Embeddings of Almost Hermitian Manifold in Almost Hyper Hermitian Manifold and Complex (Hypercomplex) Numbers in Riemannian Geometry ()
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).
Share and Cite:
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.
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