ABSTRACT

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.

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 pointis 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 subcorrespondencesobtained 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 pointsof 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 forhypersurfaces 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 followswhere 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 correspondencebetween 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 correspondenceis 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 correspondencedefine 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 tensoris 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.

REFERENCES