On spacelike hypersurfaces in anti de sitter four space

Mathematical Computation June 2014, Volume 3, Issue 2, PP.30-37

On Spacelike Hypersurfaces in Anti de Sitter Four Space Jiajing Miao School of Science, Mudanjing Normal University, Mudanjing, 157011, P.R. China #Email: jiajing0407@126.com

Abstract In this paper, contact geometry of spacelike hypersurfaces in Anti de Sitter four spacewill be studied. To do this, we revealed the relationship between the singularity Anti de Sitter height function and that of spacelike hypersurfaces. In addition the contact relations between spacelike hypersurfaces and AdS-great-hyperboloids are studied from geometrical point of view.

Keywords: Anti de Sitter Four Space, Spacelike Hyppersurfaces, Legendrian Singularities

1 INTRODUCTION Since the second half of the 20th century, the Reimannian and semi-Reimannian geometry have been active areas of research in differential geometry and its application to variety of subjects in mathematics and physics. During the mid 1970s, the interest shifted towards Lorentzian geometry, the mathematical theory used in general relativity. Since then there has been amazing leap in the depth of the connection between modern differential geometry and mathematical relativity, both from the local and the global point of view. A minor revolution in mathematical thought and technique occurred during 1960s, largely through the inventive genius of French mathematician Rene Thom. His ideas partly inspired by H. Whitney gave birth to what is called singularity theory, a term which includes catastrophes and bifurcations. Today's singularity being a direct descendant of differential calculus, is certain to have a great deal of interests to say about geometry and therefore about all the branches of mathematics, physics and other disciplines where the geometrical spirit is a guiding light. More recently developments in singularity theory have enriched the field of geometry by making possible a wealth of detail only dreamed of fifty years ago.

Anti de Sitter space is a maximally symmetric, vacuum solution of the Einstein's field equation with an attractive cosmological constant in the theory of relativity which makes it be a very important space in both the astrophysics and geometry [1-3, 8-9]. In this paper, we try to introduce a basic framework for the study of spacelike hypersurfaces in Anti de Sitter 4-space. This work can be seen as a complement work of those in [2] and some applications of the theory introduced in [1, 3-6].

The organization of this paper is as follows. In Section 2, we will develop the local differential geometry of spacelike hypersurfaces in Anti de Sitter4-space. In Section 3, we will study the properties of Anti de Sitter height function and we prove Anti de Sitter Gauss map to be a wave front set. In Sections 4, we will investigate the contact relations between spacelike hypersurfaces and AdS-great-hyperboloids. Finally, we will study the generic properties of spacelike hypersurfac-es.

We assume throughout the whole paper that all the maps and manifolds are C



unless the contrary is explicitly stated.

2 BASIC CONCEPTS AND THE LOCAL DIFFERENTIAL GEOMETRY OF SPACELIKE HYPERSURFACES In this section, we shall shortly introduce the local differential geometry of spacelike hypersurfaces in Anti de sitter 4-space.

Let R5   x1 , x2 , x3 , x4 , x5  xi  R(i  1, 2,3, 4,5)

be

a

5-dimensional

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vector

space.

For

any

vectors

x  ( x1 , x2 , x3 , x4, x5 ) and y  ( y1 , y2 , y3 , y4 , y5 ) in R 5 , the pseudo scalar product of x and y is defined to be

x, y

  x1 y1  x2 y2  x3 y3  x4 y4  x5 y5 .

2

We call ( R , , 2 ) a semi-Euclidean 5-space with index 2. We write R25 instead of ( R5 , , 2 ) . A non-zero vector 5

x  R25 is called spacelike, null or timelike if x, x 2  0 , x, x 2  0 or x, x 2  0 respectively. The norm of a vector x  R25 is defined by x 

x, x

2

.For a vector n  R25 and a real number c, we define a hyperplane with pseudo-

normal n by





HP(n, c)  x  R25 x, n 2  c

The HP(n, c) is called a Lorentz hyperplane, a semi-Euclidean hyperplane with index 2 or a null hyperplane if n is timelike, spacelike or null respectively.

We now define Anti de sitter 4-space (or AdS 4 -space) by





Ads 4  x  R25 x, x 2  1 ; Given vectors X1 , X 2 , X 3 , X 4  R , we define their wedge product X1X 2 X 3 X 4 by 5 2

e1 x11 X 1X 2 X 3 X 4  x12 x31 x14

e2 x12 x22 x32 x42

e3 x13 x23 x33 x43

e4 x14 x24 x34 x44

e5 x15 x25 x35 x45

Where e1 , e2 , e3 , e4 , e5  is a canonical basis of R25 and X i   xi1 , xi2 , xi3 , xi4 , xi5  . We can easily check that

X i , X1X 2 X 3 X 4

 det  X i , X1 , X 2 , X 3 , X 4   0

2

So that X1X 2 X 3 X 4 is pseudo orthogonal to any X i  i  1, 2,3, 4  .

We now introduce the extrinsic differential geometry of spacelike hypersurfaces in AdS 4 .Consider an embed-ding X : U  AdS 4 from an open subset U  R3 . We write M  X (U ) by identifying M with U through the embedding X . We say that X is spacelike if X u , i  1, 2,3, are always spacelike vectors. In this case, we call X (u ) i

spacelike hypersurface. We define a vector N (u ) by N (u ) 

X (u )X u (u )X u (u )X u (u ) 1

2

3

X  u  X u  u  X u  u  X u  u  1

2

3

By definition, we have

N u  , X u u  i

2

 N u  , X u  2  0

Where i  1, 2,3 .This indicates that X (u ) , N (u)  N p M , where u   u1 , u2 , u3  is a local coordinate system around p  X  u   M . We can easily check that N (u ) is timelike vector and N , N a map

N : U  Ads 4 by u

2

 1 . Therefore we can define

N u 

We call it timelike Anti de sitter Gauss image (or TADS Gauss image) of X (or M ). We now consider the geometric meaning of the TAdS Gauss image of a spacelike hypersurface. We define typical spacelike surface in AdS 4

by AH (n, c)  AdS 4  HP  n, c . We call AH  n, c  an AdS-hyperboloid in AdS 4 , if n is timelike and n  c .

Especially, we call AH  n,0  the AdS-great-hyperboloid if n is timelike. Then we have the following propositions.

Proposition 2.1 Let X : U  Ads 4 be a spacelike hypersurface in Anti de sitter 4-space. If the TAdS Gauss image N (u ) is constant, then the spacelike hypersurface X  u   M is a part of an AdS-great- hyperboloid. We can easily

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show that N u , i  1, 2,3, are tangent vectors of M . Therefore we have linear transformation i

Wp  dN  u  : Tp M  Tp M

Which is called the Anti de Sitter shape operator (or Ads-shape operator) of M  X (U ) at p  X  u  . We denote the eigenvalues of W p by ki  p  i  1, 2,3 . The Anti de sitter Gauss-Kroneker curvature (or AdS-G-K Curvature) of

M  X (U ) at p  X  u  is defined by K AdS  u   det Wp  k1  p   k2  p   k3  p  .

We call a point p  X  u  an Anti de Sitter parabolic point (or an AdS-parabolic point) of X : U  Ads 4 if K AdS  u   0 .

We call a point u U or p  X  u  an umbilic point if Wp  k  p  idT M .Especially, we call M  X (U ) totally p

umbilic if every point in M is umbilic. Then we can get the following proposition.

Proposition 2.2. Suppose that M  X (U ) is totally umbilic. Then k  p  is constant k . Under this condition we have the following classification.

(1) If k  0 , then M is a part of an AdS-great-hyperboloid HP  n, 1  AdS 4 , where n  X 

1 N is a constant k

timelike vector.

(2) If k  0 , then M is a part of an AdS-great-hyperboloid HP  n,0  AdS 4 , where n  N is a constant timelike vector.

In the last part of this section, we introduce the Anti de sitter Weingarten formula. We induce the Anti de Sitter first fundamental form ds 2   g i , j dui du j on M  X (U ) , where gij  u   X ui  u  , X u j  u  3

i , j 1

define the Anti de Sitter second fundamental invariant by hij  u    Nu  u  , X u  u  i

j

for any u U .We also 2

for any u U .

2

Proposition 2.3 Under the above notations, we have the following Anti de Sitter Weingarten formula:

Nu  u     hi j  u  X u  u  3

i

j

j 1

where  hi j  u     hik  u    g kj  u   and  g kj  u     gkj  u   . 1

As a corollary of the above proposition, we have an explicit expression for the AdS-G-K curvature in terms of the Riemannian metric and the Anti de Sitter second fundamental invariant.

Corollary 2.4 Under the above notations. The AdS-G-K curvature is given by

K Ads 

det  hij  u  

det  g  u  

We call a point p  X  u  an Anti de Sitter parabolic if K Ads  u   0 . We also call it an Anti de Sitter flat point if it is an umbilic point and K Ads  u   0 .

3 SOME PROPERTIES OF THE TIMELIKE ANTI DE SITTER HEIGHT FUNCTION In this section, we define a family of function on a spacelike hypersurface in Anti de Sitter 4-space, which are useful for the study of singularities of TAdS Gauss image. Let X : U  Ads 4 be a spacelike hypersurface. We define a family of functions.

H : U  AdS 4  R by

H  u, v   X  u  , v - 32 www.ivypub.org/mc

2

We call H a timelike Anti de Sitter height function (or an AdS-height function) on M . We denote the Hessian matrix of the AdS-height function hv  u   H  u, v0  at u0 by Hess  hv   u0  .Then we have the following 0

0

proposition.

Proposition 3.1 Let M  X (U ) be a spacelike hypersurface in Ads 4 and H  u, v  be an AdS-height function. Then (1) H  u, v  

H u, v   0 i  1, 2,3 If and only if v   N  u  ; ui

(2) Let v0  N  u0  , then det Hess  hv

0

 u   0 if and only if K u   0 .

0

Ads

Proof (1) There exist real numbers  , ,1 , 2 ,3 such that v   X   N  1 X u   2 X u  3 X u 1

Since X , X

2

 1 , we get X , X u

i

2

3

 0 . It follows that H (u, v)  0 if and only if   X  u  , v

2

0

H  u, v   X u  u  , v ui i

2

 0 . Since

3

2

  gij i j 1

And  gij  is non-degenerate, we get i  0  i  1, 2,3 . It follows that v   N . We also get v, v

2

 1 .

Therefore   1 .

(2) By definition, we get Hess  hv

0

 u    0

X u u , N  u0  i j

2

   X

ui

u  , N u  0

uj

0

2



By the AdS-Weingarten formula, we get  X u , Nu i

3

j

2

  hi X u , X u  1

3

j

2

  hi g j  hij .  1

Therefore we get K Ads 

det  hij 

det  g 



det Hess  hv

0

 u 

det  g   u0 

0

Then we complete the proof.

Corollary 3.2 Let H : U  AdS 4  R be an AdS-height function on spacelike hypersurface M  X (U ) . N is the TAdS-Gauss image and p  M . Then the following conditions are equivalent: (1) There exists v  Ads 4 such that p is a degenerate singular point of AdS-height function hv ; (2) There exists v  Ads 4 such that p is a singular point of TAdS-Gauss image N ; (3) K Ads  u   0 .

Proof By definition, (2) and (3) are equivalent. By the assertion (2) of above proposition, (1) and (3) are also equivalent.

Next, in order to study the TAdS-Gauss image of M , we should refer to the brief review on Legendrian singularity theory in [3]. Now we apply the arguments in [2-3] to our situation. We can naturally interpret the TAdS-Gauss image as a Legendrian map in the framework of Legendrian singularity theory.

Let X : U  Ads 4 be a spacelike hypersurface in Ads 4 and N be the TAdS-Gauss image of M  X (U ) . We define a mapping

L : U   by L  u    X  u  , N  u   where





   v, w v, w 2  0  AdS 4  Ads 4 .

Since - 33 www.ivypub.org/mc

X  u  , N  u  2  dX  u  , N u  2  0

The mapping L is a Legendrian embedding. We denote X  u    x1 , x2 , x3 , x4 , x5  and X  u   1 , 2 , 3 , 4 , 5  as coordinate representations. We define a smooth mapping

 :U  PT *  AdS 4  by

  u   ( N  u  ,[ x12  x21  :   x13  x31  :   x14  x41  :   x15  x51 ]) .

It is known that two fundamental notions of Legendrian singularity theory are Morse family and generating family.

We can get the following proposition on Morse family.

Proposition 3.3 The AdS-height function H : U  AdS 4  R is a Morse family.

Legendrian dualities between pseudo in general semi-Euclidean spaces have been studied in [3]. Next we will prove H to be a generating family of L(U )   . Lengendrian dualities will be used as a fundamental tool. We get the following proposition on generating family.

Proposition 3.4 For any spacelike hypersurfaces X : U  AdS 4 , the AdS-height function H : U  AdS 4  R of X is a generating family of the Legendrian embedding L .

Proof We consider a coordinate neighbourhood





W1    1 , w2 , w3 , 4 , 5  AdS 4 1  0  We also consider the contact morphism:

 : (W1 )  PT * AdS 4 W1 Since AdS-height function H is a Morse Family, we can define a Legendrian immersion

LH : * ( H ) U  W1   PT * AdS 4 W1 by

LH (u,  )  ( ,[

H H H H : : : ]) 2 3 4 5

By Proposition 3.1, we have 4 * ( H )  { u, N (u) U  AdS u U } ,

where * ( H )  {(u,  ) U  AdS 4 H  u,    Since   N  u  and 1 

H H H (u,  )  (u,  )  (u,  )  0 . u1 u2 u3

22  32  42  52  1 , we get

 H (u, N  u )   x2  u   x1  u  2 , 2 1  H (u, N  u )  x3  u   x1  u  3 , 3 1 4 H (u, N  u )  x4  u   x1  u  , 4 1  H (u, N  u )  x5  u   x1  u  5 , 5 1

where X   x1 , x2 , x3 , x4 , x5  and N  1 , 2 , 3 , 4 , 5  . It follows that

LH  u, N (u)   ( N (u),[ x12  x21 :  x13  x31 :  x14  x41 :  x15  x51 ])

 (u) Therefore we get  L(u)  (u) . This indicates that H is a generating family of L   .We can get the same - 34 www.ivypub.org/mc

relation as the above on the other local coordinates. Therefore we conclude that the TAdS-Gauss image N can be regarded as a Legendrian map.

4 CONTACT WITH ADS-GREAT-HYPERBOLOIDS In this section, we consider the geometric meaning of the singularities of the TAdS-Gauss image of spacelike hypersurface M  X (U ) in AdS 4 . We should consider the contact between spacelike hypersurface and AdS-greathyperboloid. Montaldi [7] introduce some basic notions of contact geometry and he gave a characterization of the notion of contact by using the terminology of singularity theory. For notions and basic results on the theory of Legendrian singularities, refer to [5].

We define a function H : Ads 4  Ads 4  R

by

H  u, v    u, x 2 .

For

any

v0  AdS

4

,

we

denote

v0

 u   H  u, v 

and

0

we

have

the

AdS-great-hyperboloid

M (0)  AdS  HP(v0 ,0) . We write AH  v0, 0  AdS  HP(v0 ,0) . For any u0 U , we consider the timelike 1 v0

4

4

vector v0  N (u0 ) .Then we get v0

X  u0   H

 X  id  u , v   H u , N u   0 .

Ads 4

0

0

0

0

We also get



v0

ui

X

u   0

H  u0 , N u0   0 . ui

For i  1, 2,3 . This means that the AdS-great-hyperboloid AH  v0, 0  is tangent to M at P  X (u0 ) . In this case, we call AH  v0, 0  the tangent AdS-great-hyperboloid of M  X (U ) at P  X (u0 ) and we denote it by AH  X , u0  . Let v1 , v2 be timelike vectors. If v1 and v2 are linearly dependent, then HP  v1 ,0  and HP  v2 ,0  are equal. Therefore, AH  v1 ,0  AH  v2 ,0 . We have the following simple lemma.

Lemma 4.1 Let X : U  Ads 4 be a spacelike hypersurface. For two points u1 , u2  u , N (u1 )  N (u2 ) if and only if AH  X , u1   AH  X , u2  .

We now study the contact between M and tangent AdS-great-hyperboloid at p  M as an application of Legendrian singularity theory. We should introduce an equivalence relation among Lengendrian immersion germs. Let i :  L, p    PT * R n , p  and i ' :  L' , p'    PT * R n , p'  be Legendrian immersion germs. Then we say that i and i ' are Legendrian equivalent if there exists a contact diffeomorphism germ H :  PT * Rn , P    PT * Rn , p'  such that H preserves fibres of  land H ( L)  L' . A Legendrian germ into PT * Rn at a point is said to be Legendrian stable if for every map with the given germ there are a neighbourhood in the space of Legendrian immersion (in the Whitney C  topology) and a neighbourhood has, in the second neighborhood, a point at which its germ is Legendrian equivalent to the original germ [1] The definition of  -equivalent, P    equivalent and   equivalent in [5] will be used here. We have the following theorem.

Theorem 4.2 Let X i : U , ui    AdS 4 , X i  ui    i  1, 2 be spacelike hypersurface germs such that the corresponding Legendrian embedding germs Li : U , ui    , zi  are Lengendrian stable. Then the following conditions are equivalent.

(1) TAdS-Gauss image germs N1 and N 2 are  -equivalent; (2)

H1 and H 2 are P    equivalent ;

(3)

h1,v and h2,v are   equivalent; 1

2

(4)   X1 U  , AH  X1 , u1  , v1     X 2 U  , AH  X 2 , u2  , v2  ; (5) Q  X 1 , u1  and Q  X 2 , u2  are isomorphic as R-algebras, where - 35 www.ivypub.org/mc

Q  X i , ui  

Cu

.

i

 X u  , N u 

2

C



ui

Proof By the analogous arguments in [5], it is easy to get that these conditions are equivalent.

For a spacelike hypersurface germ



X : U , u0    AdS 4 , X  u0   ,



X 1 AH  N  u0  ,0  , u0 is called the tangent AdS-great-hyperboloidic indicatrix germ of X . In general we have the following theorem: Theorem 4.3 Let X i : U , ui    AdS 4 , X i  ui    i  1, 2 be spacelike hypersurface germs such that their AdSparabolic sets have no interior an subspaces of U . If TAdS-Gauss image germs N1 and N 2 are  -equivalent, then

  X1 U  , AH  X1 , u1  , v1     X 2 U  , AH  X 2 , u2  , v2  .









In this case, X11 AH  N1  u1  ,0  , u1 and X 2 1 AH  N2  u2  ,0  , u2 are diffeomorphic as set germs.

Proof The AdS-parabolic set is the set of singular points of the TAdS-Gauss image. So the corresponding Legendrian embedding Li satisfies the conditions of legendrian stability. If TAdS-Gauss image germs N1 and N 2 are

 -equivalent, then L1 and L2 are Lengendrian equivalent, so that H1 and H 2 are P    equivalent. Therefore, h1v and h2v are   equivalent. This condition is equivalent to the condition that 1

2

  X1 U  , AH  X1 , u1  , v1     X 2 u  , AH  X 2 , u2  , v2  .

On the other hand, we have

X i1  AH ( Ni (u0 ),0  , u0 )   hiv 1  0  , u0  i

Since the   equivalent preserves the zero level X 21  AH ( N2 (u2 ),0 , u2 ) are diffeomorphic as set germs.

sets,

we

get

X i1  AH ( N1 (u1 ),0 , u1 )

that

and

Theorem 4.3 means the diffeomorphism type of the tangent AdS-great-hyperboloidic indicatrix germ is an invariant of A-classification of the TAdS-Gauss image germ of X

5 GENERIC PROPERTIES OF SPACELIKE HYPERSURFACES In this section, we consider generic properties of spacelike hypersurfaces in Ads 4 . A kind of transversality theorem will be used as a main tool. We consider the space of spacelike embeddings Embs (U , AdS 4 ) with Whitney C  topology. We also consider the function H : AdS 4  AdS 4  R defined in §4. We claim that H u is a submersion for any u  AdS 4 , where Hu  v   H  u, v  . For any X  Emb (U , AdS 4 ) , we have H  H the

l  jet

extension

j H : U  AdS  J U , R  l 1

4

l

defined

 X  id  . We also have Ads 4

by j H (u, v)  J hv u  . l 1

l

We

consider

the

trivialization J l U , R   U  R  J l  3,1 . For any submanifolds Q  J l  3,1 , we denote Q  U  {0}  Q . Then we have the following proposition and a corollary of lemma 6 of Wassermann [6]. ~

Proposition 5.1 Let Q be a submanifold of J l (3,1). then the set ~

TQ  {X  Embs(U , Ads 4 ) j1l H is transversality to Q} is a residual subset of Embs (U , AdS 4 ) . If Q is a closed subset, then TQ is open.

By the classification of stable Legendrian singularities of n  6 and Proposition 5.1, we have the following theorem.

Theorem 5.2 There exists an open dense subset o  Embs (U , AdS 4 ) such that for any X  O , the germ of the corresponding Legendrian embedding L at each point is Legendrian stable.

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ACKNOWLEDGEMENTS The author was partially supported by preparatory studies of provincial Innovation project of MNU, No.SY201225.

REFERENCES [1]

Chen L. On spacelike surfaces in Anti de Sitter 3-space from the contact viewpoint [J]. Hokkaido Mathematical Journal, 2009, 38: 701-720.

[2]

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Kong L L, Pei D H. Singularities of de Sitter Gauss map of timelike hypersurface in Minkowski 4-space [J]. Science in China Ser A, 2008, 51: 241-249.

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Izumiya S, Pei D H, Sano T. Singularities of hyperbolic Gauss maps [J]. Proceedings of the London mathematical Society, 2003, 86: 485-512.

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Wassermann G. Stability of caustics [J]. Mathematische Annalen, 1975, 210: 43-50.

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Montaldi J A. On generic composites of maps [J]. Bulletin London Mathematical Society, 1991, 23: 81-85.

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Maldacena M. The Large N Limit of Superconformal Field Theories and Supergravity [J]. Advances in Theoretical and Mathematical Physics, 1998, 2: 231-252.

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AUTHORS Jiajing Miao was born in Taonan, Jilin, China in 1982. She received the B.S and the M.S. degrees from Jilin Normal University in 2004 and 2007, respectively.

Since 2009, she has been a lecturer in Mudanjiang Normal University. Miao is mainly interested in Geometry and Lie algebra. She has published more than ten papers.

Email: jiajing0407@126.com

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