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Let Ω be a smooth real analytic submanifold of a complex manifold X. We establish and study the link between the following 3 subjects: 1) topological properties of smooth families of attached analytic discs, the manifold Ω admits, 2) lower bounds for dimensions of complex tangent spaces of Ω, 3) a generalization of the argument principle for smooth families of holomorphic mappings from the standard complex disc to X. In particular, we obtain characterization of complex manifolds and theirarXiv:0704.2871v2 fatcat:bh5cei7tm5g3jcksakxnmgqj6m