) ( ) ( ) ( )^ ( ) ( )^ ( ) ( ) { }p ( ) ( ) ( ) ( ) ( ) { }p

Embeddings of almost Hermitian manifolds in almost hyperHermitian those. Complex and hypercomplex numbers in differential geometry. Alexander A. Ermolitski Cathedra of mathematics, MSHRC, st. Nezavisimosti 62, Minsk, 220050, Belarus E-mail: [email protected] _________________________________________________________________ Abstract: Tubular neighborhoods play an important role in differential topology. We have applied these constructions to geometry of almost Hermitian manifolds. At first, we consider deformations of tensor structures on a normal tubular neighborhood of a submanifold in a Riemannian manifold. Further, an almost hyperHermitian structure has been constructed on the tangent bundle TM with help of the Riemannian connection of an almost Hermitian structure on a manifold M then, we consider an embedding of the almost Hermitian manifold M in the corresponding normal tubular neighborhood of the null section in the tangent bundle TM equipped with the deformed almost hyperHermitian structure of the special form. As a result, we have obtained that any smooth manifold M of dimension n can be embedded as a totally geodesic submanifold in a Kaehlerian manifold of dimension 2n and in a hyperKaehlerian manifold of dimension 4n. __________________________________________________________________ 1. Deformations of tensor structures neighborhood of a submanifold

on a

normal tubular

1°. Let (M ¢, g ¢) be a k–dimensional Riemannian manifold isometrically embedded in a n–dimensional Riemannian manifold (M , g ) . The restriction of g to M ¢ coincides with g' and for any p Î M ¢ . ^ T p (M ) = T p (M ¢ ) Å T p (M ¢ ) . ^ ^ So, we obtain a vector bundle M ¢ ® T (M ¢) : p ® Tp (M ¢) over the ~ submanifold M ¢ . There exists a neighborhood U 0 of the null section OM ¢ in

^ T (M ¢) such that the mapping ~ p ´ exp : v ® (p (v ), expp (v ) v ), v ÎU 0 , ~ ~ ~ is a diffeomorphism of U 0 onto an open subset U Ì M . The subset U is called a tubular neighborhood of the submanifold M ¢ in M . For any point p Î M we can consider a set {d ( p )} of positive numbers such that the mapping expU (d ( p )) is defined and injective on U (d ( p )) Ì Tp (M ) . Let

e ( p ) = sup{d ( p )}.

Lemma, [6]. The mapping M ® R+ : p ® e ( p ) is continuous on M. ~ If we take the restriction of the function e ( p ) on U then it is clear that there exists a continuous positive function e ( p ) on M ¢ such that for any p Î M ¢ open ~ æ e ( p) ö geodesic balls Bç p; ÷ Ì B( p; e ( p )) Ì U . For compact manifolds we can 2 ø è ~ ~ choose a constant function e ( p ) = e > 0 . We denote U p = exp U 0 Ç T p (M ¢)^ ,

(

)

~ æ e ( p)ö ~ æ e ( p) ö Dç p; ÷ Ç U p , D ( p;e ( p )) = B( p; e ( p )) Ç U p . It is obvious that ÷ = Bç p ; 2 ø 2 ø è è ~ dim U p = dim D ( p; e ( p )) = n - k . For any point o Î M ¢ we can consider such an orthonormal frame (X 10 ,..., X n0 ) that T0 (M=¢) L[ X 10 ,..., X k0 ] and T0 (M ¢ )= L[ X k +10 ,..., X n0 ] . There exist coordinates x1,..., xk in some neighborhood ^

~ V0 Ì M ¢ of the point o that

¶ = X i0 , i = 1, k . We consider orthonormal vector ¶xi 0

^ fields Xk+1, ..., Xn which are cross–sections of the vector bundle p ® T p (M ¢) over ~ ~ ~ V0 and the neighborhood W0 = U È U p . The basis { X k +1p ,..., X n p } defines the ~ pÎV0

~ ~ normal coordinates xk+1, ..., xn on U p [8]. For any point x Î W0 there exists such ~ ~ ^ unique point p Î V0 that x = exp p (tx ), x = 1, x Î T p (M ¢) . A point x Î W0 has the coordinates x1, ..., xk, xk+1, ..., xn where x1, ..., xk are coordinates of the point p in ~ ~ ¶ V0 and xk+1, ..., xn are normal coordinates of x in U p . We denote X i = , i = 1, n , ¶xi ~ on W0 . Thus, we can consider tubular neighborhoods e ( p) ö æ æ e ( p) ö Tbç M ¢; = ÷ U Dç p; ÷ and Tb(M ¢; e ( p )=) U D( p;e ( p )) of the 2 ø pÎM ¢ è 2 ø è pÎM ¢ submanifold M ¢ . 2°. Let K be a smooth tensor field of type (r, s) on the manifold M and for ~ x Î W0 , let

Kx = where

å

i1 ,...,i r , j1 ,..., j s

{ X 1x ,..., X xn }

is

j1 js ir k ij11,..., ,..., j s ( x )X i1 Ä ... Ä X ir Ä X x Ä ... Ä X x , x

the

dual

x

basis

of

Tx* (M ),

x = exp p (tx ),

x = 1, x Î Tp (M ¢)^ . We define a tensor field K on M in the following way.

æ e ( p)ö a) x Î Dç p; ÷ , then 2 è ø j1 js ir Kx = k ij11,..., å ,..., j s ( p ) X i1 Ä ... Ä X i r Ä X x Ä ... Ä X x ; i1 ,...,i r , j1 ,..., j s

x

x

æ e ( p)ö b) x Î D( p; e ( p )) \ Dç p; ÷ , then 2 ø è j1 js ir Kx = k ij11,..., å ,..., j s (exp p ((2t - e ( p ))x ))X i1 Ä ... Ä X i r Ä X x Ä ... Ä X x ; x

i1 ,...,i r , j1 ,..., j s

x

c) x Î M \ U D( p; e ( p )), then M¢

K x = Kx . It is easy to see the independence of the tensor field K on a choice of ~ coordinates in W0 for every point o Î M ¢ . Definition 1. The tensor field K is called a deformation of the tensor field K on the normal tubular neighborhood of a submanifold M ¢ . Remark. The obtained tensor field K is continuous but is not smooth on the e ( p) ö æ boundaries of the normal tubular neighborhoods Tbç M ¢; ÷ and Tb(M ¢; e ( p )) , 2 ø è K is smooth in other points of the manifold M. We consider a deformation g of the Riemannian metric g on the ~ normal tubular neighborhood Tb(M ¢; e ( p )) of a submanifold M ¢ . For x Î W0 , x = exp p (tx ), x = 1, x Î Tp (M ¢) , we define the Riemannian metric g by the 3°.

following way. a) g p = g p for any p Î M ¢ ; b)

g x ( X i , X j ) = g ij ( x ) = g ij ( p ) , where

Xi =

¶ , i = 1, n, ¶xi

~ æ e ( p) ö j = 1, n, on W0 , x Î Dç p; ÷; 2 ø è c) g x ( X i , X j ) = g ij ( x ) = g ij (exp p ((2t - e ( p ))x )),

æ e ( p) ö x Î D( p; e ( p )) / Dç p; ÷; 2 ø è d) = g x g x for each point x Î M \ U D( p; e ( p )) . pÎM ¢

for

Xj =

¶ , ¶x j

any

The independence of g on a choice of local coordinates follows and the correctly defined Riemannian metric g on M has been obtained. It is known from [9] that every autoparallel submanifold of M is a totally geodesic submanifold and a submanifold M ¢ is autoparallel if and only if Ñ X Y Î T (M ¢) for any X , Y Î c (M ¢) , where Ñ is the Riemannian connection of g. Theorem 1. Let M ¢ be a submanifold of a Riemannian manifold (M, g) and g be the deformation of g on the normal tubular neighborhood Tb(M ¢; e ( p )) of M ¢ constructed above. Then M ¢ is a totally geodesic submanifold of e ( p) ö ö æ æ ç Tbç M ¢; ÷, g ÷ . 2 ø ø è è

æ e ( p) ö ~ Proof. For any point x Î Dç p; ÷ Ì W0 the functions g ij ( x ) = gij ( p ) and 2 ø è

¶ æ e ( p) ö = 0, l = k + 1, n on Dç p; are ÷ because the vector fields X l = 2 ø ¶xl ¶xl è æ e ( p) ö tangent to Dç p; ÷ . By the formula of the Riemannian connection Ñ of the 2 ø è Riemannian metric g , [8], we obtain for i, j = 1, k , l = k + 1, n (1.1) 2 g p Ñ X i X j , X l = X i p g (X j , X l ) + X j p g ( X i , X l ) - X l p g ( X i , X j ) + ¶ g ij

(

)

[

]

] ¶¶gx ] = [X , X ] = [X , X ] = [

+ g p ( X i , X j , X l ) + g p ([ X l , X i ], X j ) + g p ( X i , X l , X j ) = Here we use the fact that

[X , X i

ij

= 0.

l

j

g (X j , X l ) = g ( X i , X l ) = 0 because X l Î T (M ¢) .

l

i

l

j

0 and that

^

Thus, Ñ X i X j Î T (M ¢) and from the remarks above the theorem follows.

QED. Corollary 1.1. Let R be the Riemannian curvature tensor field of Ñ . Then e ( p) ö R vanishes on every Dæç p; ÷ for p Î M ¢ . 2 ø è Proof. From the formula (1.1) it is clear that Ñ X l X m = 0 for l , m = k + 1, n . The rest is obvious.

QED. 2. Almost hyperHermitian structures (ahHs) on tangent bundles 0°. Let (M, g) be a n–dimensional Riemannian manifold and TM be its tangent bundle. For a Riemannian connection Ñ we consider the connection map K of Ñ [2], [6], defined by the formula (2.1) Ñ X Z = KZ * X , where Z is considered as a map from M into TM and the right side means a vector field on M assigning to p Î M the vector KZ * X p Î M p . If U Î TM , we denote by HU the kernel of K TM U and this n–dimensional subspace of TM U is called the horizontal subspace of TM U . Let p denote the natural projection of TM onto M, then p* is a C ¥ –map of TTM onto TM. If UÎTM, we denote by VU the kernel of p * TM and this U n–dimension subspace of TM U is called the vertical subspace of TM U (dim TM U = 2 dim M = 2 n ) . The following maps are isomorphisms of corresponding vector spaces ( p = p (U )) p * TM U : H U ® M p , K TM U : VU ® M p and we have

TM U = H U Å VU If X Î c (M ) , then there exists exactly one vector field on TM called the h

( ) v

«horizontal lift» (resp. «vertical lift») of X and denoted by X X , such that for all U Î TM : h

h

(2.2) p * X U = X p (U ) , K X U = 0p (U ) , v

v

(2.3) p * X U = 0p (U ) , K X U = X p (U ) , Let R be the curvature tensor field of Ñ, then following [2] we write v

v

(2.4) [ X , Y ] =0,

(2.5) [ X , Y ] = (Ñ X Y ) h

( K ([ X

v

v

h

h

h

h

) ) = R( X , Y )U .

(2.6) p * [ X , Y ]U = [ X , Y ] , (2.7)

, Y ]U

h

v

h

v

For vector fields X = X Å X and Y = Y Å Y on TM the natural Riemannian metric gˆ = is defined on TM by the formula (2.8) < X , Y >= g (p * X , p * Y ) + g (K X , K Y ). It is clear that the subspaces HU and VU are orthogonal with respect to < , >. h

h

h

v

v

v

It is easy to verify that X 1 , X 2 ,..., X n , X 1 , X 2 ,..., X n are orthonormal vector fields on TM if X 1 , X 2 ,..., X n are those on M i.e. g (X i , X j ) = d ij .

1°.

We define a tensor field J1 on TM by the equalities

(2.9) J1 X = X , J1 X = - X , X Î c (M ) . For X Î c (M ) we get h

v

v

((

h

h

J 12 X = J 1 J 1 X Å X

and

v

))= J (- X 1

h

ÅX

v

)= -(X

h

ÅX

v

)= - I X

J12 = - I .

For X , Y Î c (M ) we obtain h

v

h

v

h

v

v

v

< J 1 X , J 1 Y >=< - X Å X ,-Y Å Y >=< - X ,-Y > + < X , Y >, h

v

h

v

h

h

v

v

< X , Y >=< X Å X , Y Å Y >=< X , Y > + < X , Y >

and it follows that < J 1 X , J 1 Y >=< X , Y > , (TM , J 1 , < , > ) is an almost Hermitian manifold. Further, we want to analyze the second fundamental tensor field h1 of the pair (J1, ) where h1 is defined by (2.11), [3]. The Riemannian connection ш of the metric gˆ = on TM is defined by the formula (see [6])

(

1 X < Y , Z > +Y < Z , X > - Z < X , Y > + 2 + < Z , [ X , Y ] > + < Y , [ Z , X ] > + < X , [ Z , Y ] > ), X , Y , Z Î c (M ). For orthonormal vector fields X , Y , Z on TM we obtain 1 (2.11) h1XYZ =< h1X Y , Z >= < ш X Y + J1ш X J 1Y , Z >= 2 1 ˆ = < Ñ X Y , Z > - < ш X J 1Y , J1 Z > = 2 1 = ( + < [ Z , X ], Y > + < [ Z , Y ], X > – 4 – < [ X , J1Y ], J1 Z > - < [ J1 Z , X ], J1Y > - < [ J 1 Z , J1Y ], X > ) .

(2.10) < ш X Y , Z >=

(

)

Using (2.4) – (2.7) and (2.11) we consider the following cases for the tensor field h1 assuming all the vector fields to be orthonormal. h h h h h h 1 1.1°) h1 h h h = ( + < [ Z , X ], Y > + X Y Z 4 h h h h h h h h h + < [ Z , Y ], X > - < [ X , J1 Y ], J1 Z > - < [ J 1 Z , X ], J 1Y > h h h 1 - < [ J1 Z , J 1Y ], X > ) = ( g ([ X , Y ], Z ) + g ([ Z , X ], Y ) + g ([ Z , Y ], X ) 4 - < [ X , Y ], Z > - < [ Z , X ], Y > - < [ Z , Y ], X > ) = 1 1 = g (Ñ X Y , Z ) - ( g (Ñ X Y , Z ) - g (Ñ X Z , Y )) = 4 2 1 = ( g (Ñ X Y , Z ) - g (Ñ X Y , Z )) = 0. 2 h

v

v

v

h

v

v

v

h

2.1°)

h h v v h h 1 ( + < [ Z , X ], Y > + X Y Z 4 v h h h h v v h h + < [ Z , Y ], X > - < [ X , J1Y ], J 1 Z > - < [ J1 Z , X ], J1Y > -

h1 h

h

=

v

- < [ J1 Z , J1Y ], X > ) = v

h

h

)

1 (g (R( X , Y )U , Z )+ < [ Z h , X h ], Y v > = 4

1 (g (R ( X , Y )U , Z ) + g (R(Z , X )U , Y )) = 4 1 = - ( g (R ( X , Y )Z , U ) + g (R (Z , X )Y , U )). 4 By similar arguments we obtain 1 3.1°) h1 h v h = - ( g (R (Z , X )Y ,U ) + g (R ( X , Y )Z ,U )). X Y Z 4 1 4.1°) h1 v h h = - ( g (R (Z , Y ) X , U )). X Y Z 4 1 5.1°) h1 v v v = ( g (R (Z , Y )X , U )). X Y Z 4 1 6.1°) h v v h = 0. =

7.1°) 8.1°)

X Y Z 1 h v h v X Y Z 1 h h v v X Y Z

= 0. = 0.

It is obvious that ( J 1 , gˆ )

is a Kaehlerian structure if and only if h1 = 0 .

2°. Now assume additionally that we have an almost Hermitian structure J on (M, g). We define a tensor field J2 on TM by the equalities (2.12) J 2 X = (JX ) , J 2 X = - (JX ) , For X Î c (M ) we get h

h

( (

h

J 22 X = J 2 J 2 X Å X

and

v

v

v

X Î c (M ) .

)) = J ((JX ) Å -(JX ) ) = -(X h

v

2

h

ÅX

v

)- I X

J 22 = - I .

For X , Y Î c (M ) we obtain

( ) Å -(JX ) , (JY ) Å -(JY ) >=< (JX ) , (JY ) > + + < (JX ) , (JY ) >= g ( JX , JY ) + g ( JX , JY ) = g ( X , Y ) + g ( X , Y ) =

< J 2 X , J 2 Y >=< JX v

h

v

h

v

h

h

v

h

h

v

v

h

v

h

v

=< X , Y > + < X , Y >=< X Å X , Y Å Y >=< X , Y > . Further, we obtain

( ) (( ) J (J X ) = J (- X J 1 J 2 X = J 1 JX

h

h

2

1

2

( ) ) = (JX ) Å (JX ) , Å X ) = -(JX ) Å - (JX ) . Å - JX v

v

h

h

v

v

Thus, we get J 1 J 2 = - J 2 J 1 = J 3 and ahHs ( J 1 , J 2 , J 3 , ) on TM has been constructed. For orthonormal vector fields X , Y , Z on TM we obtain 1 ˆ 2 ˆ J Y , Z >= (2.13) hXYZ =< hX2 Y , Z >= < Ñ Y + J 2Ñ X X 2 2 1 ˆ J 2 Y , J 2 Z > = 1 ( + = < ш X Y , Z > - < Ñ X 2 4 + < [ Z , X ], Y > + < [ Z , Y ], X > - < [ X , J 2 Y ], J 2 Z > -

(

)

- < [ J 2 Z , X ], J 2 Y > - < [ J 2 Z , J 2 Y ], X > ).

Using (2.4) – (2.7) and (2.13) we consider the following cases for the tensor field h2 assuming all the vector fields to be orthonormal. h h h h h h 1 1.2°) h 2 h h h = ( + < [ Z , X ], Y > + X Y Z 4 h h h h h h h h h + < [ Z , Y ], X > - < [ X , J 2 Y ], J 2 Z > - < [ J 2 Z , X ], J 2 Y > h h h 1 - < [ J 2 Z , J 2 Y ], X > ) = ( g ([ X , Y ], Z ) + g ([ Z , X ], Y ) + g ([ Z , Y ], X ) 4 - g ([ X , JY ], JZ ) - g ([ JZ , X ], JY ) - g ([ JZ , JY ], X )) = 1 = ( g (Ñ X Y , Z ) - g (Ñ X JY , JZ )) = h XYZ . 2 h h v v h h 1 2.2°) h 2 h h v = ( + < [ Z , X ], Y > + X Y Z 4 v h h h h v v h h + < [ Z , Y ], X > - < [ X , J 2 Y ], J 2 Z > - < [ J 2 Z , X ], J 2 Y > v h h 1 - < [ J 2 Z , J 2 Y ], X > ) = (g (R( X , Y )U , Z ) + g (R ( X , JY )U , J Z )) = 4 1 = - ( g (R( X , Y )Z , U ) + g (R ( X , JY )J Z , U )). 4 By similar arguments we obtain 1 3.2°) h 2 h v h = - ( g (R ( X , Z )Y , U ) + g (R ( X , JZ )JY , U )). X Y Z 4 1 4.2°) h 2 v h h = - ( g (R (Z , Y ) X , U ) - g (R (JZ , J Y ) X , U )). X Y Z 4 2 5.2°) h v v v = 0 . 6.2°) 7.2°)

X Y Z h2 v v h X Y Z h2 v h v X Y Z

= 0. = 0.

1 (g (Ñ X Y , Z ) - g (Ñ X JY , JZ )) = hXYZ . X Y Z 2 Here h is the second fundamental tensor field of the pair (J, g) on M.

8.2°) h 2 h

v

v

=

3. Embeddings hyperHermitian those

of

almost

Hermitian

manifolds

in

almost

For an almost Hermitian manifold (M, J, g) we have constructed in 2 ahHs ( J1 , J 2 , J 3 , gˆ ) on TM. The manifold M can be considered as the null section OM in TM ( p « o p Î OM Ì TM ) and it is clear from (2.8) that gˆ | M = g . All the results of

1 can be applied to a submanifold M in (TM , gˆ ) , see [7]. So, we can consider the e ( p)ö normal tubular neighborhoods Tb æç M , ÷ Ì Tb (M , e ( p )) Ì TM and the 2 ø è deformations J 1 , J 2 , J 3 , g of the tensor fields J 1 , J 2 , J 3 , gˆ respectively. Theorem 2. Let (M, J, g) be an almost Hermitian manifold and Tb (M , e ( p )) be the corresponding normal tubular neighborhood with respect to gˆ =< , > on TM. Then M(OM) is a totally geodesic submanifold of the almost hyperHermitian e ( p)ö ö æ manifold ç Tb æç M , ÷, J 1 , J 2 , J 3 , g ÷ , where the ahHs (J 1 , J 2 , J 3 , g ) is the 2 ø ø è è deformation of the structure (J 1 , J 2 , J 3 , gˆ ) obtained in 2°, 1. The structure (J 1 , g ) is Kaehlerian one. Proof. It follows from theorem 1 that M is a totally geodesic submanifold of e (p)ö ö æ the Riemannian manifold ç Tb æç M , ÷, g ÷ . 2 ø ø è è ~ Let W0 be a coordinate neighborhood in TM considered in 1°, 1. A point ~ x Î W0 has the coordinates x1, …, xn, xn+1, …, x2n where x1, …, xn are coordinates ~ of the point p in V0 Ì M and xn+1, …, x2n are normal coordinates of x in æ e ( p)ö . D ç p, ÷ 2 ø è We denote k ¶ ˆ X X j å Гˆ ijk X k , = Xi i 1,2 n, Ñ Ñ JX j å J kj X k , ,= Г ij X k , Xi X j å i ¶xi k k k k JX å J j X , gˆ gˆ ( X , X ), g g (X , X ) where ш and Ñ are j

k

k

ij

i

j

ij

i

j

Riemannian connections of metrics gˆ and g , J is any tensor field from J 1 , J 2 , J 3 . Using the construction in 2°, 1 we have g ij ( x ) = gˆ ij ( p ),

e (p)ö ~ æ Tb ç M , ÷ Ç W0 . According to [8] we can write 2 ø è l 1 æ ¶ g kj ¶ g ik ¶ g ij (3.1) + å g lk Г ij = çç 2 è ¶xi ¶x j ¶x k l

ö ÷ ÷ ø

J

i j

(x ) =

J ij ( p ) on

It follows from (3.1) that Г ij ( x ) = Г ij ( p ) and Г ij ( x ) = 0 i.e. Ñ X i X j = 0 for l

l

l

i = n + 1, 2n . Further, we get

(Ñ

)

k

J X j = Ñ X i J X j - JÑ X i X j = å Ñ X i J j X k -

Xi

k

) ) - å Г J X = å (J Г - Г J + X J )X , ((Ñ J )X )(x ) = å (J Г - Г J + X J )(x )X = å ((J Г - Г J )( p ) + ( X J )( x ))X . (

(

k k k æ ö - J ç å Г ij X k ÷ = å J j Ñ X i X k + X i J j X k èk ø k l ij

k l

k ,l

Xi

l j

k

l ij

k l

k il

l ij

k l

k ,l

k il

l ij

k l

i

k

k j

i

k ,l

l j

k j

i

k ,l

l j

j

k il

k|x

k j

=

k|x

It follows that Ñ X i J = 0 for i = n + 1,2n . For i = 1, n

(X J )(x) = (X J

)( p ) and we obtain ((Ñ J )X )(x ) = å (J Гˆ - Гˆ J + X J )( p )X i

k j

Xi

i

j

k j

l j

k ,l

k il

l ij

k l

i

k j

k|x .

From the other side we can write ˆ X J X j ( p ) = å J lj Гˆ ilk - Гˆ lij J lk + X i J kj ( p )X k | p . Ñ i

((

) )

k ,l

(

(

)

)

(

)

According to [3] we have Ñ X i J X j = 2hX i JX j ( p ) where the second fundamental tensor field h is defined by (2.11). From 1.1°) – 8.1°) it follows that h1p = 0 for any p Î M (U = o p Î OM ) . Thus, we have obtained ÑJ1 = 0 and the

e ( p) ö structure J 1 , g is Kaehlerian one on Tbæç M , ÷. 2 ø è

(

)

QED. As a corollary we have got the following Theorem 3 [4]. Let (M, g) be a smooth Riemannian manifold and Tb(M , e ( p )) be the corresponding normal tubular neighborhood with respect to g = < , > on TM. Then M(OM) is a totally geodesic submanifold of the Kaehlerian e ( p)ö æ ö manifold ç Tbæç M , ÷, J 1 , g ÷ . 2 ø è è ø The classification given in [5] can be rewritten in terms of the second fundamental tensor field h, [3]. Let dimM ³ 6 and 2 b ( X ) = dF( JX ) , where F ( X , Y ) = g ( JX , Y ) , then we have

Class K U1 = NK U2 = AK U3 = SK Ç H U4

U1 Å U2 = QK U3 Å U4 = H U1 Å U3 U2 Å U4 U1 Å U4 U2 Å U3 U1 Å U2 Å U3= = SK U1 Å U2 Å U4

U1 Å U3 Å U4 U2 Å U3 Å U4 U

Defining condition h=0 hXX = 0 shXYZ = 0 hXYZ - hJXJYJZ = b (Z ) = 0 1 [< X , Y > b (Z ) - < X , Z > b (Y )- < X , JY > b ( JZ ) + 2(n - 1) + < X , JZ > b ( JY )] hXYJZ = hJXYZ N ( J ) = 0 or hXYJZ = - hJXYZ hXXY - hJXJXY = b (Z ) = 0 1 < JX , Y > b (Z )] = 0 s [h XYJZ (n - 1) 1 2 hXXY = [< X , Y > b ( X ) - X b (Y )- < X , JY > b ( JX )] 2(n - 1) s [ hXYJZ + hJXYZ ] = b (Z ) = 0 b =0

h XYZ =

1 [< X , Y > b (JZ )- < X , Z > b ( JY ) + (n - 1) + < X , JY > b (Z )- < X , JZ > b (Y )] hXJXY + hJXXY = 0 s [hXYJZ + hJXYZ ] = 0 No condition

hXYJZ - hJXYZ =

Proposition 4. Let (J, g) be from some class from the table above. Then the e ( p)ö structure (J 2 , g ) has the analogous class on Tb æç M , ÷. 2 ø è 2 Proof. From 1.2°) – 8.2°) it follows that h XYZ = 2h XYZ . The rest is obvious from the table.

4. Complex and hypercomplex numbers in differential geometry For the manifold M we consider the products M 2 = M ´ M = = {(x; y) | x; y Î M}, M 4 = M 2 ´ M 2 = {(x; y; u; v) | x; y, u; v Î M} and the diagonals D (M 2) = {(x; x) Î M 2}, D (M 4) = {(x; x; x; x) Î M 4}. It is obvious that the manifold D (M 2) and D (M 4) are diffeomorphic to M ( D (M 2) @ D (M 4) @ M).

Theorem 5 [6]. Let (M, Ñ) be a manifold with a connection Ñ and p : TM ® M be the canonical projection. Then there exists such a neighborhood N0 of the null section OM in TM that the mapping j : p ´ exp : X ® (p ( X ), exp p ( X ) X ) is the diffeomorphic of N0 on a neighborhood N D of the diagonal D (M 2). Further, Ñ is a Riemannian connection of the Riemannian metric g. Combining the theorems 3, 5 we have obtained the following. Theorem 6. The diffeomorphism j induces the Kaehlerian structure (J 1 , g ) on the neighborhood N D of the diagonal D (M 2) and D (M 2) @ M is a totally geodesic submanifold of the Kaehlerian manifold (N D , J 1 , g ) . Remark. Generally speaking, the complex structure of the Kaehlerian manifold (N D , J 1 , g ) is not compatible with the product structure of M 2. It means that if zl , l = 1, n are the complex coordinates of a point (x; y) Î N D , then, generally speaking, we can not find such real coordinates xl , yl , l = 1, n of the points x, y Î M respectively that zl = xl + iyl where i 2 = -1 . Combining the theorems 2, 3, 4, 5, 6 we have obtained the following. Theorem 7. There exists the hyperKaehlerian structure (J 1 , J 2 , J 3 , g ) on a neighborhood N D of the diagonal D (M 4) and D (M 4) @ M is a totally geodesic submanifold of the hyperKaehlerian manifold (N D , J 1 , J 2 , J 3 , g ). Remark. Generally speaking, the hypercomplex structure of the hyperKaehlerian manifold (N D , J 1 , J 2 , J 3 , g ) is not compatible with the product structure of M 4. It means that if ql , l = 1, n are the hypercomplex coordinates of a point (x; y; u; v) Î N D , then, generally speaking we can not find such real coordinates xl , yl , ul , vl , l = 1, n of the points x; y; u; v Î M respectively that 2 2 2 ql = xl + iyl + jul + kvl where i = j = k = –1, ij = –ji = k. References 1. 2. 3. 4. 5. 6.

S.A. Bogdanovich, A.A. Ermolitski, On almost hyperHermitian structures on Riemannian manifolds and tangent bundles, CEJM, 2(5) (2004) 615–623. P. Dombrowski, On the geometry of the tangent bundle, J. Reine und Angew. Math., 210 (1962) 73–78. A.A. Ermolitski, Riemannian manifolds with geometric structures, BSPU, Minsk, 1998 (in Russian), (English version: Internet, Google, Ermolitski ) A.A. Ermolitski, Deformations of structures, embedding of a Riemannian manifold in a Kaehlerian one and geometric antigravitation, Banach Center Publicantions, V. 76, Warszawa 2007, 505–514. A. Gray, L. M. Herwella, The sixteen classes of almost Hermitian manifolds and their linear invariants, Ann. Mat. pura appl., 123 (1980) 35–58. D. Gromoll, W. Klingenberg, W. Meyer, Riemannsche geometrie im grossen, Springer, Berlin, 1968 (in German).

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M.W. Hirsch, Differential topology. Graduate texts in mathematics, 33, Springer, N.Y., 1976. S. Kobayashi, K. Nomizu, Foundations of differential geometry, V. 1, Wiley, N.Y., 1963. S. Kobayashi, K. Nomizu, Foundations of differential geometry, V. 2, Wiley, N.Y., 1969.