ON SINGULAR PROJECTIVE DEFORMATIONS OF TWO SECOND CLASS TOTALLY FOCAL PSEUDOCONGRUENCES OF PLANES

Let C: L L be a projective deformation of the second order of two totally pn focal pseudocongruences L and L of (m-l)-planes in projective spaces and n, 2m-i < n < 3m-l, and let K be a collineation realizing such a C. The deformation C is said to be weakly singular, singular, or a-strongly singular, s 3,4,..., if the collineation K gives projective deformations of order i, 2 or of all corresponding focal surfaces of L and L. It is proved that C is weakly singular and conditions are found for C to be singular. The pseudocongruences L and L are identical if and only f C is 3-strongly singular.


INTRODUCTION.
Let L and L be totally focal pseudocongruences of (m-l)-planes in projective spaces pn and n and let C: L be a correspondence between planes of L and .In the case of pseudocongruences of straight lines (m 2) regular and singular projective deformations C were studied in many papers (see Svec [i] where one can find further references) In the present paper we will suppose that m > 2 and 2m-i < n < 3m-l.The last restriction means that L and L are of second class, i.e. lie in their second osculating spaces provided that their first osculating spaces are tangent spaces.
The author (see Goldberg [2]) found necessary and sufficient conditions for C to be a projective deformation of order 1,2, and 3.However, conditions under which the pseudocongruences L and L are identical were not found in [2].
In the present paper we will indicate such a condition in terms of singular projective deformations.Note that second and third order singular projective deforma- tions were studied by the author for every n > 3m-i (see Goldberg [3]) and for every n > 4m-i (see Goldberg [4]).Note also that second order singular projective deforma- tions in odd-dimensional projective spaces were considered by Krelzlik [5,6].
If K is a collineation realizing a projective deformation C of second order, and at the same time K realizes projective deformations of order i, 2, or , ==3,4,..., of all corresponding focal surfaces of L and , then C is called weakly singular, singular, or n-strongly singular respectively.
In the present paper it is proved that a second order projective deformation C is weakly singular, necessary and sufficient conditions for C are found to be singular, and the following condition of identity of L and L is obtained; pseudocongruences L and L related by a second order projective deformation C: L L are identical if and only if C is 3-strongly singular.
Note that the author proved in [2] that if L C pn and C: L is a projective deformation, then C n.Because of this, we suppose from the beginning that L C pn and c n.A family L of planes of an n-dimensional projective space pn is said to be a pseudocorence if each hyperplane of pn contains locally a unique plane of Lo A pseudocongruence L of (ml)-planes is a family of m parameters.The admissible m-tuples (Ul,...,um) are taken from an open neighborhood of C m (C complex numbers).
A one-parameter subfamily of L is said to be ocaZ o ordP r if infinitesimally close planes of L have an r-dimensional intersection.Focal subfamilies of maximum order m-2 are called deop sPs of L. A pseudocongruence of (m-l)-planes possessing the maximum number m of developable surfaces is called oa.
In general, (m-2)-dimensional characteristics of each of these m developable surfaces forms a symplex in a plane Pm-I L. The vertices of this symplex are of Pm-l" Each focus generates the oc s of L of dimension mo A plane Pm-i belongs to the tangent m-plane of each of the m focal surfaces.
It was shown by Geidelman [7] that focal pseudocongruences can be of three types: (a) Pseudocongruences whose (m-l)-planes belong to an (m+l)-plane; (b) Pseudocongruences foliating into =b subfamilies of m-b parameters, 1 b m, where all (m-l)-planes of each of these subfamilies belong to an m-plane; (c) Pseudocongruences possessing m systems of integrable (m-l)-parameter focal subfamilies of order zero.
Pseudocongruences of the third type are called o (abbreviated t.f.).
Each of the m focal surfaces of a t.f.pseudocongruences is an m-conjugate system (see Geidelman [7]).
Let L be a t.f.pseudocongruence of (m-l)-planes Pm-i in pn.To each plane Pm-i L we associate a moving frame consisting of linearly independent analytic points AI,...,An+I, such that [AI, An+1 i In this paper we will suppose that 2m-i !n < 3m-l.In this case we can specialize the moving frames in such a way that (i) the vertices A i i l,...,m, are foci of Pm-i (ii) the line A.AIm+i is tangent to the line Yi of the conjugate net (A i) which is not tangent to Pm-1 (iii) the points A2m+l,...,A2m+o, where 2m+o n+l, are chosen arbitrarily (of course, (2.1) is supposed to be satisfied).
Under such a choice of vertices A of the moving frames the developable surfaces U m+i of L are determined by equations m.
0. Since all foci are supposed to be linearly 1 m+i independent, forms .are also linearly independent.We will take them as forms of I i the dual cobasis and will denote them by m m+i i m_ m (2.5) 1 In (2.5) and in what follows there is no summation of the indices i,j,k l,.o.,m, unless it is indicated by the summation sign.If the moving frames are specialized in the above described manner, we have: . 0 r i, o.
Because of this we have 2m+r.
rank(a i o (2.14) In other words (2.14) means that the second osculating space of L is the whole space pn, i.eo L is of second class.
Further exterior differentiation of (2o10), (2o11), and (2o12) and application of Cartan's lemma give rise to the following Pfaffian equations: (2.18) In the following we will need the differential extensions of (2.15) and (2.16)   which have the following form: 2. FIRST AND SECOND ORDER PROJECTIVE DEFORMATIONS OF T.F.PSEUDOCONGRUENCES.
It is well known that (m-l)-planes Pm-i of the space pn can be represented as points of the Grassmannlan G(m-l,n), dim G re(n-re+l), in a projective space g(pn)_ pN ,%).If {A u} is a moving frame in pn, then {[Aul ,...,%]} is a moving frame of pNo Let pn and n be two n-dimensional projective spaces with moving frames {A u} and {A--and K: pn / n be a co llineatlon given by The collineation K induces the collineation g(K): g(pn) g (n) given by A pseudocongruence L is represented in g(pn) by some surface belonglng to O(m-l,n).
Ne will denote it also by L.
A correspondence C: L / between two t.f.pseudocongruences L and of pn and n is said to be a projective deformation of order h if for any plane Pm-I 6 L there exists a colllneation K: pn / n such that surfaces g(K)g(L) and g() have the analytic contact of order h at the point g(pm_l ), i.e. if S KdS[A I'''%] Z (h)0 dh-[l''" L =0 (3.3)where s O,l,...,h and 0 are f-forms.Suppose that the moving frames {Au associated to the planes Pm-I 6 L are speci- alized similarly to the moving frames associated to the planes Pm-i 6 L. We will denote all expressions connected with L by suppressing the overbar.Then we have equations According to (3.3), the correspondence C: L / L is a projective deformation of order one if for any Pm-I E L there exists a colllneatlon K: pn -n such that Using notations (3.5), we can write (3.4) in the form Bymeans of (2.1), (2.3), and (2.4) one obtains diAl mi The author proved in [4] the following theorem: KA2m+r a2m+rAu r 1,..., o, u 1,..., n+l.
Although restrictions for n are different in [4] and in the present paper (n > 4m-i and 2m-i < n < 3m-i respectively), in the proof presented in [4] one needs to have n > 2m-1 only.
Note that in the proof as consequences of (3.6) the author obtained the form (3.8) of the collineation K, equalities i --i m m (3.9) giving developability of C, and the following form for the 1-form e I in (3.6): i i -i i.

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(-i + am+i0i m In addition, in what follows we will need the differential extension of equation (3.9) that has the following form: m+i i i Tm+i i tim (3.12)In the case of a projective deformation of second order one obtains from (3.Proof of Theorem 2 is computational and follows proof of a similar theorem in [4] where one should use (3.13) and (3.14).
Note also that in [4] the author proved that if C: L L is a projective deforma- tion of second order, then the following identities hold: 0 J 0j bj b(pjtj a j i ij ij m+j J .aJ..-J.oi..They can be obtained from (3.15) and (2o15)o (3.17) (3.18) realized by the same collineation Ko THEOREM 3. A second order projective deformation C: L L is weakly singular.
PROOF.Suppose that C: L + L is a projective deformation of second order, Joe.
we have (3.8) and (3.15)-(3o18)It follows from ( 26) and (2o10) that i dA i miAi + Using (3.8), one finds from (4.2) that i bJmJA.+ m Am+i (4.2) singular, singular, or a-strongly singular, a 3,4   if the correspondences C i induced by C are projective deformations of order one, two or a respectively 4. SINGULAR PROJECTIVE DEFORMATIONS OF T.F.PSEUDOCONGRUENCES.
A correspondence C: L + L induces the correspondences Ci: (Ai) / (A i) of focal surfaces of L and L. Suppose that there exists a collineation H such that S HdSAi (h dh-i =0 where @i are -formso In this case we will say that C i is a pro$etiue deformation omdem h between (A i) and (Ai).A second order projective deformation C: L -L realized by the collineation K which is determined by ( 38) is said to be weakZy According to (4ol), it means that K realizes a projective deformation of first order of (A i) and (A i) for any i and therefore the correspondence C: L L is weakly singular Note also that ( 43)and (4.1) show that 0 1 THEOREM 4 A second order projective deformation C: L L is singular if and only if the following equations hold: PROOF Suppose again that C: L L is a second order projective deformation.
THEOREM 5. Suppose that L and L are second class t.f.pseudocongruences of pn planes in projective spaces and n, 2m-I <_ n < 3m-l, and suppose that they are related by a second order projective deformation C: L L. The pseudocongruences L and L are identical if and only if the deformation C is 3-strongly singular PROOF.Suppose again that C: L L is a second order projective deformation between L and .The deformation C is 3-strongly singular if (4.1) holds for s 1,2,3o We already showed that for s 1 equation (4ol) holds automatically and for s 2 it holds if and only if conditions (4.5) and (4.6) are satisfied For a 3-strongly singular deformation C we additionally have K d3A i O i d}i + 3i 1 1 2 3 where #i and i are i-and 2-forms determined by ( 44) and (4.9) and #i is a 3-form.
2o A SPECIALIZATION OF MOVING FRAMES ASSOCIATED WITH A TOTALLY FOCAL PSEUDOCONGRUENCE AND FUNDAMENTAL EQUATIONS.