Cmjv03i02p0221

J. Comp. & Math. Sci. Vol.3 (2), 221-224 (2012)

On Singularities KULKARNI VENKATESH and SHABBIR AHMED Department of Mathematics Gulbarga University, Gulbarga – 586105, India.

(Received on : March 30, 2012) ABSTRACT In this present paper we investigate the algebraic singularities of the differential forms of degree (n-1), further we exhibit the singularities without zero on one variety of dimension ‘n’.

Singularities introduced by Saloman and Zaare have studied the generic singularities in the year 2005. We studied the generic singularities and computer applications.

Keywords: One variety M of dimensions n, folio ℑ, transverse volume ω.

INTRODUCTION The concept of singularities first observed by J. Martineat5. Our aim is to study the problems of stability on local models for the generic singularities and computer applications.

Variety M of dimensions n we define a folio (ℑ) of dimension one of M varieties and (n-1) form ω without zero.

One generalization of the problem is studied.

We consider of the study of the singularities of structures (M, ℑ, ω), where ℑ is a folio of co-dimension p of the variety M and ω a pform representing a transverse volume at ℑ.

Geometrically the classification of the generic singularities rests on the behaviour of the hyper surface ∑ 1 (ω) set of the equation ω = 0; with the couple (ℑ, ∑ 1

(ω)) is associated in natural way as the germ of applications of ℜn-1 under ℜn-1 The corresponding applications on generic singularities are the germs of the type ∑ 1 ….1,0 of which the stability is of GRAS3.

Further the structure has wider applications.

We have shown that the stability of singularities of (n-1) forms, without zero, consists of an inversion of a differential operator of order one and of one homomorphism of modules over a ring of functions.

It is therefore necessary to employ the theorem of preparation4 and the resolution of a system of partial differential equations.

The various applications lie on essential differences in case of the similarities of differential.

Journal of Computer and Mathematical Sciences Vol. 3, Issue 2, 30 April, 2012 Pages (131-247)

222

Kulkarni Venkatesh, et al., J. Comp. & Math. Sci. Vol.3 (2), 221-224 (2012)

The singularities of order inferior or equal to ‘n’ are stable and we give them the local models and the singularities of order “n+1” are stable.

We define the singularities on the space of jet forms and we write the generic singularities using the theorem of transverseability.

The singularities posed by V.I. Arnold1 we define the stability of germs of the forms of degree (n-1) and we show that the singularities of order inferior or equal “n” are stable. Then, we deduce the local models for the singularities. In the end, we show that the singularities of order (n+1) are instable.

Generally all objects considered shall be C∞ Suppose M is a variety of n dimension, ℑ a folio of M of dimension (np), which is transversably orientable. This signifies that ℑ is defined by a system of pfaff E on M of rank p (that is to say a subfiber E of cotangent fiber T*M), completely interable such that the fiber on the right, Λp E multi vector of dimension p be trivial.

We call transverse volume of folio ℑ all sections without zero of fiber Λp E .

One transverse volume is thus a p-form on M, without zero, completely decomposable, integrable and defining the folio ℑ.

Let us suppose ω is a transverse volume on ℑ which is fixed. At a point x ∈ M where the differential dω of ω is not zero, one can always choose a system of local coordinates (x1,……..,xn) under which ω can be written as ω = (1+X1) dXn-p+1 Λ ………. Λ dXn5

where Xn-p+1,……….Xn are the local first integrals of.ℑ.

If dω is identically zero (one writes dω = 0).

This signifies that the volume ω is invariant by the actions of the field vectors tangent to the folio ℑ (i.e. whatever the field of vectors X, tangent to ℑ the Lie derivative of ω with respect to X, written θ (X). ω is zero).

If dω ≠ 0 it is natural to call all points X ∈ M singular points where dω is zero.

In the general case it is difficult to make a study of the singularities of the structures (M, ℑ, ω). A particular case where the dimension of ℑ is one will be treated here shown in6.

Once for all fix on the variety M a folio ℑ of dimension one, transversably orientable.

One should note the fiber Jn+1 Λn−1E on M of (n+1) jets of sections of

(

)

(

)

fiber Λn −1 E . One should construct on Jn+1 Λn−1E a set of sub-verities which shall be the singularities of transverse volume on ℑ.

We consider a system of local coordinates (X1,…………,Xn) under M such that at X2,…………..Xn be the first local integrals of the system E (or of folio ℑ); under such a system, called the adopted system of local coordinates on ℑ, all ω ∈ Jn+1 Λn−1 E will be written in a unique manner as:

(

)

ω = fdX 2 Λ........ΛdX n Where f ∈ Jn+1 (ℜn), are the (n+1) jet fibers of the function of n variables.

Journal of Computer and Mathematical Sciences Vol. 3, Issue 2, 30 April, 2012 Pages (131-247)

223

Kulkarni Venkatesh, et al., J. Comp. & Math. Sci. Vol.3 (2), 221-224 (2012)

By the choice of the local coordinates (X1,………Xn) one has an isomorphism of Jn+1 Λn−1 E /U under Jn+1 (ℜn)/U where U is the domain of the system of the coordinates considered.

All f ∈ Jn+1(ℜn) of source a = (a1,…….,an) ∈ U identify with a polynomial of degree n+1, in the variables XI = Xi – ai, i = 1,……,n we write

(

f=

∑

α ≤ n +1

)

Aα .Xα

…… +αn The functions (αi, Aα) contribute a system of coordinates on Jn+1(ℜn)/U, and therefore on Jn+1 Λn−1 E /U through the isomorphism of the choice of coordinates (α1,…..αn),

(

We define ∑ i = E

(

)

)

1,.....,1 123 i

, i = 1,…..,n,

under J Λ E /U as the sub-variety of the equations n−1

A1,0,…..,0 = A2,0,……,0= ….. = Ai,0,…..0, = 0 A0,0,….…0 ≠ 0 Trivially it can be verified that this definition does not depend on local system of coordinates (X1,….,Xn) adopted on the considered ℑ. Thus, we define on Jn+1 Λn−1 E a sequence of sub-varieties

(

1

)

12

As per the construction the sub1i

varieties ∑ are invariant (globally) by all diffeomorphisms of M, leaving the folio ℑ an invariant.

One will write

∑

by Jn+1

where according to the usage, α = (α1,……….,αn) is a multi index such that for all i, αi > 0. One has posed α = α1 + α2 +

n+1

1i

such that the codimension ∑ = i

1n

∑ 1 ⊃ ∑ ⊃ ..... ⊃ ∑

by J

∑

1i , 0

= ∑ 1i − ∑ 1i +1 , i ≤ n

If ω is a section of Λn −1E we design ω the section of Jn+1 Λn −1E defined

n +1 x

1i , 0

(

)

ω = (n+1) jet of ω on X. A volume ω presents the singularity of X or rather X is a singular point of th

ω of type ∑ 1i ,0 and observe X ∈ ∑ 1i ,0 (ω) if J nx +1 ω ∈ ∑ 1i ,0

We study the following analytical description If one writes ω = fdX 2 , Λ....ΛdX n under a local system of coordinates adapted with ℑ, then ω presents at the origin the singularity ∑ 1i ,0 , i < n, if and only if. i i +1 ∂f (0) = ..... = ∂ fi (0) = 0, ∂ i +f1 (0) ≠ 0 (1) ∂x ∂x ∂x1

Taking into consideration the conditions (1) it deals to the differential forms

 ∂f   ∂ i −1f    d ,......, d  ∂x i −1  ∂ x  1  i  being linearly independent at the origin.

Journal of Computer and Mathematical Sciences Vol. 3, Issue 2, 30 April, 2012 Pages (131-247)

224

Kulkarni Venkatesh, et al., J. Comp. & Math. Sci. Vol.3 (2), 221-224 (2012)

If J 0n +1 ω ∈ ∑

1i , 0

In general, if J 0n +1 ω ∈ ∑

and if Jn+1ω is

1i , 0

transverse in O at ∑ , it will signifies geometrically that the restriction Π : ∑ 1 (ω) ℜn-1 presents contraction in O i.e. a 1 ,0 singularity ∑ 2 in the sense of6.

1i , 0

and if

it will J ω is transverse in O at ∑ signifies geometrically that the restriction Π 1 −1, 0 signifies a singularity ∑ i in the sense of7 in O. 1i , 0

n+1

In the end we give the following examples of each type of singularities Let

ω1 ω2 : : ωI : : ωn

=

(1 + x )dX Λ.....ΛdX (1 + x x + x )dX Λ.....ΛdX

=

(1 + x x

=

(1 + x

=

2 1

2

1

1

1

n

n

n

3 1

2

n

)

+ x12 x n −1 + ... + x1i −1 x n − i + 2 + x ii +1 dX 2 Λ.....ΛdX n

)

x n + x 12 x n −1 + .. + x 1n −1 x 2 + x 1n dX 2 Λ.....ΛdX n

It is easily verified that ωi presents transversally a singularity of type ∑ 1i ,0 in O.

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Russian Mathematical Surveys Vol 23, n0j, pp-1-42 (Jan & Feb 1988).

2. Clint McCrory, Theodore Shifrin and Robertvarley: The Gauss map of a generic Hyper surfaces in P4.

J. Differential Geometry 30, 689-759 (1989).

3. G.R.A.S: Canonical forms of the singularities of differentiable application 260, pp 5662-5665 (1985).

4. Malgrange, B: Ideals of Differentiable functions Landon Oxford Univ Press (1995).

5. Martin Golubistky and David Tischler: On the local stability of differential forms. Transactions of the American Mathematical Society, Volume 223, (1976).

6. Martin, B: Modular deformation and space curves singularities, Rev. Mat. Iberoamericana. 19, No. 2 613-621 (2003).

7. Martinet, J: SUR les singularities des formes differentiells, Ann. Inst. Fourier (Grenoble) 20, fasc.1, 95-178 (1979).

8. Zhitomirskii M: Finite determinancy of vector fields, diffeomorphisms and exterior differential 1-forms Docl. Acad Nauk Ukar. SSR. Ser. A1, 6-9 (1987).

Journal of Computer and Mathematical Sciences Vol. 3, Issue 2, 30 April, 2012 Pages (131-247)