In the present article we initiate the study of submanifolds in normal complex contact metric manifolds. We define invariant and anti-invariant (
-totally real) submanifolds in such manifolds and start the study of their basic properties. Also, we establish the Chen first inequality and Chen inequality for the invariant
-totally real submanifolds in a normal complex contact space form and characterize the equality cases. We also prove the minimality of
-totally real submanifolds of maximum dimension satisfying the equalities.
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