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 ![](https://www.scirp.org/html/htmlimages\6-7402351x\27e4faf4-3be6-426c-a7c3-fe3164f1c7b2.png)
[2] . For any point
there exists such unique point
that
. A point
has the coordinates
where
are coordinates of the point p in ![](https://www.scirp.org/html/htmlimages\6-7402351x\2852ec8d-ac65-4fd9-a369-ccfed820293e.png)
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 ![](https://www.scirp.org/html/htmlimages\6-7402351x\93d7cb26-2c8f-4ce2-a3b7-3b086f3dcccc.png)
(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
![](https://www.scirp.org/html/htmlimages\6-7402351x\95203160-176c-492c-9f62-815612a5364f.png)
. The following maps are isomorphisms of corresponding vector spaces ![](https://www.scirp.org/html/htmlimages\6-7402351x\a2e0233b-4ca5-4a3e-9e44-712dd86b765b.png)
![](https://www.scirp.org/html/htmlimages\6-7402351x\b92422eb-455a-4eb0-91b8-e481bad59d60.png)
and we have
![](https://www.scirp.org/html/htmlimages\6-7402351x\59b6a88f-1a46-436e-be3a-8c9ec2212740.png)
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
![](https://www.scirp.org/html/htmlimages\6-7402351x\6e33e369-dff8-4fc1-a1bd-94cda942fe98.png)
and
.
For
we obtain
![](https://www.scirp.org/html/htmlimages\6-7402351x\a3f5ba5d-cd11-469d-b5dd-70be192206cf.png)
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
![](https://www.scirp.org/html/htmlimages\6-7402351x\85492a58-7239-4fe5-9771-34d9821bd51d.png)
and
![](https://www.scirp.org/html/htmlimages\6-7402351x\ba97da1b-42ed-4979-9cbb-6beff5dbfb61.png)
For
we obtain
![](https://www.scirp.org/html/htmlimages\6-7402351x\170943ad-c5c6-425c-8f5a-7e05a3ec2b39.png)
Further, we obtain
![](https://www.scirp.org/html/htmlimages\6-7402351x\af93d861-0e12-46d0-9941-23f89491c79e.png)
![](https://www.scirp.org/html/htmlimages\6-7402351x\cc6b47ab-e952-4b12-9b40-d5ef94f84dc0.png)
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
![](https://www.scirp.org/html/htmlimages\6-7402351x\46fcd3d4-2a11-4e7a-a930-df77bf882399.png)
![](https://www.scirp.org/html/htmlimages\6-7402351x\e4d4b831-b060-4470-a924-04f7545eb938.png)
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
![](https://www.scirp.org/html/htmlimages\6-7402351x\1d90adba-c36a-40db-970c-94335b608ecf.png)
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)
![](Images/Table_Tmp.jpg)
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
![](https://www.scirp.org/html/htmlimages\6-7402351x\071d0d94-5ffe-40de-8a08-1d7bfcad858c.png)
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 ![](https://www.scirp.org/html/htmlimages\6-7402351x\9f775f5e-5d72-4fc9-a361-3431ac060b95.png)
. It is known [3] that the Kaehlerian metric
has on
the following decomposition
where u is a real-valued function on
.
We have
![](https://www.scirp.org/html/htmlimages\6-7402351x\e29a4a72-8ae1-4fb1-8eba-9fbfbc659f4f.png)
![](https://www.scirp.org/html/htmlimages\6-7402351x\224bac65-eaea-4787-8cd5-24eff5a514c7.png)
It follows that
.
Further, we obtain
![](https://www.scirp.org/html/htmlimages\6-7402351x\bcefffa6-8321-4a5b-b9c9-a4a2469de212.png)
![](https://www.scirp.org/html/htmlimages\6-7402351x\e80bf30c-2989-405d-b8f6-7b7b8ed1184b.png)
Finally, we get
![](https://www.scirp.org/html/htmlimages\6-7402351x\140c806f-8de9-4c4b-99db-553c1bb48efd.png)
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.
![](https://www.scirp.org/html/htmlimages\6-7402351x\73e0e8dd-d8b7-4589-a924-5a035f4cb4d5.png)
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.