Point Correspondences between N + 1 Hypersurfaces of Projective Spaces and ( N + 1 )-Webs

For a correspondence in question we establish a sequence of fundamental geometrical objects of the correspondence and find invariant normalizations of the first and second orders of all hupersurfaces under the correspondence. We single out main tensors of the correspondence and establish a connection between the geometry of point correspondences between n + 1 hypersurfaces of projective spaces and the theory of multidimensional (n + 1)-webs.


Introduction
Differentional geometry of point correspondences between projective, affine and euclid spaces of equal dimensions were studied and were studing by scientists till 1920.One can finds the analysis of obtained results to 1964 in the paper [1] by Ryzhkov.Among all papers devoted to the theory of point correspondences between two three-dimensional spaces we must note papers [2] written by Svec, [3] written by Murracchini,[4] written by Mihailescu and [5] by Vranceanu.They introduce characteristic directions of point correspondences, consider some special classes of correspondences, show connections of point correspondences between spaces with different parts of differentional geometry.
A straight line passing through the point 0 , is called a first order normal of a hypersurface of -dimensional projective space in the point 0 , M  if the straight line has no other points with the tangent hyperplane of the hupersurface [9].We call a -dimensional plane as the second order normal of the hypersur- It is known that the main problem of nonmetric differentional geometry of a surface is a construction of invariant normalization of this surface.To construct an invariant first normal in a point of a surface it is necessary to use third-order differential neighbourhood of the point [10].In our previous papers we showed that to construct an invariant first normal in points of two surfaces under point correspondences it is sufficient to use a second-order differential neighbourhood of corresponding points, but to construct an invariant second normal in points of two surfaces under point correspondences it is necessary to use third-order differential neighbourhood of the point. .In the current paper we will find invariant normalizations of the first and second orders of all hupersurfaces under the correspondence.
There exists a connection between the geometry of point correspondences between three spaces or surfaces and the theory of multidimensional 3-webs (Akivis [11]).We showed it in papers [12,13], devoted point correspondences between three projective spaces and between three hupersurfaces of projective spaces.
The theory of of multidimensional (n + 1)-webs is constructed in the paper [14] by Goldberg.In the current paper we will consider a connection between the geometry of point correspondences between 1 n  hypersurfaces of projective spaces and the theory of multidimensional (n + 1)-webs.
In the way of the investigation we use the exterior differentiation, tensor analysis and G.F.Laptev invariant methods [15].

Main Equations of Correspondence, the Sequence of Main Geometrical Objects
Let us consider n + 1 smooth hypersurfaces of projective spaces and a point correspondence between these hypersurfaces.
obtained by fixation of n  2 corresponding points and generates point mappings by fixation of n1 corresponding points.

Mappings
must be regular in neighbourhoods of points under correspondences of surfaces and have the inverse mappings.The equations of infinitesimal displacement of our projective frames have the form: where , , 0,1, , 1 .
We can write equations of hypersurfaces n V  as follows: The Pfaffian forms   3) relations between forms   take the simplest case.Let us suppose that necessary transformations of frames are done and we can write relations between To find equations of a mapping we fix Using Equations ( 2), (4), we have Consider projective mappings K  , where By Equations (1), (5) the following relations satisfy projective mappings: where 1   -a quantity of the first order according to v u   .The projective mapping K  : T V has a first order tangency with the mapping in correspond- re m s (2), (4) a ain equations of our problem.With the help of exterior differentiation of these equations and applying Cartan's lemma we obtain 1 0 0 0 , , where , Note that quadratic forms are asympto ave equations of ma tic quadratic forms of hypers Now in the family of frames we h pping 0 T  in the way and similar for .
where and To c ions (6 xte ontinue the system of Equat ) we use erior 0 i l differentiation of these equations and Cartan's lemma.We obtain new equations: To write these equations we used operators  and   .Operator  is defined by forms 0 i j  and we have and similarly operators   are defined by forms .8), we obtain the system of di

Le
If frames are etrical object according to G.F.Laptev invariant methods [15].This object is the fundamental geometrical object of second order of point correspondence fferentional equations of a sequence of fundamental geometrical objects of point correspondence under consideration , , ,

Correspondences
t us consider a mapping : .
 define the quadratic trans on of tangent direc s of hypersurfaces 0 00 .
In geometry of point correspondences [1] directions are said to be characteristic if they are invariant according to these quadratic transformations.They must satisfy a system of equations A geodesic curve of hypersurfa ith the family of first order normals, is called a curve, whose 2-dimensional osculant plane passes through corresponding first order normals of hypersurface in every point (see for exsample [9]).If Pfaffian forms has the similar property in the point  by the corresponding characteristic direction.
From geometric meaning of characteristic directions it is no clear, that they depend on the choice of first order rmals of a hypersurface and do not depend on the choice of second order normals.
We can rewrite Equations ( 9) in this way We obtained equations of cubic c nes. Characteristic directions are common generatrices of these cones.
After calculations we get the relations: , are arbitrary quantities.
We denote the valu it es quant ies  ust also satisfy relations (10).
Let us consider the object .
After substituting the values and considering similar terms we obtain These relations must be true for any va Contructing these relations with respect to indices the and , we arrive at the equation i l 0 0 jk   for .a sim lar way we get It is known that hypersurfaces degenerate into hyperlanes if the asymptotic tensors ij p 0.

  
In this case a point correspondence

Invariant Normalizations of Hypersurfaces under Point Correspondences
we can write Equation (8) for the follows: , (11) where n that main problem l geom ry of a surface is a construction o variant normalization of this surface.According to eory [10] for a hypersurface it is necessary to construct on the basis of the sequence of fundamental geometrical objects of the correspondence under consideration some quantities.These quantities must satisfy the following equations: For the invariant first order normal (straight line) For the point on the invariant first order normal For the second order normal (   By Equation (11) we have differential equations: By Equation (11) we obtai satisfy equations the invariant first order normal geometrical object of the hypersurface From Equation ( 11) we have Therefore, by Equation ( 12) the quantities satisfy Equation ( 12) and define the invariant first order normal geometrical objects of the hypersurface To construct the invariant second order normal geometrical object of the hypersurface Calculations show that quantities satisfy Equation (14).
Thus, it is proved.

The Main Tensors of the Point Correspondence between n + 1 Hypersurfaces
Let us use the quantities p , , , s of infinitesim f these frames as We denote Pfaffian form al displacement o . can be written as follows By Equations ( 12), ( 14) quantities eters, therefo depend on re we can differentials of principal param write forms 0 i   and i By new frames Equations (4), ( 6) of the corresponddence C can be written in the form: Calculations show, that quantities , of a secon a a are absolute tensors d-order differential neighbourhood of the correspondence.They satisfy some additional conditions: By relations ( 7), (7'), ( 19) in the family of new frames we have equations of mapping 0 T  in the way 0 , and similar for  : : Conversely, if we use invariant first and second order norm te (21) then relations (10) are true.

The Whole Projectiv Normalization of Hypersurfaces under the
To finish normalizations of hypersurfaces under consideration it is e t objects satisfying ry to construc rolong Equatio help of exterior differentiations and applyin lemma we obtain new equations: We construct quantities These quantities satisfy Equations (13) and define invariant points on the first order normals of hypersurfaces Let us find a geometrical meaning of chosen invariant po ces .We fix the hy ints.We consider hypersurfa - , To get values y  , defined focuses on the straight line We can define the harmonic pole [16] on each straight of the congruence according to the po ation int 0 N and n focuses by the rel Let points  : ood of corresponding points. .

Point Correspondences between (n + 1) Hyp P Multidime
C will be determined as an 2 n -dimesional smooth sub nifold.There exist 1 n  foliations of codimensio determ n on this submanifold.Each foliation is ined by the hypersurface n V  .These foliations define   We introduce additional forms where    .By relations (11) we h e av Therefore, quantities 0. .
The uations show that forms i eq j  are the forms of an affine connection assosiated to the web W and te It is known that parallelizable webs [11] are the simplest class of (n + 1)-webs.A corre on ten spondence between (n + 1) hypesurfaces of projective spaces is said to be parallelizable if the (n + 1)-web of this corr pondence is parallelizable.The necessary and sufficient conditions for correspondence to be parallelizable are relations n n pesurfa es 0. In paper [11] specific classes of webs are introduced called a class of (2n + 2)-adric webs.For these classes the following relations are true Comparing these relations with conditions (21), we no o iv te that they are true for geodesic correspondences, that's why the (n + 1)-web adjoined to the geodesic correspondence between (n + 1) hypersurfaces of pr ject e spaces is always (2n + 2)-adric web of type 2.
A point correspodence

Conclusions
We write main equations of a point correspondence between 1 n  hypersurfaces of projective spaces and construct the sequence of main geometrical objects f the correspondence.we define characteristic directions of a correspondence and prove that there exist invariant characteristic directions.[14] V. V. Goldberg, "On (n + 1)-Webs of Multidimensional Surfaces," Doklady Akademii Nauk SSSR, Vol.210, No. 4 define some partial cases of correspondences.We lish a connection between the geometry of point correspondences between 1 n  hypersurfaces of projective spaces and the theory of multidimensional (n + 1)-webs.In particular we prove that the (n + 1)-web adjoined to the geode persurfaces of projective spaces is always (2n + 2)-adric web of type 2.

.
The geometry of correspondences under consideration will be studied according to the transformation group, which is a direct product of projective transformation groups of spaces n P

1 -
forms containing parameters, on which the family of frames in question depends, and their differentials.The forms v u   satisfy the structural equations of projective space:


satisfy the following linear relations: of the geometrical second order tangency of the curve  and a geodesic curve having the same tangent direction in this point 0 Really, let conditions of the th he correspondenc ecomes Godeux's homography.eorembe true in corres- ), then mappings T  tive mappings.Correspondences between projective spaces having similar properties are called Gode ography.


depend linearly on difrentials of principal parameters differentials of other parameters.The other parameters define trasformations fe and of ng moving frames for fixi points 0


becomes charactiristic by invariant first order normals in corresponding points of hypersurface s.It follows the point correspondence n is geodesic.
set of invariant first order normals of the hypersurface n V gener- ates -parametrical fimily of straight lines.This set is ca n lled as a congruence of straight lines.
It is proved.orrespondencen define the whole projective-invariant normalization of hypersurfaces in the third differentia ourh differential n -quasigroup from the algebraic point of view.There exists an   1 n  -web connected with this n-quasigroup.To find this web it is sufficient to consider a new mani- of the correspondence.It can be written as e a tensor of a second-order diffe   2 1 of   1 n  -web adjoined to correspondence C we use Equations (4), (22) and structural equations of projective spaces.We obtain 0 k sor of W [14].
if hypersurfaces are give parallelizable correspondences between ( + 1) hy ces of projective spaces exist and depend on   1 n n  functions in n variables.


Let us find equations of correspondences C  and equations of three-webs joined to them.Equations of correspondences 0 tensor.If we take a corres- then the torsion tensor of three-web adjoined to C  can be written as follows paratactical three-webs [11] In with this, point correspondences between (n + 1) hypersurfaces of projective spaces are calle There exist the soaccordance .d paratactical, if all their subcorrespondences C hupersurfaces and prove that invariant first and second orders normals for all hypersurfaces (n > 2