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It seems that there are two common meanings for a submanifold of an almost-complex Riemannnian manifold to be "totally real": one says that the almost-complex structure takes the tangent space into the normal space, and the other says that the image of the tangent space under the almost-complex structure has trivial intersection with the tangent space.

The former definition appears in section 16 of Bang-Yen Chen's survey "Riemannian submanifolds" and in chapter 6 of Krishan Duggal and Bayram Sahin's book "Differential geometry of lightlike submanifolds"

The latter definition appears in Jason Lotay and Tommaso Pacini's article "Complexified diffeomorphism groups, totally real submanifolds and Kähler-Einstein geometry" and in section 2.3 of Gromov's "Partial differential relations" and his "Pseudo holomorphic curves in symplectic manifolds."

Is one of these more standard than the other? And under the former definition, is there any specific language to refer to the latter definition, perhaps as a "real submanifold"?

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  • $\begingroup$ Nice question Bendir. I wonder what's the idea behind using the terminology "totally real". What's the motivation and intuition behind sending the tangent space to its orthogonal complement by the complex structure, call it $J$? What idea is it capturing? Is this another way Lagrangian manifolds arise? $\endgroup$
    – Rachid Atmai
    Commented Nov 26, 2020 at 19:09
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    $\begingroup$ I've always thought of it as "totally real" = "not at all complex". $\endgroup$
    – Paul Reynolds
    Commented Nov 26, 2020 at 19:11
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    $\begingroup$ The second definition does not use the Riemannian metric, so it is the only definition you can use if you want to study almost complex manifolds not arising equipped with a Riemannian metric. I have only ever made use of the second definition, for that reason. $\endgroup$
    – Ben McKay
    Commented Nov 26, 2020 at 19:14
  • $\begingroup$ The first definition is for a manifold with a Riemannian metric and a compatible almost complex structure, which means that $J: T_xM \rightarrow T_xM$ is orthogonal. The second is for any almost complex manifold. The former implies the latter. $\endgroup$
    – Deane Yang
    Commented Nov 26, 2020 at 19:42
  • $\begingroup$ If the metric is almost Kahler then requiring J TL to be orthogonal to TL is actually the Lagrangian condition. The standard usage is that J TL is a complement to TL (i.e. what Gromov said). As Deane Yang says, this makes sense even without a metric, and the concept actually originated in complex geometry (no metric in sight). $\endgroup$
    – Jonny Evans
    Commented Nov 26, 2020 at 20:07

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