There are some familiar definition about these two notions and also there are some different definitions about them. Could you advise any reliable references?
The following two books are good sources:
Bejancu, Aurel, Geometry of CR-submanifolds. D. Reidel Publishing Co., Dordrecht, 1986. MR0861408 (87k:53126)
Yano, Kentaro; Kon, Masahiro, Structures on manifolds.
World Scientific Publishing Co., Singapore, 1984. MR0794310 (86g:53001)
Traditionally, the name CR-submanifold is reserved for a Hermitian ambient, contact-CR or semi-invariant are the corresponding names in almost contact manifolds etc.
Contact CR-submanifolds and semi-invariant submanifolds in a Sasakian manifold N are different.
1) A submanifold M of N is called a contact CR-submanifold of a Sasakian manifold N (with structure vector field \xi) if the tangent bundle TM is the direct sum of an invariant subbundle D1 of TM and the orthogonal subbundle D2 of D1 which is anti-invariant. For such a submanifold \xi may lies in D1 or in D2.
2) A submanifold M of N is called a semi-invariant submanifold if TM is the direct sum of 3 sub bundles, namely, an invariant subbundle D1, an anti-invariant subbundle D2, and the line bundle spanned by \xi.
Clearly, for a semi-invariant submanifold, xi does not lie in D1 nor in D2.
This may be a long answer. In fact, the name "semi-invariant" seems to be more natural. A submanifold M of an almost Hermitian manifold should be called a semi-invariant submanifold (A. Bejancu (1978) calls it a CR-submanifold) if the tangent bundle TM is the direct sum of an invariant subbundle D1 of TM and the orthogonal subbundle D0 of D1 which is anti-invariant. A well known result of Blair and Chen (1979) shows that a semi-invariant submanifold (which they call CR-submanifold) of a Hermitian manifold carries a CR-structure. I gave an example (1994), where the ambient manifold is not Hermitian but semi-invariant submanifold still carries a CR-structure. I am sure that one can find an example, where the ambient manifold is just an almost Hermitian manifold and its semi-invariant submanifold does not carry a CR-structure (D1,J). Later Chen gave another nice generalization of an invariant submanifold namely, slant submanifold. In his monograpgh (1990) slant submnifolds got big attention. Also the monograph and another paper by Ronsse include the concept of generic submanifold of an almost Hermitian manifold. This generic submanifold contains most of the classes of submanifold of an almost Hermitian manifold, namely slant, semi-slant, anti-slant, hemi-slant, pointwise slant, pointwise semi-slant etc given afterwards. I also studied generic submanifolds of an almost Hermitian manifold in 1997 and also gave example of a proper generic submanifold and a skew-CR submanifold. I think Ronsse knew his theory in 1985. to be continued...
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