Embeddings of Almost Hermitian Manifold in Almost Hyper Hermitian Manifold and Complex (Hypercomplex) Numbers in Riemannian Geometry ()
1. Deformations of Tensor Structures on a Normal Tubular Neighborhood of a Submanifold
1˚. Let 
be a k-dimensional Riemannian manifold isometrically embedded in a n-dimensional Riemannian manifold
. The restriction of g to 
coincides with g′ and for any
.
.
So, we obtain a vector bundle 
over the submanifold
. There exists a neighborhood 
of the null section 
in 
such that the mapping
is a diffeomorphism of 
onto an open subset
. The subset 
is called a tubular neighborhood of the submanifold 
in
.
For any point 
we can consider a set 
of positive numbers such that the mapping 
is defined and injective on
. Let
.
Lemma [1] . The mapping 
is continuous on M.
If we take the restriction of the function 
on 
then it is clear that there exists a continuous positive function 
on 
such that for any 
open geodesic balls
. For compact manifolds we can choose a constant function
. We denote
, 
,
. It is obvious that
. For any point 
we can consider such an orthonormal frame
that 
and
. There exist coordinates
in some neighborhood 
of the point o that
. We consider orthonormal vector fields 
which are cross-sections of the vector bundle 
over 
and the neighborhood
. The basis 
defines the normal coordinates 
on 
[2] . For any point 
there exists such unique point 
that
. A point 
has the coordinates 
where 
are coordinates of the point p in 
and 
are normal coordinates of x in
. We denote
, on
. Thus, we can consider tubular neighborhoods 
and 
of the submanifold
.
2˚. Let K be a smooth tensor field of type (r, s) on the manifold M and for
, let
where 
is the dual basis of 
. We define a tensor field 
on M in the following way.
a)
, then
;
b)
, then
;
c)
, then
.
It is easy to see the independence of the tensor field 
on a choice of coordinates in 
for every point
.
Definition 1. The tensor field 
is called a deformation of the tensor field K on the normal tubular neighborhood of a submanifold
.
Remark. The obtained tensor field 
is continuous but is not smooth on the boundaries of the normal tubular neighborhoods 
and
; 
is smooth in other points of the manifold M.
3˚. We consider a deformation 
of the Riemannian metric g on the normal tubular neighborhood
of a submanifold
. For
, 
, we define the Riemannian metric 
by the following way.
a) 
for any
;
b)
, where 
on
,
;
c)
, for any
;
d) 
for each point
.
The independence of 
on a choice of local coordinates follows and the correctly defined Riemannian metric 
on M has been obtained.
It is known from [3] that every autoparallel submanifold of M is a totally geodesic submanifold and a submanifold 
is autoparallel if and only if 
for any
, where 
is the Riemannian connection of g.
Theorem 1. Let 
be a submanifold of a Riemannian manifold (M, g) and 
be the deformation of g on the normal tubular neighborhood 
of 
constructed above. Then 
is a totally geodesic submanifold of
.
Proof. For any point 
the functions 
and 
on
because the vector fields 
are tangent to
. By the formula of the Riemannian connection 
of the Riemannian metric
, [2] , we obtain for 
(1.1)
Here we use the fact that 
and that 
because
.
Thus, 
and from the remarks above the theorem follows.
QED.
Corollary 1.1. Let 
be the Riemannian curvature tensor field of
. Then 
vanishes on every
for
.
Proof. From the formula (1.1) it is clear that 
for
. The rest is obvious.
wang#title3_4:spQED.
2. Almost Hyper Hermitian Structures (ahHs) on Tangent Bundles
0˚. We follow especially close to [4] .
Let (M, g) be a n-dimensional Riemannian manifold and TM be its tangent bundle. For a Riemannian connection 
we consider the connection map K of 
[5] , [1] , defined by the formula
, (2.1)
where Z is considered as a map from M into TM and the right side means a vector field on M assigning to 
the vector
.
If
, we denote by HU the kernel of 
and this n-dimensional subspace of 
is called the horizontal subspace of
.
Let π denote the natural projection of TM onto M, then π* is a 
-map of TTM onto TM. If
, we denote by VU the kernel of 
and this n-dimension subspace of 
is called the vertical subspace of
. The following maps are isomorphisms of corresponding vector spaces 
and we have
If
, then there exists exactly one vector field on TM called the “horizontal lift” (resp. “vertical lift”) of X and denoted by
, such that for all
:
, (2.2)
(2.3)
Let R be the curvature tensor field of
, then following [5] we write
, (2.4)
(2.5)
, (2.6)
. (2.7)
For vector fields 
and 
on TM the natural Riemannian metric 
is defined on TM by the formula
. (2.8)
It is clear that the subspaces HU and VU are orthogonal with respect to
.
It is easy to verify that 
are orthonormal vector fields on TM if 
are those on M i.e.
.
1˚. We define a tensor field J1 on TM by the equalities
(2.9)
For 
we get
and
.
For 
we obtain
and it follows that 
is an almost Hermitian manifold.
Further, we want to analyze the second fundamental tensor field h1 of the pair 
where h1 is defined by (2.11), [6] .
The Riemannian connection 
of the metric 
on TM is defined by the formula (see [1] )
(2.10)
For orthonormal vector fields 
on TM we obtain
(2.11)
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.
(1.1˚)
(2.1˚)
By similar arguments we obtain
(3.1˚)
(4.1˚)
(5.1˚)
. (6.1˚)
. (7.1˚)
. (8.1˚)
It is obvious that 
is a Kaehlerian structure if and only if
.
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)
For 
we get
and
For 
we obtain
Further, we obtain
Thus, we get 
and ahHs 
on TM has been constructed.
For orthonormal vector fields 
on TM we obtain
(2.13)
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.
(1.2˚)
(2.2˚)
By similar arguments we obtain
(3.2˚)
(4.2˚)
. (5.2˚)
. (6.2˚)
. (7.2˚)
(8.2˚)
Here h is the second fundamental tensor field of the pair (J, g) on M.
3. Embeddings of Almost Hermitian Manifolds in Almost Hyper Hermitian Those
For an almost Hermitian manifold (M, J, g) we have constructed in Section 2 ahHs 
on TM. The manifold M can be considered as the null section OM in TM 
and it is clear from (2.8) that
. All the results of 1 can be applied to a submanifold M in
, see [7] . So, we can consider the normal tubular neighborhoods 
and the deformations 
of the tensor fields 
respectively.
Theorem 2. Let (M, J, g) be an almost Hermitian manifold and 
be the corresponding normal tubular neighborhood with respect to 
on TM. Then M(OM) is a totally geodesic submanifold of the almost hyper Hermitian manifold
, where the ahHs 
is the deformation of the structure 
obtained in 2˚, Section 1. The structure 
is Kaehlerian one.
Proof. It follows from Theorem 1 that M is a totally geodesic submanifold of the Riemannian manifold
.
Let 
be a coordinate neighborhood in TM considered in 1˚, Section 1. A point 
has the coordinates 
where 
are coordinates of the point p in 
and 
are normal coordinates of x in
.
We denote 
where 
and 
are Riemannian connections of metrics 
and
, J is any tensor field from
.
Using the construction in 2˚, Section 1 we have 
on
. According to [2] we can write
(3.1)
It follows from (3.1) that 
and 
i.e. 
for
. Further, we get
It follows that 
for
.
For 
and we obtain
.
From the other side we can write
.
According to [6] we have 
where the second fundamental tensor field h is defined by (2.11). From (1.1˚)-(8.1˚) it follows that 
for any
. Thus, we have obtained 
and the structure 
is Kaehlerian one on
.
QED.
As a corollary we have got the following:
Theorem 3 [8] . Let (M, g) be a smooth Riemannian manifold and 
be the corresponding normal tubular neighborhood with respect to 
on TM. Then M(OM) is a totally geodesic submanifold of the Kaehlerian manifold
.
The classification given in [9] can be rewritten in terms of the second fundamental tensor field h (Table 1)
Table 1. Classification of almost Hermitian structures.
see chapter 5 of monograph [6] .
Let dimM ≥ 6 and
, where
, then we have Table1
Proposition 4. Let (J, g) be from some class from the Table1 Then the structure 
has the analogous class on
.
Proof. From (1.2˚)-(8.2˚) it follows that
. The rest is obvious from the table.
wang#title3_4:spQED.
4. Complex and Hypercomplex Numbers in Differential Geometry
For the manifold M we consider the products
,
and the diagonals
,
. It is obvious that the manifold 
and 
are diffeomorphic to M
.
Theorem 5 [1] . Let (M,
) be a manifold with a connection 
and π: TM → M be the canonical projection. Then there exists such a neighborhood N0 of the null section OM in TM that the mapping
is the diffeomorphic of N0 on a neighborhood 
of the diagonal
.
Further, 
is a Riemannian connection of the Riemannian metric g. Combining the Theorems 3 and 5 we have obtained the following.
Theorem 6. The diffeomorphism φ induces the Kaehlerian structure 
on the neighborhood 
of the diagonal 
and 
is a totally geodesic submanifold of the Kaehlerian manifold 
.
Remark. Generally speaking, the complex structure of the Kaehlerian manifold 
is not compatible with the product structure of M2. It means that if 
are the complex coordinates of a point
, then, generally speaking, we can not find such real coordinates 
of the points 
respectively that 
where
.
Combining the Theorems 2, 3, 4, 5 and 6 we have obtained the following.
Theorem 7. There exists the hyper Kaehlerian structure 
on a neighborhood 
of the diagonal 
and 
is a totally geodesic submanifold of the hyper Kaehlerian manifold
.
Remark. Generally speaking, the hypercomplex structure of the hyper Kaehlerian manifold 
is not compatible with the product structure of M4. It means that if 
are the hypercomplex coordinates of a point
, then, generally speaking we can not find such real coordinates
of the points x; y; u; 
respectively that 
where i2 = j2 = k2 = –1, ij = –ji = k.
5. A Local Construction of Kaehlerian and Riemannian Metrics
1˚. We consider a Riemannian manifold (M, g) as a totally geodesic subanifold of the Kaehlerian manifold
(see Theorem 3) then
.
Let 
be coordinates in some coordinate neighborhood 
and 
be the corresponding vector fields. We can choose a neighborhood 
where 
for every point
. It is clear from 3o, 1 that 
is a Riemannian product with respect the metric
. For every point 
where 
we denote 
and the vector fields 
define the coordinates 
on 
hence 
is tangent to 
for
.
So, 
is an coordinate neighborhood of the Kaehlerian manifold
, with complex coordinates
, and the vector fields 
. It is known [3] that the Kaehlerian metric 
has on 
the following decomposition
where u is a real-valued function on
.
We have
It follows that
.
Further, we obtain
Finally, we get
We can consider the restriction of 
and the function u on the neighborhood U. So, we have obtained.
Theorem 8. Let (M, g) be a Riemannian manifold and 
be coordinates is some coordinate neighborhood
. There exists a smooth function u: 
that 
on U.
2˚. Let (M, J, g) be a Kaehlerian manifold
, 
, be coordinates is some coordinate neighborhood
, where
. We consider a function u: 
from Theorem 5. Then, we have the following conditions on this function.
6. Conclusion
We consider such mappings in the category of Riemannian manifolds that metrics are invariant with respect to them. It follows that only totally geodesic submanifolds are “naturally good”. Theorems 6 and 7 allow considering any Riemannian manifold as a totally geodesic submanifold of a Kaehlerian (hyper Kaehlerian) one i.e. to apply the results of Kaehlerian (hyper Kaehlerian) geometry to Riemannian metrics. We remark that Whitnies embeddings are not suitable in this context.