Peterson's Deformations of higher dimensional quadrics

We provide the first explicit examples of deformations of higher dimensional quadrics: a straightforward generalization of Peterson's explicit 1-dimensional family of deformations in C³ of 2-dimensional general quadrics with common conjugate system given by the spherical coordinates on the comp...

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Дата:	2010
Автор:	Dincă, I.I.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут математики НАН України 2010
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
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Цитувати:	Peterson's Deformations of higher dimensional quadrics / Dincă I.I. // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 7 назв. — англ.

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spelling	irk-123456789-1461152019-02-08T01:23:07Z Peterson's Deformations of higher dimensional quadrics Dincă, I.I. We provide the first explicit examples of deformations of higher dimensional quadrics: a straightforward generalization of Peterson's explicit 1-dimensional family of deformations in C³ of 2-dimensional general quadrics with common conjugate system given by the spherical coordinates on the complex sphere S² ⊂ C³ to an explicit (n–1)-dimensional family of deformations in C²ⁿ⁻¹ of n-dimensional general quadrics with common conjugate system given by the spherical coordinates on the complex sphere Sⁿ ⊂ Cⁿ⁺¹ and non-degenerate joined second fundamental forms. It is then proven that this family is maximal. 2010 Article Peterson's Deformations of higher dimensional quadrics / Dincă I.I. // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 7 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 53A07; 53B25; 35Q58 http://dspace.nbuv.gov.ua/handle/123456789/146115 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We provide the first explicit examples of deformations of higher dimensional quadrics: a straightforward generalization of Peterson's explicit 1-dimensional family of deformations in C³ of 2-dimensional general quadrics with common conjugate system given by the spherical coordinates on the complex sphere S² ⊂ C³ to an explicit (n–1)-dimensional family of deformations in C²ⁿ⁻¹ of n-dimensional general quadrics with common conjugate system given by the spherical coordinates on the complex sphere Sⁿ ⊂ Cⁿ⁺¹ and non-degenerate joined second fundamental forms. It is then proven that this family is maximal.
format	Article
author	Dincă, I.I.
spellingShingle	Dincă, I.I. Peterson's Deformations of higher dimensional quadrics Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Dincă, I.I.
author_sort	Dincă, I.I.
title	Peterson's Deformations of higher dimensional quadrics
title_short	Peterson's Deformations of higher dimensional quadrics
title_full	Peterson's Deformations of higher dimensional quadrics
title_fullStr	Peterson's Deformations of higher dimensional quadrics
title_full_unstemmed	Peterson's Deformations of higher dimensional quadrics
title_sort	peterson's deformations of higher dimensional quadrics
publisher	Інститут математики НАН України
publishDate	2010
url	http://dspace.nbuv.gov.ua/handle/123456789/146115
citation_txt	Peterson's Deformations of higher dimensional quadrics / Dincă I.I. // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 7 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT dincaii petersonsdeformationsofhigherdimensionalquadrics
first_indexed	2025-07-10T23:11:58Z
last_indexed	2025-07-10T23:11:58Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 6 (2010), 006, 13 pages Peterson’s Deformations of Higher Dimensional Quadrics Ion I. DINCĂ Faculty of Mathematics and Informatics, University of Bucharest, 14 Academiei Str., 010014, Bucharest, Romania E-mail: dinca@gta.math.unibuc.ro Received July 13, 2009, in final form January 16, 2010; Published online January 20, 2010 doi:10.3842/SIGMA.2010.006 Abstract. We provide the first explicit examples of deformations of higher dimensional quadrics: a straightforward generalization of Peterson’s explicit 1-dimensional family of deformations in C3 of 2-dimensional general quadrics with common conjugate system given by the spherical coordinates on the complex sphere S2 ⊂ C3 to an explicit (n−1)-dimensional family of deformations in C2n−1 of n-dimensional general quadrics with common conjugate system given by the spherical coordinates on the complex sphere Sn ⊂ Cn+1 and non- degenerate joined second fundamental forms. It is then proven that this family is maximal. Key words: Peterson’s deformation; higher dimensional quadric; common conjugate system 2010 Mathematics Subject Classification: 53A07; 53B25; 35Q58 1 Introduction The Russian mathematician Peterson was a student of Minding’s, who in turn was interested in deformations (through bending) of surfaces1, but unfortunately most of his works (including his independent discovery of the Codazzi–Mainardi equations and of the Gauß–Bonnet theorem) were made known to Western Europe mainly after they were translated in 1905 from Russian to French (as is the case with his deformations of quadrics [7], originally published in 1883 in Russian). Peterson’s work on deformations of general quadrics preceded that of Bianchi, Calapso, Darboux, Guichard and Ţiţeica’s from the years 1899–1906 by two decades; in particu- lar Peterson’s 1-dimensional family of deformations of surfaces admitting a common conjugate system (u, v) (that is the second fundamental form is missing mixed terms du � dv) are as- sociates (a notion naturally appearing in the infinitesimal deformation problem) to Bianchi’s 1-dimensional family of surfaces satisfying (log K)uv = 0 in the common asymptotic coordi- nates (u, v), K being the Gauß curvature (see Bianchi [2, Vol. 2, §§ 294, 295]). The work of these illustrious geometers on deformations in C3 of quadrics in C3 (there is no other class of surfaces for which an interesting theory of deformation has been built) is one of the crowning achievements of the golden age of classical geometry of surfaces and at the same time it opened new areas of research (such as affine and projective differential geometry) continued later by other illustrious geometers (Blaschke, Cartan, etc.). Peterson’s 1-dimensional family of deformations of 2-dimensional quadrics is obtained by imposing an ansatz naturally appearing from a geometric point of view, namely the constraint that the common conjugate system of curves is given by intersection with planes through an axis and tangent cones centered on that axis; thus this result of Koenigs (see Darboux [5, §§ 91–101]) was again (at least when the cones are tangent along plane curves) previously known to Peterson. Note also that Calapso in [3] has put Bianchi’s Bäcklund transformation of 1See Peterson’s biography at http://www-history.mcs.st-and.ac.uk/Biographies/Peterson.html. mailto:dinca@gta.math.unibuc.ro http://dx.doi.org/10.3842/SIGMA.2010.006 http://www-history.mcs.st-and.ac.uk/Biographies/Peterson.html 2 I. Dincă deformations in C3 of general 2-dimensional quadrics with center in terms of common conjugate systems (the condition that the conjugate system on a 2-dimensional quadric is a conjugate system on one of its deformations in C3 was known to Calapso for a decade, but the Bäcklund transformation for general quadrics eluded Calapso since the common conjugate system was best suited for this transformation only at the analytic level). Although this is the original approach Peterson used to find his deformations of quadrics, other features of his approach will make it amenable to higher dimensional generalizations, namely the warping of linear element (the warping of the linear element of a plane curve to get the linear element of a surface of revolution (d(f cos(u1)))2 + (d(f sin(u1)))2 = (df)2 + f2(du1)2 for f = f(u2) is such an example) and separation of variables; post-priori the common conjugate system property may be given a geometric explanation analogous to that in dimension 3. In 1919–1920 Cartan has shown in [4] (using mostly projective arguments and his exterior differential systems in involution and exteriorly orthogonal forms tools) that space forms of dimension n admit rich families of deformations (depending on n(n−1) functions of one variable) in surrounding (2n−1)-dimensional space forms, that such deformations admit lines of curvature (given by a canonical form of exteriorly orthogonal forms; thus they have flat normal bundle; since the lines of curvature on n-dimensional space forms (when they are considered by definition as quadrics in surrounding (n + 1)-dimensional space forms) are undetermined, the lines of curvature on the deformation and their corresponding curves on the quadric provide the common conjugate system) and that the co-dimension (n−1) cannot be lowered without obtaining rigidity as the deformation being the defining quadric. In 1983 Berger, Bryant and Griffiths [1] proved (including by use of tools from algebraic geometry) in particular that Cartan’s essentially projective arguments (including the exterior part of his exteriorly orthogonal forms tool) can be used to generalize his results to n-dimensional general quadrics with positive definite linear element (thus they can appear as quadrics in Rn+1 or as space-like quadrics in Rn × (iR)) admitting rich families of deformations (depending on n(n−1) functions of one variable) in surrounding Euclidean space R2n−1, that the co-dimension (n − 1) cannot be lowered without obtaining rigidity as the deformation being the defining quadric and that quadrics are the only Riemannian n-dimensional manifolds that admit a family of deformations in R2n−1 as rich as possible for which the exteriorly orthogonal forms tool (naturally appearing from the Gauß equations) can be applied. Although Berger, Bryant and Griffiths [1] do not explicitly state the common conjugate system property (which together with the non-degenerate joined second fundamental forms assumption provides a tool similar to the canonical form of exteriorly orthogonal forms), this will turn out to be the correct tool of differential geometry needed to attack the deformation problem for higher dimensional quadrics; also at least for diagonal quadrics without center Peterson’s deformations of higher dimensional quadrics will turn out to be amenable to explicit computations of their Bäcklund transformation2. All computations are local and assumed to be valid on their open domain of validity without further details; all functions have the assumed order of differentiability and are assumed to be invertible, non-zero, etc when required (for all practical purposes we can assume all functions to be analytic). Here we have the two main theorems concerning the (n−1)-dimensional family of deformations of higher dimensional general quadrics and respectively its maximality: Theorem 1. The quadric n∑ j=0 (xj)2 aj = 1, aj ∈ C∗ 2See Dincă I.I., Bianchi’s Bäcklund transformation for higher dimensional quadrics, arXiv:0808.2007. http://arxiv.org/abs/0808.2007 Peterson’s Deformations of Higher Dimensional Quadrics 3 distinct parameterized with the conjugate system (u1, . . . , un) ⊂ Cn given by the spherical coor- dinates on the unit sphere Sn ⊂ Cn+1: X = √ a0C0e0 + n∑ k=1 √ akCk sin ( uk ) ek, Ck := n∏ j=k+1 cos(uj) and the sub-manifold Xz = n−1∑ k=1 Ckfk ( z, uk )( cos ( gk ( z, uk )) e2k−2 + sin ( gk ( z, uk )) e2k−1 ) + h ( z, un ) e2n−2 of C2n−1 depending on the parameters z = (z1, z2, . . . , zn−1) ∈ Cn−1, z0 := 1 and with fk ( zk−1, zk, u k ) := √ (zk−1 − zk)a0 + (ak − zk−1a0) sin2(uk), k = 1, . . . , n− 1, gk ( zk−1, zk, u k ) := ∫ uk 0 √ (zk−1 − zk)a0ak + (ak − zk−1a0)zka0 sin2(t) (zk−1 − zk)a0 + (ak − zk−1a0) sin2(t) dt, h ( zn−1, u n ) := ∫ un 0 √ an − (an − zn−1a0) sin2(t)dt (1) have the same linear element \|dX\|2=\|dXz\|2. For z1 = · · · = zn−1 = 0 we get g2 = · · · = gn−1 = 0, X = X0 with Cn+1 ↪→ C2n−1 as (x0, x1, . . . , xn) 7→ (x0, x1, x2, 0, x3, 0, . . . , xn−1, 0, xn). For z1 = · · · = zn−1 = 1 we get X1 = (x0, . . . ,x2n−2) given by Peterson’s formulae√ (x2k−2)2 + (x2k−1)2 = √ ak − a0Ck sin ( uk ) , tan−1 ( x2k−1 x2k−2 ) = √ a0√ ak − a0 tanh−1 ( cos ( uk )) , k = 1, . . . , n− 1, x2n−2 = ∫ un 0 √ an − (an − a0) sin2(t)dt. (2) Moreover (u1, . . . , un) form a conjugate system on Xz with non-degenerate joined second fun- damental forms (that is [d2X T N d2X T z Nz] is a symmetric quadratic Cn-valued form which contains only (duj)2 terms for N normal field of X and Nz = [N1 . . . Nn−1] normal frame of Xz and the dimension n cannot be lowered for z in an open dense set). Theorem 2. For x ⊂ C2n−1 deformation of the quadric x0 ⊂ Cn+1 (that is \|dx\|2 = \|dx0\|2) with n ≥ 3, (u1, . . . , un) common conjugate system and non-degenerate joined second fundamental forms, NT 0 d2x0 =: n∑ j=1 h0 j (duj)2 second fundamental form of x0 we have Γl jk = 0 for j, k, l distinct and such deformations are in bijective correspondence with solutions {aj}j=1,...,n ⊂ C∗ of the differential system ∂uk log aj = Γj jk, j 6= k, n∑ j=1 (h0 j )2 a2 j + 1 = 0. In particular this implies that for (u1, . . . , un) being the conjugate system given by spherical coordinates on Sn ⊂ Cn+1 the above explicit (n− 1)-dimensional family of deformations Xz is maximal. The remaining part of this paper is organized as follows: in Section 2 we shall recall Peterson’s deformations of quadrics; the proof of Theorem 1 appears in Sections 3, 4 and the proof of Theorem 2 appears in Sections 5, 6. 4 I. Dincă 2 Peterson’s deformations of quadrics Although Peterson [7] discusses all types of quadrics in the complexified Euclidean space( C3, 〈·, ·〉 ) , 〈x, y〉 := xT y, \|x\|2 := xT x for x, y ∈ C3 and their totally real cases, we shall only discuss quadrics of the type 2∑ j=0 (xj) 2 aj = 1, aj ∈ C∗ distinct, since the remaining cases of quadrics should follow by similar computations. Their totally real cases (that is (xj)2, aj ∈ R) are discussed in detail in Peterson [7], so we shall not insist on this aspect. Remark 1. It is less known since the classical times that there are many types of quadrics from a complex metric point of view, each coming with its own totally real cases (real valued (in)definite linear element); among these quadrics for example the quadric (x0 − ix1)x2 − (x0 + ix1) = 0 is rigidly applicable (isometric) to all quadrics of its confocal family and to all its homothetic quadrics. It is Peterson who first introduced the idea of ideal applicability (for example a real surface may be applicable to a totally real space-like surface ⊂ R2 × (iR) of a complexified real ellipsoid, so it is ideally applicable on the real ellipsoid). With {ej}j=0,1,2, eT j ek = δjk the standard basis of C3 and the functions f = f(z, u1), g = g(z, u1), h = h(z, u2) depending on the parameter(s) z = (z1, z2, . . . ) to be determined later we have the surfaces Xz := cos ( u2 ) f ( z, u1 )( cos ( g ( z, u1 )) e0 + sin ( g ( z, u1 )) e1 ) + h ( z, u2 ) e2. (3) Note that the fields ∂u1Xz\|u1=const, ∂u2Xz\|u2=const generate developables (cylinders with gene- rators perpendicular on the third axis and cones centered on the third axis), so (u1, u2) is a conjugate system on Xz for every z; in fact all surfaces have conjugate systems arising this way and can be parameterized as x = f ( u1, u2 )( cos ( u1 ) e0 + sin ( u1 ) e1 ) + g ( u1, u2 ) e2, ∂u1 ( ∂u2 ( g f )/ ∂u2 ( 1 f )) = 0. The quadric 2∑ j=0 (xj) 2 aj = 1 is parameterized by the spherical coordinates X = √ a0 cos ( u2 ) cos ( u1 ) e0 + √ a1 cos ( u2 ) sin ( u1 ) e1 + √ a2 sin ( u2 ) e2. We have \|dXz\|2 = cos2 ( u2 )( f ′2 ( z, u1 ) + f2 ( z, u1 ) g′2 ( z, u1 ))( du1 )2 + 1 2d ( cos2 ( u2 )) d ( f2 ( z, u1 )) + ( f2 ( z, u1 ) sin2 ( u2 ) + h′2 ( z, u2 ))( du2 )2 , \|dX\|2 = cos2 ( u2 )( a1 − (a1 − a0) sin2 ( u1 ))( du1 )2 + 1 2d ( cos2 ( u2 )) d ( a0 + (a1 − a0) sin2 ( u1 )) + ( a2 − ( a2 − a0 − (a1 − a0) sin2 ( u1 )) sin2 ( u2 ))( du2 )2 . Thus the condition \|dXz\|2 = \|dX\|2 becomes f2 ( z, u1 ) + ( a2 − a0 − (a1 − a0) sin2 ( u1 )) = const = a2 − h′2(z, u2) sin2(u2) , f ′2 ( z, u1 ) + f2 ( z, u1 ) g′2 ( z, u1 ) = a1 − (a1 − a0) sin2 ( u1 ) , Peterson’s Deformations of Higher Dimensional Quadrics 5 from where we get h ( z1, u 2 ) := ∫ u2 0 √ a2 − (a2 − z1a0) sin2(t)dt, f ( z1, u 1 ) := √ (1− z1)a0 + (a1 − a0) sin2(u1), g ( z1, u 1 ) := ∫ u1 0 √ (1− z1)a0a1 + (a1 − a0)z1a0 sin2(t) (1− z1)a0 + (a1 − a0) sin2(t) dt. (4) Note that f ( 0, u1 ) cos ( g ( 0, u1 )) = √ a0 cos ( u1 ) , f ( 0, u1 ) sin ( g ( 0, u1 )) = √ a1 sin ( u1 ) , (5) (we assume simplifications of the form √ a √ b ' √ ab with √ · having the usual definition √ reiθ := √ re iθ 2 , r > 0, −π < θ ≤ π, since the possible signs are accounted by symmetries in the principal planes for quadrics and disappear at the level of the linear element for their deformations), so X = X0. The coordinates x0, x1, x2 of X1 satisfy (modulo a sign at the second formula) Peterson’s formulae:√ (x0)2 + (x1)2 = √ a1 − a0 cos ( u2 ) sin ( u1 ) , tan−1 ( x1 x0 ) = √ a0√ a1 − a0 tanh−1 ( cos ( u1 )) , x2 = ∫ u2 0 √ a2 − (a2 − a0) sin2(t)dt. (6) More generally h ( z1, u 2 ) := ∫ u2 0 √ h′2(t)− z1 sin2(t)dt, f ( z1, u 1 ) := √ z1 + f2(u1), g ( z1, u 1 ) := ∫ u1 0 √ z1(f ′2(t) + f2(t)g′2(t)) + f4(t)g′2(t) z1 + f2(t) dt (7) give Peterson’s 1-dimensional family of deformations (3) with common conjugate system (u1, u2). 3 Peterson’s deformations of higher dimensional quadrics Again we shall discuss only the case of quadrics with center and having distinct eigenvalues of the quadratic part defining the quadric, without insisting on totally real cases and deformations (when the linear elements are real valued). Remark 2. A metric classification of all (totally real) quadrics in Cn+1 requires the notion of symmetric Jordan canonical form of a symmetric complex matrix (see, e.g. [6]). The symmetric Jordan blocks are: J1 := 0 = 01,1 ∈ M1(C), J2 := f1f T 1 ∈ M2(C), J3 := f1e T 3 + e3f T 1 ∈ M3(C), J4 := f1f̄ T 2 + f2f T 2 + f̄2f T 1 ∈ M4(C), J5 := f1f̄ T 2 + f2e T 5 + e5f T 2 + f̄2f T 1 ∈ M5(C), J6 := f1f̄ T 2 + f2f̄ T 3 + f3f T 3 + f̄3f T 2 + f̄2f T 1 ∈ M6(C), 6 I. Dincă etc., where fj := e2j−1−ie2j√ 2 are the standard isotropic vectors (at least the blocks J2, J3 were known to the classical geometers). Any symmetric complex matrix can be brought via conju- gation with a complex rotation to the symmetric Jordan canonical form, that is a matrix block decomposition with blocks of the form ajIp + Jp; totally real quadrics are obtained for eigen- values aj of the quadratic part defining the quadric being real or coming in complex conjugate pairs aj , āj with subjacent symmetric Jordan blocks of same dimension p. Consider the quadric n∑ j=0 (xj) 2 aj = 1, aj ∈ C∗ distinct with parametrization given by the spherical coordinates on the unit sphere Sn ⊂ Cn+1 X = √ a0C0e0 + n∑ k=1 √ akCk sin ( uk ) ek, Ck := n∏ j=k+1 cos ( uj ) . The correct generalization of (3) allows us to build Peterson’s deformations of higher dimensional quadrics. With an eye to the case n = 2 we make the natural ansatz Xz = n−1∑ k=1 Ckfk ( z, uk )( cos ( gk ( z, uk )) e2k−2 + sin ( gk ( z, uk )) e2k−1 ) + h ( z, un ) e2n−2 (8) with the parameter(s) z = (z1, z2, . . . ) to be determined later. We have \|dXz\|2 = n−1∑ k=1 [ C2 k ( f ′2k ( z, uk ) + f2 k ( z, uk ) g′2k ( z, uk ))( duk )2 + 1 2d ( C2 k ) d ( f2 k ( z, uk )) + f2 k (z, uk)(dCk)2 ] + h′2 ( z, un )( dun )2 , \|dX\|2 = a0(dC0)2 + n∑ k=1 ak ( d ( Ck sin ( uk )))2 . Comparing the coefficients of (dun)2 from \|dXz\|2 = \|dX\|2 we get 1 cos2(un) [ C2 1 ( f2 1 ( z, u1 ) − a0 − (a1 − a0) sin2 ( u1 )) + n−1∑ k=2 C2 k ( f2 k ( z, uk ) − ak sin2 ( uk ))] = const = an cos2(un)− h′2(z, un) sin2(un) from where we get with z0 := 1: f2 k ( zk−1, zk, u k ) := (zk−1 − zk)a0 + (ak − zk−1a0) sin2 ( uk ) , k = 1, . . . , n− 1, h′2 ( zn−1, u n ) := an − (an − zn−1a0) sin2 ( un ) . Now we have (dC0)2 = n−1∑ k=1 [ zk−1(dCk−1)2 − zk(dCk)2 ] + zn−1(dCn−1)2 = n−1∑ k=1 [ zk−1 ( C2 k sin2 ( uk )( duk )2 − 1 2d ( C2 k ) d ( sin2(uk) ) + cos2 ( uk )( dCk )2)− zk ( dCk )2]+ zn−1 ( dCn−1 )2 ,( d ( Ck sin ( uk )))2 = C2 k cos2 ( uk )( duk )2 + 1 2d ( C2 k ) d ( sin2 ( uk )) + sin2 ( uk )( dCk )2 , Peterson’s Deformations of Higher Dimensional Quadrics 7 so \|dX\|2 = n−1∑ k=1 [ C2 k ( ak − (ak − zk−1a0) sin2 ( uk ))( duk )2 + 1 2(ak − zk−1a0)d ( C2 k ) d ( sin2 ( uk )) + ( (zk−1 − zk)a0 + (ak − zk−1a0) sin2 ( uk ))( dCk )2] + ( an − (an − zn−1a0) sin2 ( un ))( dun )2 , 0 = \|dXz\|2 − \|dX\|2 = n−1∑ k=1 C2 k ( f ′2k ( z, uk ) + f2 k ( z, uk ) g′2k ( z, uk ) − ak + (ak − zk−1a0) sin2 ( uk ))( duk )2 , so we finally get (1). For z1 = z2 = · · · = zn−1 = 0 we get g2 = · · · = gn−1 = 0 and using (5) we get X = X0 with Cn+1 ↪→ C2n−1 as (x0, x1, . . . , xn) 7→ (x0, x1, x2, 0, x3, 0, . . . , xn−1, 0, xn). For z1 = z2 = · · · = zn−1 = 1 we get X1 = (x0, . . . ,x2n−2) given by Peterson’s formulae (2). More generally and with z0 := 0 fk ( zk−1, zk, u k ) := √ zk + f2 k (uk)− zk−1 cos2(uk), k = 1, . . . , n− 1, gk ( zk−1, zk, u k ) := ∫ uk 0 √ f ′2k (t) + f2 k (t)g′2k (t)− (f ′2k (zk−1, zk, t) + zk−1 sin2(t)) fk(zk−1, zk, t) dt, h ( zn−1, u n ) := ∫ un 0 √ h′2(t)− zn−1 sin2(t)dt (9) give an (n− 1)-dimensional family of deformations (8); for gk(uk) = 0, k = 2, . . . , n− 1 we have X0 ⊂ Cn+1. 4 The common conjugate system and non-degenerate joined second fundamental forms The fact that (u1, . . . , un) is a conjugate system on X0 is clear since we have ∂uk∂ujX0 = − tan ( uj ) ∂ukX0, 1 ≤ k < j ≤ n. With the normal field N̂0 := ( √ a0)−1C0e0 + n∑ k=1 ( √ ak)−1Ck sin ( uk ) ek we have N̂T 0 d2X0 = − n∑ k=1 C2 k(duk)2. To see that (u1, . . . , un) is a conjugate system on X = (x0, . . . , x2n−2) := n−1∑ k=1 Ckfk ( uk )( cos ( gk ( uk )) e2k−2 + sin ( gk ( uk )) e2k−1 ) + h ( un ) e2n−2 we have again ∂uk∂ujX = − tan(uj)∂ukX , 1 ≤ k < j ≤ n; again the n− 1 fields ∂u1X\| u1,u2,...,ûk,...,un=const , ∂u2X\| u1,u2,...,ûk,...,un=const , . . . , ∂̂ukX\| u1,u2,...,ûk,...,un=const , . . . , ∂unX\| u1,u2,...,ûk,...,un=const , k = 1, . . . , n 8 I. Dincă generate ruled n-dimensional developables in C2n−1 because the only term producing shape is ∂uk∂ukX . For the non-degenerate joined second fundamental forms property we have un = h−1(x2n−2), h′ ( un ) dun = dx2n−2, uk = g−1 k ( tan−1 ( x2k−1 x2k−2 )) , C2 kf 2 k (uk)g′k(u k)duk = x2k−2dx2k−1 − x2k−1dx2k−2, k = 1, . . . , n− 1 and X is given implicitly by the zeroes of the functionally independent Fk := (x2k−2)2 + (x2k−1)2 −C2 kf 2 k ( uk ) , k = 1, . . . , n− 1. We have the natural linearly independent normal fields Nk := ∇Fk = 2(x2k−2e2k−2 + x2k−1e2k−1)− 2f ′k(u k)(−x2k−1e2k−2 + x2k−2e2k−1) fk(uk)g′k(u k) + 2C2 kf 2 k (uk)  n−1∑ j=k+1 tan(uj)(−x2j−1e2j−2 + x2j−2e2j−1) C2 jf 2 j (uj)g′j(uj) + tan(un)e2n−2 h′(un)  , k = 1, . . . , n− 1, and ∂ul∂ulX = − l−1∑ j=1 (x2j−2e2j−2 + x2j−1e2j−1) + ( f ′′l (ul) fl(ul) − g′2l ( ul )) (x2l−2e2l−2 + x2l−1e2l−1) + g′l ( ul )(2f ′l (u l) fl(ul) + g′′l (ul) g′l(u l) ) (−x2l−1e2l−2 + x2l−2e2l−1), l = 1, . . . , n− 1, ∂un∂unX = − n−1∑ l=1 (x2l−2e2l−2 + x2l−1e2l−1) + h′′ ( un ) e2n−2, and the second fundamental form NT k d2X = 2C2 kf 2 k [( f ′′k (uk) fk(uk) − g′2k ( uk ) − f ′k(u k) fk(uk) ( 2f ′k(u k) fk(uk) + g′′k(uk) g′k(u k) ))( duk )2 + n−1∑ l=k+1 ( tan ( ul )(2f ′l (u l) fl(ul) + g′′l (ul) g′l(u l) ) − 1 )( dul )2 + ( tan(un)h′′(un) h′(un) − 1 )( dun )2] , k = 1, . . . , n− 1. For Peterson’s deformations of higher dimensional quadrics we have NT k d2X = −2a0C2 kf 2 k ( akzk−1(duk)2 g′2k (uk)f4 k (uk) + n−1∑ l=k+1 al(zl−1 − zl)(dul)2 g′2l (ul)f4 l (ul) + an(dun)2 a0h′2(un) ) . It is now enough to check the open non-degenerate joined second fundamental forms property Peterson’s Deformations of Higher Dimensional Quadrics 9 only for z = (1, 1, . . . , 1). Thus with δ := an a0 sin2(un)+an cos2(un) we need 0 6= ∣∣∣∣∣∣∣∣∣∣∣∣∣∣ C1 C2 C3 . . . Cn−1 δ−1Cn a1 a1−a0 0 0 . . . 0 sin2 ( u1 ) 0 a2 a2−a0 0 . . . 0 sin2 ( u2 ) 0 0 a3 a3−a0 . . . 0 sin2 ( u3 ) ... ... ... · · · ... ... 0 0 0 . . . an−1 an−1−a0 sin2 ( un−1 ) ∣∣∣∣∣∣∣∣∣∣∣∣∣∣ almost everywhere, which is straightforward. Remark 3. Note that a-priori X1 comes close to lie in a degenerate deformation of Cn+1 in C2n−1: N̂T 0 d2X0 − ( n−1∑ k=1 1 2ak Nk )T d2X1 depends only on (dun)2 and this is as closest to 0 as we can get. 5 Conjugate systems Consider the complexified Euclidean space( Cn, 〈·, ·〉), 〈x, y〉 := xT y, \|x\|2 := xT x, x, y ∈ Cn with standard basis {ej}j=1,...,n, eT j ek = δjk. Isotropic (null) vectors are those vectors v of length 0 (\|v\|2 = 0); since most vectors are not isotropic we shall call a vector simply vector and we shall only emphasize isotropic when the vector is assumed to be isotropic. The same denomination will apply in other settings: for example we call quadric a non-degenerate quadric (a quadric projectively equivalent to the complex unit sphere). For n ≥ 3 consider the n-dimensional sub-manifold x = x ( u1, u2, . . . , un ) ⊂ Cn+p, du1 ∧ du2 ∧ · · · ∧ dun 6= 0 such that the tangent space at any point of x is not isotropic (the scalar product induced on it by the Euclidean one on Cn+p is not degenerate; this assures the existence of orthonormal normal frames). We shall always have Latin indices j, k, l,m, p, q ∈ {1, . . . , n}, Greek ones α, β, γ ∈ {n + 1, . . . , n + p} and mute summation for upper and lower indices when clear from the context; also we shall preserve the classical notation d2 for the tensorial (symmetric) second derivative. We have the normal frame N := [Nn+1 . . . Nn+p], NT N = Ip, the first \|dx\|2 = gjkduj � duk and second d2xT N = [hn+1 jk duj � duk . . . hn+p jk duj � duk] fundamental forms, the Christoffel symbols Γl jk = glm 2 [∂ukgjm + ∂ujgkm − ∂umgjk], the Riemann curvature Rjmkl = gmpR p jkl = gmp[∂ulΓp jk − ∂ukΓp jl + Γq jkΓ p ql − Γq jlΓ p qk] tensor, the normal connection NT dN = {nα βjduj}α,β=n+1,...,n+p, nα βj = −nβ αj and the curvature rβ αjk = ∂uknβ αj−∂ujn β αk +nγ αjn β γk−nγ αkn β γj tensor of the normal bundle. We have the Gauß–Weingarten (GW) equations ∂uk∂ujx = Γl jk∂ulx + hα jkNα, ∂ujNα = −hα jkg kl∂ulx + nβ jαNβ and their integrability conditions ∂ul(∂uk∂ujx) = ∂uk(∂ul∂ujx), ∂uk(∂ujNα) = ∂uj (∂ukNα), from where one obtains by taking the tangential and normal components (using −∂ulgjk = 10 I. Dincă gjmΓk ml + gkmΓj ml and the GW equations themselves) the Gauß–Codazzi–Mainardi(–Peterson)– Ricci (G-CMP-R) equations Rjmkl = ∑ α ( hα jkh α lm − hα jlh α km ) , ∂ulhα jk − ∂ukhα jl + Γm jkh α ml − Γm jl h α mk + hβ jkn α βl − hβ jln α βk = 0, rβ αjk = hα jlg lmhβ mk − hα klg lmhβ mj . If we have conjugate system hα jk =: δjkh α j , then the above equations become: Rjkjk = −Rjkkj = ∑ α hα j hα k , ∂ukhα j = Γj jkh α j − Γk jjh α k − hβ j ηα βk, j 6= k, Rjklm = 0 otherwise, Γl jkh α l = Γk jlh α k , j, k, l distinct, rβ αjk = (hα j hβ k − hβ j hα k )gjk. (10) In particular for lines of curvature parametrization (gjk = δjkgjk) we have flat normal bundle, so one can choose up to multiplication on the right by a constant matrix ∈ Op(C) normal frame N with zero normal connection NT dN = 0. This constitutes a differential system in the np unknowns hα j and the yet to be determined coefficients ηα βk; according to Cartan’s exterior differential systems in involution tools in order to study deformations of n-dimensional sub-manifolds of Cn+p in conjugate system parame- terization one must iteratively apply compatibility conditions (commuting of mixed derivatives) to the equations of this system and their algebraic-differential consequences, introducing new variables as necessary and assuming only identities obtained at previous iterations and general identities for the Riemann curvature tensor (symmetries and Bianchi identities): Rjklm = −Rkjlm = −Rjkml = Rlmjk, Rjklm + Rjlmk + Rjmkl = 0, Rjklm;q + Rjkmq;l + Rjkql;m = 0, Rjklm;q := ∂uqRjklm − Γr qjRrklm − Γr qkRjrlm − Γr qlRjkrm − Γr qmRjklr until no further conditions appear from compatibility conditions. However one cannot use in full the Cartan’s exterior differential forms and moving frames tools (see, e.g. [1]), since they are best suited for arbitrary (orthonormal) tangential frames and orthonormal normal ones and their corresponding change of frames; thus one loses the advantage of special coordinates suited to our particular problem. In our case we only obtain ∂ulRjkjk = ( Γj jl + Γk kl ) Rjkjk − Γl kkRjljl − Γl jjRklkl, j, k, l distinct, Γm lkRjmjm − Γl mkRjljl = 0, j, k, l,m distinct. (11) Remark 4. Differentiating the first equations of (10) with respect to ul, l 6= j, k and using (10) itself we obtain ∂ulRjkjk = ∑ α ( ∂ulhα j hα k + hα j ∂ulhα k ) = ∑ α [( Γj jlh α j − Γl jjh α l − hβ j ηα βl ) hα k + hα j ( Γk klh α k − Γl kkh α l − hβ kηα βl )] = ( Γj jl + Γk kl ) Rjkjk − Γl kkRjljl − Γl jjRklkl, that is the first equations of (11), so the covariant derivative of the Gauß equations become, via the G-CMP equations, the Bianchi second identity (the second equations of (11) being consequence of the CMP equations is obvious; see also [1]). Peterson’s Deformations of Higher Dimensional Quadrics 11 6 The non-degenerate joined second fundamental forms assumption With an eye towards our interests (deformations in C2n−1 of quadrics in Cn+1 and with com- mon conjugate system) we make the genericity assumption of non-degenerate joined second fundamental forms of x0, x: with d2xT 0 N0 =: h0 j (duj)2 being the second fundamental form of the quadric x0 ⊂ Cn+1 whose deformation x ⊂ C2n−1 is (that is \|dx0\|2 = \|dx\|2) the vectors hj := [ih0 j hn+1 j . . . h2n−1 j ]T are linearly independent. From the Gauß equations we obtain h0 jh 0 k = Rjkjk = ∑ α hα j hα k , j 6= k ⇔ hT j hk = δjk\|hj \|2; thus the vectors hj ⊂ Cn are further orthogonal, which prevents them from being isotropic (should one of them be isotropic, by a ro- tation of Cn one can make it f1 and after subtracting suitable multiples of f1 from the remaining ones by another rotation of Cn the remaining ones linear combinations of e3, . . . , en, so we would have n− 1 linearly independent orthogonal vectors in Cn−2, a contradiction), so aj := \|hj \| 6= 0, hj =: ajvj , R := [v1 . . . vn] ⊂ On(C). Thus with ηα 0j = −η0 αj := 0, (ηα βj)α,β=0,n+1,...,2n−1 =: Υj = −ΥT j we have reduced the problem to finding R = [v1 . . . vn] ⊂ On(C), aj ⊂ C∗, Υj ⊂ Mn(C), Υj = −ΥT j , Υje1 = 0 satisfying the differential system ∂uk log aj = Γj jk, ∂ukvj = −Γk jj ak aj vk −Υkvj , ∂ukΥj − ∂ujΥk − [Υj ,Υk] = −gjkajak ( In − e1e T 1 )( vjv T k − vkv T j )( In − e1e T 1 ) , j 6= k,∑ j (h0 j ) 2 a2 j + 1 = 0 (12) derived from the CMP-R equations and ajv 1 j = ih0 j , ∑ j(v 1 j ) 2 = 1 and with the linear element further satisfying the condition Γl jk = 0, j, k, l distinct (13) derived from the CMP equations. First we shall investigate the consequences of (13), via the properties of the Riemann cur- vature tensor, on the other Christoffel symbols. For j, k, l distinct we have 0 = gpmRjmkl = Rp jkl = ∂ulΓp jk − ∂ukΓp jl + Γq jkΓ p ql − Γq jlΓ p qk; thus for p = k we obtain ∂ulΓk kj = Γk klΓ l lj + Γk kjΓ j jl − Γk klΓ k kj , j, k, l distinct. (14) We also have Rp jjl = gpmRjmjl = gplRjljl, j 6= l, so gplRjljl = ∂ulΓp jj + Γp jj ( Γp pl − Γj jl ) + Γl jjΓ p ll, j, l, p distinct, gjlRjljl = ∂ulΓj jj − ∂ujΓj jl + Γl jjΓ j ll − Γl ljΓ j jl, gllRjljl = ∂ulΓl jj − ∂ujΓl lj + Γq jjΓ l lq − Γj jlΓ l jj − ( Γl lj )2 , j 6= l. (15) Conversely, (13) and (14) imply Rjklm = 0 for three of j, k, l, m distinct. Remark 5. Note that (13) are valid for orthogonal coordinates, so conjugate systems with the property (13) are a natural projective generalization of lines of curvature on n-dimensional sub-manifolds x ⊂ Cn+p (to see this first ∂uk∂ujx = Γj jk∂ujx+Γk kj∂ukx, j 6= k is affine invariant (thus Γj jk, Γk kj are also affine invariants) and further ∂uk∂uj x ρ = (Γj jk − ∂uk log ρ)∂uj x ρ + (Γk kj − ∂uj log ρ)∂uk x ρ , j 6= k for ρ ⊂ C∗ with ∂uk∂ujρ = Γj jk∂ujρ + Γk kj∂ukρ, j 6= k). 12 I. Dincă Imposing the compatibility conditions ∂ul(∂uk log aj) = ∂uk(∂ul log aj), ∂ul(∂ukvj) = ∂uk(∂ulvj), j, k, l distinct we obtain ∂ulΓj jk − ∂ukΓj jl = 0 = ∂ulΓk jj + Γk jj ( Γk kl − Γj jl ) + Γl jjΓ k ll − gklh0 jh 0 l , j, k, l distinct, (16) which are consequences of (14) and the first equations of (15). From the first equations of (12) we get by integration a precise determination of aj up to multiplication by a function of uj ; thus the aj part of the solution depends on at most n functions of one variable (with the last equation of (12) also to be taken into consideration); we shall see later that the remaining part of the differential system involves the normal bundle and its indeterminacy, so it will not produce a bigger space of solutions (no functional information is allowed in the normal bundle). From the CMP equations of x0 and the second equations of (16) we obtain with γjk := Γk jj h0 k h0 j , j 6= k: ∂ulγjk = γjkγjl − γjlγlk − γjkγkl + gklh0 kh 0 l , j, k, l distinct. (17) From the first equations of (12) and the CMP equations of x0 we obtain with bj := h0 j aj ∂uk log bj = −γjk, j 6= k, (18) so differentiating the last equation of (12) we obtain ∂uj log bj = b−2 j ∑ l 6=j b2 l γlj . (19) This assures that Υk := −∂ukRRT − ∑ j 6=k γjk bj bk ( vkv T j − vjv T k ) (20) satisfies eT 1 Υk = 0. Thus we have reduced the problem to finding bj satisfying (18), (19) (in this case ∑ j b2 j + 1 = 0 becomes a prime integral of (18), (19) and removes a constant from the space of solutions) and then completing v1 j = −ibj to R = [v1 . . . vn] ⊂ On(C) in an arbitrary manner (that is undetermined up to multiplication on the left with R′ ⊂ On(C), R′e1 = e1); with the second fundamental form of x found one finds x by the integration of a Ricatti equation and quadratures (the Gauß–Bonnet(–Peterson) theorem). Υj given by (20) will satisfy ∂ukΥj − ∂ujΥk − [Υj ,Υk] = − gjkh0 jh 0 k bjbk ( In − e1e T 1 )( vjv T k − vkv T j )( In − e1e T 1 ) , j 6= k. (21) Imposing the compatibility condition ∂uk(∂uj log bj) = ∂uj (∂uk log bj), j 6= k on (18), (19) we obtain ∂uk ( γkj bk bj ) + ∂uj ( γjk bj bk ) − ∑ l 6=j,k ( γljγlk − gjkh0 jh 0 k ) b2 l bjbk = 0, j 6= k. (22) Now (21) becomes∑ m6=j ∂uk ( γmj bm bj )( vjv T m − vmvT j ) − ∑ l 6=k ∂uj ( γlk bl bk )( vkv T l − vlv T k ) + ∑ m6=j, l 6=k γmjγlk bmbl bjbk [ δmk ( vjv T l − vlv T j ) + δml ( vkv T j − vjv T k ) − δjl ( vkv T m − vmvT k )] = gjkh0 jh 0 k bjbk ( In − e1e T 1 )( vjv T k − vkv T j )( In − e1e T 1 ) , j 6= k, Peterson’s Deformations of Higher Dimensional Quadrics 13 which boils down to (22) and ∂uk ( γlj bl bj ) + ( γlkγkj − gjkh0 jh 0 k )bl bj = 0, j, k, l distinct, which follows from (17) and (18). Note that (22) can be written as b2 j (∂ujγjk + 2γjkγkj) + b2 k(∂ukγkj + 2γkjγjk) + ∑ l 6=j,k b2 l ( ∂ukγlj + 2(γlkγkj + γljγjk − γljγlk) ) = 0, j 6= k, so the differential system (18), (19) is in involution (completely integrable) for ∂ujγjk = ∂ukγkj = −2γjkγkj , ∂ukγlj = 2(γljγlk − γlkγkj − γljγjk), j, k, l distinct.(23) Thus if (23) holds, then the solution of (18), (19) is obtained by integrating n first order ODE’s (namely finding the functions of uj upon whose multiplication with aj depends), so the space of solutions depends on (n− 1) constants (the prime integral ∑ j b2 j + 1 = 0 removes a constant from the space of solutions); if (23) does not hold, then the space of solutions depends on less than (n − 1) constants. Since for Peterson’s deformations of higher dimensional quadrics (or more generally for deformations of sub-manifolds of the type (8) with fk, gk, h given by (9) with gk(uk) = 0, k = 2, . . . , n − 1) we already have an (n − 1)-dimensional explicit family of deformations, we conclude that this family is maximal and that (23) holds in these cases. Remark 6. Note that (23) generalizes the case n = 2 condition ∂u1γ12 = ∂u2γ21 = −2γ12γ21 that the conjugate system (u1, u2) is common to a Peterson’s 1-dimensional family of deformations of surfaces (see Bianchi [2, Vol. 2, §§ 294, 295]), so conjugate systems of n-dimensional sub- manifolds in Cn+1 satisfying (13) and (23) are a natural generalization of Peterson’s approach in the deformation problem. Acknowledgements I would like to thank the referees for useful suggestions. The research has been supported by the University of Bucharest. References [1] Berger E., Bryant R.L., Griffiths P.A., The Gauss equations and rigidity of isometric embeddings, Duke Math. J. 50 (1983), 803–892. [2] Bianchi L., Lezioni di geometria differenziale, Vols. 1–4, Nicola Zanichelli Editore, Bologna, 1922, 1923, 1924, 1927. [3] Calapso P., Intorno alle superficie applicabili sulle quadriche ed alle loro transformazioni, Annali di Mat. 19 (1912), no. 1, 61–82. Calapso P., Intorno alle superficie applicabili sulle quadriche ed alle loro transformazioni, Annali di Mat. 19 (1912), no. 1, 107–157. [4] Cartan É., Sur les variétés de courboure constante d’un espace euclidien ou non-euclidien, Bull. Soc. Math. France 47 (1919), 125–160. Cartan É., Sur les variétés de courboure constante d’un espace euclidien ou non-euclidien, Bull. Soc. Math. France 48 (1920), 132–208. [5] Darboux G., Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal, Vols. 1–4, Gauthier-Villars, Paris, 1894–1917. [6] Horn R.A., Johnson C.R., Matrix analysis, Cambridge University Press, Cambridge, 1985. [7] Peterson K.-M., Sur la déformation des surfaces du second ordre, Ann. Fac. Sci. Toulouse Sér. 2 7 (1905), no. 1, 69–107. http://dx.doi.org/10.1215/S0012-7094-83-05039-1 http://dx.doi.org/10.1215/S0012-7094-83-05039-1 http://dx.doi.org/10.1007/BF02419392 http://dx.doi.org/10.1007/BF02419394 http://www.numdam.org/item?id=BSMF_1919__47__125_1 http://www.numdam.org/item?id=BSMF_1919__47__125_1 http://www.numdam.org/item?id=BSMF_1920__48__132_1 http://www.numdam.org/item?id=BSMF_1920__48__132_1 http://afst.cedram.org/afst-bin/item?id=AFST_1905_2_7_1_69_0 1 Introduction 2 Peterson's deformations of quadrics 3 Peterson's deformations of higher dimensional quadrics 4 The common conjugate system and non-degenerate joined second fundamental forms 5 Conjugate systems 6 The non-degenerate joined second fundamental forms assumption References

Peterson's Deformations of higher dimensional quadrics

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