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MATHEMATICS
S. L. PEVZNER
INVARIANTS AND CANONICAL REPRESENTATIONS OF QUADRICS IN QUASI-ELLIPTIC SPACES
(Presented by Academician P. S. Novikov on 24 X 1962)
In the present note we find invariants and canonical representations of quadrics in quasi-elliptic spaces \((^{1},\,^{2})\), and a substantial part of the results is proved for spaces of a more general kind.
- Consider the Cayley–Klein space whose absolute in the projective space \(P_n\) consists of a cone of index \(l_0\)
\[ \sum_{a,b} h_{ab}x^a x^b=0,\qquad a,\ b=0,\ 1,\ldots,m, \tag{1} \]
in an \((n-m-1)\)-dimensional flat vertex in which a nondegenerate quadric of index \(l_1\) is specified,
\[ x^a=0,\qquad \sum_{u,v} g_{uv}x^u x^v=0,\qquad u,\ v=m+1,\ldots,n. \tag{2} \]
We shall call such a space quasi-Euclidean (cf. \((^{1},\,^{2})\)) and denote it by \({}_{l_0l_1}S_n^m\). For \(l_0=l_1=0\) we arrive at a quasi-elliptic space \({}_{00}S_n^m\) (or \(S_n^m\)); if \(l_0^2+l_1^2\ne0\), then the space is called quasi-hyperbolic. For \(m=0\) we obtain a generalized Euclidean (or pseudo-Euclidean) space \({}_{0l_1}S_n^0\) (or \({}_{l}S_n^0\)). For \(m=l_1=0\) we obtain the proper Euclidean space \(S_n^0\).
In the quasi-Euclidean space \({}_{l_0l_1}S_n^m\) consider the quadric
\[ x^T Mx=0 \tag{3} \]
with coefficient matrix
\[ M=\left\|\begin{array}{cc} C & B\\ B^T & A \end{array}\right\|\left\{\begin{array}{l} m+1\\ n-m; \end{array}\right. \]
where \(T\) denotes transposition and \(x\) is the column matrix of the coordinates of the current point. Introduce the matrix
\[ M(\mu,\lambda)= \left\|\begin{array}{cc} C-\mu H & B\\ B^T & A-\lambda G \end{array}\right\|, \]
where \(H=\|h_{ab}\|\), \(G=\|g_{uv}\|\), and, of course, \(\det H\ne0\) and \(\det G\ne0\). This matrix may be called the characteristic matrix of the quadric.
The following theorems on invariants of quadrics with respect to the motions (see \((^{3})\)) of the space \({}_{l_0l_1}S_n^m\) have been proved.
Theorem 1. If the quadric (3) is specified up to motions of the space \({}_{l_0l_1}S_n^m\), then the matrices \(A-\lambda G\) and \(M(\mu,0)\) are determined up to congruent transformations.
This theorem strengthens the results of \((^1,{}^4)\). From it there follows not only the invariance of the elementary divisors of the matrices \(A-\lambda G\) and \(M(\mu,0)\), but also the invariance of the discrete characteristics of these real \(\lambda\)-matrices (see \((^5,{}^6)\)).
The invariant polynomial \(\det(A-\lambda G)\) gives \(n-m\) invariants of the equation of the quadric. The polynomial \(\det M(\mu,0)\) can give \(m+2\) invariants, of which \(m+1\) invariants, independent of the preceding ones, will be the ones lacking for determining the equation of the quadric. However, if the degree of the polynomial \(\det M(\mu,0)\) is less than \(m\), then it gives \(<m+1\) independent invariants; if the rank of the matrix \(\|B^T A\|\) is less than \(n-m\), then this polynomial vanishes identically and provides no invariants.
Denote by \(q\) the defect of the matrix \(\|B^T A\|\) with respect to rows (the number of linear dependencies among the rows of the matrix), and by \(p\) the defect of the matrix \(A\); \(p\ge q\).
Theorem 2. The polynomial
\[ m_q(\mu)=\left.\frac{\det M(\mu,\lambda)}{\lambda^q}\right|_{\lambda=0} \]
is invariant with respect to motions of the space \({}^{\iota}\!{}^{\iota}_{1}S_n^m\).
It can be shown that the degree of the polynomial \(m_q(\mu)\) for \(p\ne0\) (for \(p=q=0\) the degree of the polynomial \(m_0(\mu)\) is equal to \(m+1\)) is equal to \((m+1)-(p-q)\). In this case all coefficients of the polynomial \(m_q(\mu)\) are independent of the invariants found earlier, and we obtain \((m+2)-(p-q)\) new invariants.
Theorem 3. The coefficients of the polynomial \(\det M(\mu,\lambda)\) at the terms with
\[ \mu^{m+1-s}\lambda^{p-s},\qquad s=1,2,\ldots,p-q, \]
are invariant with respect to motions of the space \({}^{\iota}\!{}^{\iota}_{1}S_n^m\).
In the case \(p\ne0\), this theorem gives \(p-q-1\) invariants independent of the preceding invariants. Theorems 1–3 in all cases provide \(n+1\) independent invariants. Discrete invariants are not involved in the count.
The proof of Theorems 2 and 3 rests essentially on the following lemma.
Lemma 1. Suppose there are two square matrices consisting of identical blocks:
\[ \left\|\begin{array}{cc} X & Y\\ Z & T \end{array}\right\|,\qquad \left\|\begin{array}{cc} A & B\\ C & D \end{array}\right\|, \]
where \(X,T,A,D\) are square blocks and \(D\) is nonsingular. Suppose also that the defect of the matrix \(\|Z\ T\|\) with respect to rows and the defect of the matrix \(\left\|\begin{array}{c}Y\\ T\end{array}\right\|\) with respect to columns are equal to \(q\). Then, in the polynomials
\[ P_1(\lambda)= \left| \begin{array}{cc} X-\lambda A & Y-\lambda B\\ Z-\lambda C & D-\lambda T \end{array} \right|, \qquad P_2(\lambda)= \left| \begin{array}{cc} X & Y\\ Z & D-\lambda T \end{array} \right|, \]
the coefficients of \(\lambda^q\) are equal, and the terms with \(\lambda^0,\lambda^1,\ldots,\lambda^{q-1}\) are absent.
- Let us now consider quadrics in the quasi-elliptic space \(S_n^m\). All conclusions are formulated under the assumption that in the space
such a coordinate system is introduced in which the equations of the absolute (1) and (2) have, respectively, the form
\[ \sum_a (x^a)^2=0, \]
\[ x^a=0,\qquad \sum_u (x^u)^2=0. \]
By a successive simplification of the equation of the quadric, the following theorem can be proved:
Theorem 4. By motions of the quasielliptic space \(S_n^m\), the equation of a quadric can be brought to such a form that the matrix of the coefficients of the quadric will have the form
\[ M_0= \left\| \begin{array}{ccc|c|c|c} \mu_0 & & 0 & 0 & 0 & 0\\ & \ddots & & & & \\ 0 & & \mu_{m-(p-q)} & & & \\ \hline 0 & & & 0 & 0 & h_{p-q}\\ & & & & \ddots & \\ & & & h_1 & 0 & 0\\ \hline & & 0 & 0 & h_1 & \\ & & & \ddots & 0 & 0\\ & & h_{p-q} & 0 & & \\ \hline 0 & & & 0 & \lambda_{m+p+1} & 0\\ & & & & 0 & \ddots\\ & & & & & \lambda_n \end{array} \right\|. \]
Here \(\lambda_{m+p+1},\ldots,\lambda_n\) are the nonzero roots of the polynomial \(\det|A-\lambda G|\); \(\mu_0,\mu_1,\ldots,\mu_{m-(p-q)}\) are the roots of the polynomial \(m_q(\mu)\); \(h_1,\ldots,h_{p-q}\) are determined on the basis of Theorem 3.
It follows from this theorem that, in the case of a quasielliptic space, the system of invariants of quadrics determined by Theorems 1—3 is complete.
For \(m=0\), the quasielliptic space becomes the proper Euclidean space \(S_n^0\), and Theorem 4 reduces to known results (see, for example, (7)); Theorem 3 in this case is not needed (it gives no new invariants).
Received
19 X 1962
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