International Journal of Mathematics and Computer Science, 11(2016), no. 2, 151–172 M CS Riemannian foliation with dense leaves on a compact manifold Cyrille Dadi, Adolphe Codjia Fundamental Mathematics Laboratory University Felix Houphouet-Boigny , ENS 08 PO Box 10 Abidjan, Ivory Coast. email: cyriledadi@yahoo.fr, ad wolf2000@yahoo.fr (Received August 5, 2016, Accepted September 12, 2016) Abstract In this paper, we show that if G = Lie(G) is the Lie structural algebra of a Riemannian foliation with dense leaves (M, F ) on a com- pact manifold M, there exists a representation ρ : H 0 → Diff (V ) where V is an open subset of G such as: (a) There exists a biunivocal correspondence between the Lie subal- gebras of G invariant by Ad (ρa(v)) −1 .v for every (a, v) ∈ H 0 × V and F extensions. (b) An extension is a Lie foliation if the subalgebra corresponding is an ideal of G . (c) Every extension F ′ of F is a Riemannian foliation and there exists a common bundle-like metric for the foliations F and F ′ . (d) If F H is an extension of F corresponding to a subalgebra H of G , then to isomorphism nearly of Lie algebras we have ℓ(M, F H )= {u ∈H ⊥ /∀ (h, a, v) ∈ H×H 0 ×V ,[u, h]=0 and Ad (ρa(v)) −1 .v (u)= u}. 1 Introduction The purpose of this article is to generalize the following results in ([6], [8]) and [7] respectively to Riemannian foliations with dense leaves on a compact manifold: Key words and phrases: Lie foliation, Riemannian foliation, foliation with dense leaves, extension of a foliation. AMS (MOS) Subject Classifications: 53C05, 53C12, 58A30. ISSN 1814-0432, 2016, http://ijmcs.future-in-tech.net