with respect to any pair of lower indices.
indices and satisfy the identities
\[ a^{\alpha_1\ldots\alpha_k} b^i_{\alpha_1\ldots\alpha_k}=0,\qquad a^{\alpha_1\ldots\alpha_k}\,[\Delta b^i_{\alpha_1\ldots\alpha_k}\omega]=0, \]
where \(a^{\alpha_1\ldots\alpha_k}\) are the algebraic complements of the elements \(a_{\alpha_1\ldots\alpha_k}\) of the hyperdeterminant \(a\).
From (3) it follows:
Theorem 1. The intrinsic fundamental object
\[
\Gamma=\{a^j_i,\; da_{\alpha_1\ldots\alpha_k},\; b^i_{\alpha_1\ldots\alpha_k}\}
\]
is the fundamental object of the manifold \(V_m^k\). The proper specification of the components of its first prolongation determines the manifold \(V_m^k\) up to constants \((^2)\).
The systems of quantities \(a^j_i\) and \(a_{\alpha_1\ldots\alpha_k}\) form subobjects of the object \(\Gamma\), and \(a_{\alpha_1\ldots\alpha_k}\) is a symmetric tensor. If \(m<n\), then the geometric object \(a^j_i\) determines in the hyperplane of the locally algebraic element of the manifold \(V_m^k\) an \((n-m-1)\)-dimensional characteristic subspace
\[ x^i+x^i a^j_i=0,\qquad x^{n+1}=0, \tag{4} \]
consisting of points which, under transition to a neighboring algebraic element, remain in the same hyperplane \(x^{n+1}=0\). On the manifold \(V_m^2\) the system of quantities \(a^{\alpha\beta}\) determines a twice contravariant symmetric tensor.
- Consider the manifold \(V_n^k\). The system of quantities \((a_{\alpha_1\ldots\alpha_k}, b_{\alpha_1\ldots\alpha_{k-1}})\), where \(b_{\alpha_1\ldots\alpha_{k-1}}=b^\beta_{\alpha_1\ldots\alpha_{k-1}\beta}\), determines a linear homogeneous object enveloping the tensor \(a_{\alpha_1\ldots\alpha_k}\). For manifolds \(V_n^2\) this object has a simple geometric characteristic. It determines in the space \(P_n\) an invariant pencil of hyperquadrics
\[ a_{\alpha\beta}x^\alpha x^\beta+ \frac{2n}{(n-1)(n+2)}\,b_\alpha x^\alpha x^{n+1} +\lambda (x^{n+1})^2=0, \tag{5} \]
containing the local quadratic element. The quasitensor \(b^\alpha=a^{\alpha\beta}b_\beta\) determines the invariant point
\[
\overline{B}=b^\alpha \overline{A}_\alpha+
\frac{2-n(n+1)}{n}\,\overline{A}_{n+1},
\]
which is the vertex of the hypercone of the pencil (5).
The system of quantities
\[ b_{\alpha_1\ldots\alpha_{2k-1}} = a_{\beta(\alpha_1\ldots\alpha_{k-1}} b^\beta_{\alpha_k\ldots\alpha_{2k-1})} - \frac{k}{n+k}\, b_{(\alpha_1\ldots\alpha_{k-1}} a_{\alpha_k\ldots\alpha_{2k-1})}, \tag{6} \]
where the parentheses denote cyclic alternation, forms a \((2k-1)\)-times covariant symmetric tensor determining, in the hyperplane of each locally algebraic element, an invariant \((n-2)\)-dimensional algebraic surface of order \(2k-1\) (the attached surface)
\[ b_{\alpha_1\ldots\alpha_{2k-1}}x^{\alpha_1}\cdots x^{\alpha_{2k-1}}=0,\qquad x^{n+1}=0. \]
The common points of the locally algebraic element (1) and of the attached surface form the \(t\)-focal manifold of the algebraic element. For \(n=3\) this manifold consists of \(k(2k-1)\) \(t\)-focal points of a plane algebraic curve, which have a simple geometric characteristic.
Theorem 2. In order that a point \(M\) of the curve (1) be a \(t\)-focal point of this curve, it is necessary and sufficient that it be a focus of the nonholonomic congruences (curves) determined by all points of the tangent to the curve (1) at \(M\).
Remark. If, in the plane \(x^4=0\) of the curve (1), one takes an arbitrary point \(N\,(q^\alpha,0)\) \((\alpha=1,2,3)\), then all displacements of this point lying in the plane of the curve satisfy the equation \(q^\alpha\omega_\alpha=0\), which determines a nonholonomic congruence of curves belonging to the complex \(V_3^k\). Consequently, each point in the plane of the curve uniquely determines a nonholonomic curvilinear congruence.
- On the manifold \(V_n^2\) of quadratic elements there are defined the tensors
\(b_{\alpha\beta}=a^{\mu\eta}a^{\nu\varkappa}b_{\mu\nu}b_{\eta\varkappa\beta}\) and
\(c^\alpha=a^{\alpha\mu}a^{\beta\nu}a^{\gamma\eta}b_{\mu\nu}b_{\beta\gamma}\), and the following system of relative and absolute invariants:
\[ b_0=a^{\alpha\beta}b_{\alpha\beta},\quad c_0=a_{\alpha\beta}c^\alpha c^\beta,\quad \hat c_0=b_{\alpha\beta}c^\alpha c^\beta,\quad \hat c=b_{\alpha\beta\gamma}c^\alpha c^\beta c^\gamma,\quad b=\det(b_{\alpha\beta}); \tag{7} \]
\[ B_0=\frac{b_0^n}{b},\quad C_0=\frac{c_0^n}{b^3},\quad \hat C_0=\frac{\hat c_0^n}{b^4},\quad \hat C=\frac{\hat c^n}{b^5}. \tag{8} \]
For such a manifold the following “principle of transfer” is valid.
Theorem 3. The differential geometry of an \(n\)-dimensional manifold \(V_n^2\) of quadratic elements of the space \(P_n\) may be regarded as the geometry of a certain regular hypersurface of the \((n+1)\)-dimensional tangential centroprojective space \(P_0^{\,n+1}\), in which the original \(n\)-dimensional point space \(P_n\) plays the role of a fixed point.
- Let us consider a manifold \(V_m^2\) of dimension \(m<n\). In the general case the characteristic subspace (4) does not intersect its polar subspace, and we can place the vertices \(A_{\breve i}\) of the frame in the subspace (4), and the vertices \(A_i\) in the polar subspace. Then
\(a_{\breve i}^{\,i}=0,\ a_{\breve i\breve j}=0\), and the Pfaff forms \(\omega_{\breve i}^{\,i},\ \omega_i^{\,\breve i}\) become the principal forms of the manifold \(V_m^2\), and we may put
\(\omega_{\breve i}^{\,j}=b_{\breve i}^{\,jp}\omega_p,\ \omega_i^{\,\breve i}=a^{\breve i j}(b_{\breve i j}^{\,p}+a_{iq}b_{\breve j}^{\,qp})\omega_p\).
On the manifold \(V_m^2\) there exist the following geometric objects:
1) The tensors \(a_{ij},\ a_{\breve i\breve j},\ a^{ij},\ a^{\breve i\breve j}\). The equations \(x^{n+1}=0,\ a_{ij}x^i x^j=0\) and \(x^{n+1}=0,\ a_{\breve i\breve j}x^{\breve i}x^{\breve j}=0\) define invariant hypercones whose vertices are, respectively, the characteristic subspace and the polar subspace.
2) The quasitensor \(\hat a^i=a^{jp}b_{jp}^{\,i}\), which determines the \((n-m)\)-dimensional invariant plane
\(x^i+\dfrac{n}{2(n-m)}\hat a^i x^{n+1}=0\).
3) The quasitensors \(\hat b^{\,i}=a^{ij}b_{jp}^{\,p},\ \hat b^{\,\breve i}=a^{\breve i\breve j}b_{p\breve j}^{\,p}\), which determine the invariant point
\[
\bar B=\bar A_{n+1}-\frac{n}{n(m+1)-2}\hat b^{\,i}\bar A_i-\frac{1}{m}\hat b^{\,\breve i}\bar A_{\breve i}
\]
of the space \(P_n\).
4) The vector
\[
a^i=\hat a^i-\frac{2(n-m)}{n(m+1)-2}\hat b^{\,i},
\]
which determines in the polar subspace the invariant point \(\bar A=a^i\bar A_i\).
5) The thrice covariant tensor
\[
b_{ijp}=a_{q(i}b_{jp)}^{\,q}-\frac{2(n-1)}{n(m+2)-1}\hat b^{\,q}a_{q(i}a_{jp)}.
\]
It determines the associated \(t\)-focal manifold
\[
a^{ij}x^i x^j=0,\quad b_{ijp}x^i x^j x^p=0,\quad x^{n+1}=0,
\]
consisting of an \((m-3)\)-parameter family of \((n-m)\)-dimensional plane generators passing through the characteristic subspace (4). For \(m=3\) the polar subspace (plane) contains, in the general case, 6 \(t\)-focal points belonging to the invariant conic
\[ a_{ij}x^i x^j=0,\quad x^{\breve i}=0,\quad x^{n+1}=0 \tag{9} \]
and possessing a simple geometric characteristic.
Theorem 4. In order that a point \(M\) of the conic \(a_{ij}x^i x^j=0,\ x^{n+1}=0\) of the manifold \(V_3^2\) be \(t\)-focal, it is necessary and sufficient that it be projected from the characteristic subspace to a point \(M^*\) of the conic (9) that is the focus of all nonholonomic congruences (cones) determined by the points of the tangent to the conic (9) at \(M^*\).
Tomsk State University
named after V. V. Kuibyshev
Received
6 V 1963
References Cited
- N. P. Sokolov, Spatial matrices and their applications, Moscow, 1960.
- G. F. Laptev, Trudy Moskov. matem. obshch., 2, 275 (1953).