1. 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.
Properties of point correspondences between n-dimensional projective, affine and euclid spaces are studied by Ryzhkov [6], Sokolova [7] and Pavljuchenko [8].
A straight line 
passing through the point
is called a first order normal of a hypersurface of 
-dimensional projective space in the point 
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 hypersurface in the point 
if the tangent hyperplane of the hypersurface in the point 
includes this 
dimensional plane and this 
-dimensional plane does not pass through the point
.
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 
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].
2. 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.
Let 
be corresponding points of hypersurfaces
. A correspondence 
generates 
families point subcorrespondences
obtained by fixation of n - 2 corres- ponding 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.
We will assume, that surfaces 
belong to different projective spaces
. 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
.
With any point 
we associate a projective moving frame consisting of the point 
points 
of the tangent hyperplane of the hypersurface 
in the point 
and a point 
outside the tangent hyperplane.
The equations of infinitesimal displacement of our projective frames 
have the form:
(1)
where 
are 1-forms containing parameters, on which the family of frames in question depends, and their differentials. The forms 
satisfy the structural equations of projective space:
We can write equations of hypersurfaces 
as follows:
(2)
The Pfaffian forms 
define displacements of corresponding points 
of hypersurfaces
. It follows that the forms 
satisfy the following linear relations:
(3)
Since for 
forms 
are linearly independent, therefore the following conditions are true:
We can transform all frames of projective spaces in points 
by setting
. For new frames 
we will have 
By Equations (3) relations between forms 
take the simplest case. Let us suppose that necessary transformations of frames are done and we can write relations between forms 
of frames 
as follows
(4)
Geometrically Equations (4) mean that frames in points
of spaces 
are chosen so that directions in points
, 
are corresponding by mappings
.
To find equations of a mapping 
we fix points
, where
. Using Equations (2), (4), we have
(5)
Consider projective mappings
, where
By Equations (1), (5) the following relations satisfy projective mappings:
where
—a quantity of the first order according to
. The projective mapping 
has a first order tangency with the mapping 
in corresponding points 
Equations (2), (4) are main equations of our problem. With the help of exterior differentiation of these equations and applying Cartan’s lemma we obtain
(6)
where 
Note that quadratic forms 
are asymptotic quadratic forms of hypersurfaces 
Now in the family of frames we have equations of mapping 
in the way
(7)
and similar for 
(7’)
where 
and
.
To continue the system of Equations (6) we use exterior differentiation of these equations and Cartan’s lemma. We obtain new equations:
(8)
To write these equations we used operators 
and
. Operator 
is defined by forms 
and we have
and similarly operators 
are defined by forms 
Quantities 
are symmetric with respect to the indices i, j and k, for quantities 
some additional finite conditions are true.
The system of quantities 
define the geometrical object according to G.F.Laptev invariant methods [15]. This object is the fundamental geometrical object of second order of point correspondence
.
If we continue Equations (8), we obtain the system of differentional equations of a sequence of fundamental geometrical objects of point correspondence under consideration
3. Characteristic Directions of Point Correspondences
Let us consider a mapping 
If frames are fixed in corresponding points of hypersurfaces 
then the object 
define the quadratic transformation of tangent directions of hypersurfaces
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
(9)
A geodesic curve of hypersurface 
connected with 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 
define a tangent direction to a curve 
in a point 
then relations
are the condition of the geometrical second order tangency of the curve 
and a geodesic curve having the same tangent direction in this point
.
Characteristic directions have the following property. If a curve 
and a geodesic curve have second order tangency along a characteristic direction in the point 
then the image 
of the curve under 
has the similar property in the point 
by the corresponding characteristic direction. It follows from Equations (7,) (7’), (9) and relations
From geometric meaning of characteristic directions it is clear, that they depend on the choice of first order normals 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 cones. Characteristic directions are common generatrices of these cones.
Let us assume, that any direction 
in a point 
by some choice of a first order normal on hypersurfaces 
is characteristic for a mapping
. Then the last equations must be sutisfied for any magnitudes
. Therefore, the following conditions are true for simillar correspondences
After calculations we get the relations:
(10)
where
.
Theorem 1. If any direction 
in a point 
by any choice of first order normals on hypersurfaces 
is characteristic for a mapping
, then for
hypersurfaces 
degenerate into hyperplanes and the correspondence becomes Godeux’s homography.
Really, let conditions of the theorem be true in corresponding points 
of all hypersurfaces 
according to some first order normals
, then relations (10) are satisfied. We transform first order normals on hypersurfaces 
as follows
where 
are arbitrary quantities.
We denote the values quantities 
for new frames 
of hypersurfaces of the correspondence as 
Calculations show that
Since any direction 
is characteristic according to first order normals on hypersurfaces 
then quantities 
must also satisfy relations (10).
Let us consider the object 
We have
After substituting the values 
and considering similar terms we obtain
These relations must be true for any values 
then
Contructing these relations with respect to the indices 
and
, we arrive at the equation 
for
.
In a similar way we get
.
It is known that hypersurfaces degenerate into hyperplanes if the asymptotic tensors 
In this case a point correspondence
between hypersurfaces transforms into a point correspondence 
between hyperplanes. Since quantities 
satisfy relations (10), then mappings 
degenerate in projective mappings. Correspondences between projective spaces having similar properties are called Godeux’s homography.
4. Invariant Normalizations of Hypersurfaces under Point Correspondences
Moving frames of hypersurfaces 
under the correspondence depend on parameters of two types. There exsist principal parameters determined displacements of corresponding points 
of hypersurfaces
. Since points 
are connected by the correspondence the number of independent principal parameters is equal to
. By the Equations (4) 1-forms 
are independent linear combinations of differentials of principal parameters.
The Pfaffian forms 
depend linearly on differentials of principal parameters and differentials of other parameters. The other parameters define trasformations of moving frames for fixing points
. We denote values of forms 
as 
for fixing principal parameters.
We denote as 
values of operators 
and denote as 
values of the Pfaffian forms 
for fixing principal parameters.
By Equation (6) we have:
it follows 
With the help of the operator 
we can write Equation (8) for the case 
as follows:
(11)
where
.
It follows from relations (11) that quantities 
are relative tensors.
It is known that the main problem of nonmetric differentional geometry of a surface is a construction of invariant normalization of this surface. According to theory [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)
(12)
For the point on the invariant first order normal
(13)
For the second order normal (
-dimensional plane inside the tangent hyperplane)
(14)
Below we will assume, that asymptotic quadratic forms of hypersurfaces 
are nondegenerate. By virtue of this,
. It follows there exsist tensors 
symmetric with respect to the indices
,
. These tensors sutisfy conditions 
By Equation (11) we have differential equations:
By Equation (11) we obtain:
where
. Note that for 
quantities
(15)
satisfy equations
Therefore, by Equation (12) the quantities 
define the invariant first order normal geometrical object of the hypersurface 
From Equation (11) we have
It follows that quantities
(15’)
satisfy Equation (12) and define the invariant first order normal geometrical objects of the hypersurfaces 
To construct the invariant second order normal geometrical object of the hypersurface 
we consider quantities
(16)
Calculations show that quantities 
satisfy Equation (14).
Thus, it is proved.
Theorem 2. If asymptotic quadratic forms of 
hypersurfaces 
are nondegenerate and
, then a point correspondence 
between these hypersurfaces determine invariant first and second orders normals for all hypersurfaces in a second-order differential neighbourhood of corresponding points.
Note that to find necessary objects we used quantities
. A quantity 
may be used instead of the previous one. In general cases there exist 
different quantities 
Therefore, different invariant normalizations of hypersurfaces exist. In the paper we used a symmetrical case.
Below we will suppose that
. The case 
is considered in paper [12].
5. The Main Tensors of the Point Correspondence between n + 1 Hypersurfaces
Let us use the quantities 
for construction of invariant frames of the correspondence. We introduce an invariant family of frames 
defined by points
We denote Pfaffian forms of infinitesimal displacement of these frames as 
Then relations between 1-forms 
and 
can be written as follows
(17)
By Equations (12), (14) quantities
depend on differentials of principal parameters, therefore we can write forms 
and 
as follows
(18)
By new frames Equations (4), (6) of the corresponddence 
can be written in the form:
(19)
where 
and
(20)
Calculations show, that quantities 
satisfy equations
Therefore, quantities 
are absolute tensors of a second-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 
in the way
and similar for 
where 
and
.
We will call tensors 
as main tensors of the correspondence. Tensors 
define quadratic transformations 
generated invariant charactiristic directions in corresponding points of hypersurfaces.
Let us consider correspondences 
if there are relations
A point correspondence 
is called geodesic, if any tangent directions of hypersurfaces 
in corresponding points 
became charactiristic for mappings 
by some choice of the first order normals in these points.
It is true.
Theorem 3. For 
a point correspondence 
will be geodesic if ahd only if main tensors 
Really, let there exist 
families of the first order normals of hypersurfaces under correspondence by them a point correspondence 
is geodesic. Then relations (10) must be true. In this case as follows from Equations (15), (15’) the first order normal objects of hypersurfaces 
By setting 
in relations (16), we get values of second order normal objects of hypersurfaces under correspondence in this way:
If we substitute values 
in Equation (20) and use relations (10), then we obtain
Conversely, if we use invariant first and second order normals in all hypersurfaces under correspondence and tensors
(21)
then relations (10) are true.
Any tangent direction 
becomes charactiristic by invariant first order normals in corresponding points of hypersurfaces. It follows the point correspondence
is geodesic.
6. The Whole Projective-Invariant Normalization of Hypersurfaces under the Point Correspondence
To finish normalizations of hypersurfaces under consideration it is necessary to construct objects satisfying Equations (13). We prolong Equations (18). With the help of exterior differentiations and applying Cartan’s 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 points. We consider hypersurfaces
. We fix the hypersurface
, then
. The set of invariant first order normals of the hypersurface 
generates 
-parametrical fimily of straight lines. This set is called as a congruence of straight lines.
Let point 
be a focus of the congruence of the straight lines 
then infinitesimal displacement of focus 
must belong to the straight line 
Since
then focuses 
are obtained by conditions
or
To get values
, defined focuses on the straight line 
we consider the equation
For roots of this equation we have
We can define the harmonic pole [16] on each straight line 
of the congruence according to the point 
and 
focuses by the relation
Let points 
of frames coinside with invariant points 
where quantities 
are defined by values 
Other points of frames we leave without changing. After these transformations quantities 
become absolute tensors and quantities 
become relative tensors of the correspondence. Some relations are true
Forms 
will depend only on differentials of principal parameters, that’s why they can be written as follows 
It is proved.
Theorem 4. For 
a point correspondence
define the whole projective-invariant normalization of hypersurfaces in the third differential neighbourhood of corresponding points.
7. Point Correspondences between (n + 1) Hypersurfaces of Projective Spaces and Multidimensional (n + 1)-Webs
A point correspondence 
between 
hyperspaces 
of projective spaces 
is a local differential
-quasigroup from the algebraic point of view. There exists an 
-web connected with this n-quasigroup. To find this web it is sufficient to consider a new manifold constructed as
A correspondence C will be determined as an 
-dimesional smooth submanifold. There exist 
foliations of codimension 
on this submanifold. Each foliation is determined by the hypersurface
. These foliations define 
web W(n + 1, n) on the 
-dimensional submanifold.
We introduce additional forms
(22)
and quantities
where
.
By relations (11) we have
Therefore, quantities 
determine a tensor of a second-order differential neighbourhood of the correspondence. It can be written as
Using relations (17) we obtain
Thereforeforms 
do not depend on a choice of frames in corresponding points of hypersurfaces.
To write equations of 
-web adjoined to correspondence 
we use Equations (4), (22) and structural equations of projective spaces. We obtain
The equations show that forms 
are the forms of an affine connection assosiated to the web 
and tensors 
are the torsion tensor of 
[14].
It is known that parallelizable webs [11] are the simplest class of (n + 1)-webs. A correspondence between (n + 1) hypesurfaces of projective spaces is said to be parallelizable if the (n + 1)-web of this correspondence is parallelizable. The necessary and sufficient conditions for correspondence to be parallelizable are relations
Calculations show that if hypersurfaces are given then parallelizable correspondences between (n + 1) hypesurfaces of projective spaces exist and depend on 
functions in 
variables.
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 note that they are true for geodesic correspondences, that’s why the (n + 1)-web adjoined to the geodesic correspondence between (n + 1) hypersurfaces of projective spaces is always (2n + 2)-adric web of type 2.
A point correspodence 
generates 
families point subcorrespondences
obtained by fixation of n − 2 corresponding points. We can adjoin the web 
to each subcorrespondence 
Let us find equations of correspondences 
and equations of three-webs joined to them. Equations of correspondences 
can be written in the following way
Substituting these values into equations of (n + 1)-web we have after transformations
The forms
are connection forms of this three-web and the tensor
is the torsion tensor. If we take a correspondence 
then the torsion tensor of three-web adjoined to 
can be written as follows
There exist the so-called paratactical three-webs [11]. In accordance with this, point correspondences between (n + 1) hypersurfaces of projective spaces are called paratactical, if all their subcorrespondences 
are paratactical ones (torsion tensors are equal zero). The following relations
are conditions of the existence of paratactical correspondences.
8. Conclusions
We write main equations of a point correspondence between 
hypersurfaces of projective spaces and construct the sequence of main geometrical objects of the correspondence. we define characteristic directions of a correspondence and prove that there exist invariant characteristic directions.
We construct whole projective-invariant normalizations of all hupersurfaces and prove that invariant first and second orders normals for all hypersurfaces (n > 2) under point correspondences are determined in a secondorder differential neighbourhood of corresponding points. We single out main tensors of the correspondence and define some partial cases of correspondences.
We establish a connection between the geometry of point correspondences between 
hypersurfaces of projective spaces and the theory of multidimensional (n + 1)-webs. In particular we prove that the (n + 1)-web adjoined to the geodesic correspondence between (n + 1) hypersurfaces of projective spaces is always (2n + 2)- adric web of type 2.