**
Applied Mathematics** Vol.5 No.16(2014), Article
ID:49372,12
pages
DOI:10.4236/am.2014.516238

Embeddings of Almost Hermitian Manifold in Almost Hyper Hermitian Manifold and Complex (Hypercomplex) Numbers in Riemannian Geometry

Alexander A. Ermolitski

IIT-BSUIR, Minsk, Belarus

Email: ermolitski@mail.by

Copyright © 2014 by author and Scientific Research Publishing Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY).

http://creativecommons.org/licenses/by/4.0/

Received 10 July 2014; revised 11 August 2014; accepted 22 August 2014

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 hyper Hermitian 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 hyper Hermitian structure of the special form. As a result, we have obtained that any Riemannian manifold M of dimension n can be embedded as a totally geodesic submanifold in a Kaehlerian manifold of dimension 2n (Theorem 6) and in a hyper Kaehlerian manifold of dimension 4n (Theorem 7). Such embeddings are “good” from the point of view of Riemannian geometry. They allow solving problems of Riemannian geometry by methods of Kaehlerian geometry (see Section 5 as an example). We can find similar situation in mathematical analysis (real and complex).

**Keywords:**Riemannian Manifolds, Almost Hermitian and Almost Hyper Hermitian Structures, Tangent Bundle

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 H_{U} 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 V_{U} 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 H_{U} and V_{U} 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 J_{1} 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 h^{1} of the pair where h^{1} 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 h^{1} 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 J_{2} 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 h^{2} 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 O_{M} 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(O_{M}) 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(O_{M}) 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 N_{0} of the null section O_{M} in TM that the mapping

is the diffeomorphic of N_{0} 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 M^{2}. 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 M^{4}. 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 i^{2} = j^{2} = k^{2} = –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 3^{o}, 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.

References

- Gromoll, D., Klingenberg, W. and Meyer, W. (1968) Riemannsche Geometrie im Grossen. Springer, Berlin. http://dx.doi.org/10.1007/978-3-540-35901-2
- Kobayashi, S. and Nomizu, K. (1963) Foundations of Differential Geometry. Vol. 1, Wiley, New York.
- Kobayashi, S. and Nomizu, K. (1969) Foundations of Differential Geometry. Vol. 2, Wiley, New York.
- Bogdanovich, S.A. and Ermolitski, A.A. (2004) On Almost Hyper Hermitian Structures on Riemannian Manifolds and Tangent Bundles. Central European Journal of Mathematics, 2, 615-623. http://dx.doi.org/10.2478/BF02475969
- Dombrowski, P. (1962) On the Geometry of the Tangent Bundle. Journal für die Reine und Angewandte Mathematik, 210, 73-78.
- Ermolitski, A.A. (1998) Riemannian Manifolds with Geometric Structures. Monograph, BSPU, Minsk. arXiv:0805.3497.
- Hirsch, M.W. (1976) Differential Topology. Graduate Texts in Mathematics. Springer, New York, 33.
- Ermolitski, A.A. (2007) Deformations of Structures, Embedding of a Riemannian Manifold in a Kaehlerian One and Geometric Antigravitation. Vol. 76, Banach Center Publicantions, Warszawa, 505-514.
- Gray, A. and Herwella, L.M. (1980) The Sixteen Classes of Almost Hermitian Manifolds and Their Linear Invariants. Annali di Matematica Pura ed Applicata, 123, 35-58. http://dx.doi.org/10.1007/BF01796539