Totally real submanifolds in 𝑆⁶(1) satisfying Chen’s equality

Dillen, Franki; Vrancken, Luc

doi:10.1090/S0002-9947-96-01626-1

Totally real submanifolds in $S^6(1)$ satisfying Chen’s equality
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by Franki Dillen and Luc Vrancken

Trans. Amer. Math. Soc. 348 (1996), 1633-1646

DOI: https://doi.org/10.1090/S0002-9947-96-01626-1

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Abstract:

In this paper, we study 3-dimensional totally real submanifolds of $S^{6}(1)$. If this submanifold is contained in some 5-dimensional totally geodesic $S^{5}(1)$, then we classify such submanifolds in terms of complex curves in $\mathbb {C}P^{2}(4)$ lifted via the Hopf fibration $S^{5}(1)\to \mathbb {C}P^{2}(4)$. We also show that such submanifolds always satisfy Chen’s equality, i.e. $\delta _{M} = 2$, where $\delta _{M}(p)=\tau (p)-\inf K(p)$ for every $p\in M$. Then we consider 3-dimensional totally real submanifolds which are linearly full in $S^{6}(1)$ and which satisfy Chen’s equality. We classify such submanifolds as tubes of radius $\pi /2$ in the direction of the second normal space over an almost complex curve in $S^{6}(1)$.

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