Some Riemannian manifolds are expressed as a product manifold. Recently, I have read two articles about space-times. In both articles, the authors prove that a Riemannian manifold \bar{M}^n is expressed as a product of the form I×M^{n−1}. Both authors use similar techniques, namely integrable distribution, in this decomposition. Really, I do not understand this technique. But it is enough to know a characterization of Riemannian manifolds which we can express it as a product manifold M^1\times M^2.
Q1 Does this characterization exist?(if yes, a reference is required)
Q2 What conditions and proof hints could one think of to characterize these manifolds?
You can find interesting answers to your well posed question by the following article:
http://www.scielo.cl/pdf/proy/v27n2/art01.pdf
best regards
Professor Massimiliano Ferrara
Thank you Prof. Massimilianp Ferrara, a quick reading of the article shows that there many characterizations of riemannian manifolds, which is not my my question. I am looking for those manifolds which we can write them as a product manifold.
This is a good question.
In addition to the help observation by @Massimiliano Ferrara, consider
Y. Gunduzalp, Slant submersions from almost product Riemannian manifolds, Turkish J. of Math. 37, 2013, 863-873:
http://journals.tubitak.gov.tr/math/issues/mat-13-37-5/mat-37-5-13-1205-64.pdf
This is a wide-range introduction to a form of product Riemannian manifolds. See Section 3, starting on page 865, on almost product Riemannian manifolds.
More to the point is
A. Olteanu, A general inequality for doubly warped product sub manifolds, Math. J. Okayama. Univ. 52, 2010, 133-142:
http://www.math.okayama-u.ac.jp/mjou/mjou52/_11_olteanu.pdf
See Section 3, starting on page 135, on doubly warped product sub manifolds in arbitrary Riemannian manifolds. A good introduction to Riemannian as well as Finslerian manifolds is given in
P.I. Radu, Generalizations of some results from Riemannian to Finsler geometry, Ph.D. thesis:
https://dea.lib.unideb.hu/dea/bitstream/handle/2437/78487/de_1572.pdf;jsessionid=008BD97D6669706109D4269B0461A1AF?sequence=2
See Section 6, starting on page 36, ends of submanifolds.
Thank you Prof. James F Peters. I will see this articles and then back to you again
Thank you Prof. Cenap Ozel. Kindly give me key words to search for. I really searched for this and found many articles about product manifolds but no one of them consider this problem.
Dear sameh recently there are many published articles on the characterization of product manifolds
Dear all, Prof. James F Peters. and Prof. Cenap Ozel. My question is not about characterization of manifolds product but is about riemannian manifold M that splits as a product of two manifolds M_1 and M_2. Not all manifolds split or expressed as product of two subsets. This is what am looking for.
Dear Professor Shenawy,
Please search for your answer in the following references:
1. Bang Yen Chen: Differential geometry of semiring of immersions I: general theory, Bull. Inst. Mathemat. Academia Sinica, 21 (1993)
2. I. Mihai, L. Verstraelen : Introduction to tensor products of submanifolds, Geometry and Topology of Submanifolds VI, World Scientific Publ. 1994.
Thank you Professor Emilija Nesovic. The best answer for my question is de Rham decomposition theorem. But this theorem uses some difficult concepts to understand or consider in my topic. So I am looking foe conditions on the topology and curvature of the manifold.
I will consider these references then I will return to you. Thank you again.
A Riemannian manifold $M$ is a product (al least locally) if the tangent space $T_oM$ of a point $o$
admits a proper subbspace $V$ invariant w.r.t. the holonomy group $H$. In pseudo-Riemannian case you need also that that $V$ will be not degenerate. i.e.
$T_oM = V \oplus V^{\perp}$. The nesessary condition is that $V$ is invariant w.r.t.
the curvature operator $R(X,Y),\, X,Y \in T_oM$ and its covariant derivatives
$\nabla_X_1 \cdots \nable_{X_k}R(X,Y)$. In analytic case, this condition is also sufficient.
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