Journal of the
Korean Mathematical Society JKMS

We study an $(n+3)(n\geq 7)$-dimensional real submanifold of a $(4m+3)$-unit sphere $S^{4m+3}$ with Sasakian 3-structure induced from the canonical quaternionic K\"ahler structure of quaternionic $(m+1)$-number space $Q^{m+1}$, and especially determine contact three $CR$-submanifolds with $(p-1)$ contact three $CR$-dimension under the equality conditions given in $(4.1)$, where $p=4m-n$ denotes the codimension of the submanifold. Also we provide necessary conditions concerning sectional curvature in order that a compact contact three $CR$-submanifold of $(p-1)$ contact three $CR$-dimension in $S^{4m+3}$ is the model space $S^{4n_1+3}(r_1) \times S^{4n_2+3}(r_2)$ for some portion $(n_1, n_2)$ of $(n-3)/4$ and some $r_1, r_2$ with $r_1^2+r_2^2=1$.

Keywords: contact three $CR$-submanifold, contact three $CR$-dimension, Sasakian 3-structure.

MSC numbers: 53C40, 53C15