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

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).

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Ermolitski, A. (2014) Embeddings of Almost Hermitian Manifold in Almost Hyper Hermitian Manifold and Complex (Hypercomplex) Numbers in Riemannian Geometry. Applied Mathematics, 5, 2464-2475. doi: 10.4236/am.2014.516238.

 [1] 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 [2] Kobayashi, S. and Nomizu, K. (1963) Foundations of Differential Geometry. Vol. 1, Wiley, New York. [3] Kobayashi, S. and Nomizu, K. (1969) Foundations of Differential Geometry. Vol. 2, Wiley, New York. [4] 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 [5] Dombrowski, P. (1962) On the Geometry of the Tangent Bundle. Journal für die Reine und Angewandte Mathematik, 210, 73-78. [6] Ermolitski, A.A. (1998) Riemannian Manifolds with Geometric Structures. Monograph, BSPU, Minsk. arXiv:0805.3497. [7] Hirsch, M.W. (1976) Differential Topology. Graduate Texts in Mathematics. Springer, New York, 33. [8] 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. [9] 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