Immersions into Sasakian space forms (Q6564154)

A Sasakian manifold can be shortly described as a contact manifold \(M\) that is endowed with a compatible Riemannian metric such that its Riemannian cone \(M\times \mathbb{R}_{>0}\) is a Kähler manifold. In this paper, the authors study immersions of Sasakian manifolds into Sasakian space forms. The analogous problem in the Kähler case has a long tradition starting from \textit{E. Calabi} [Ann. Math. (2) 58, 1--23 (1953; Zbl 0051.13103)]. Calabi's pioneering work allowed him to obtain necessary and sufficient conditions for a neighborhood of a point to be locally Kähler immersed into a finite or infinite-dimensional complex space form. This led to a classification of finite-dimensional complex space forms admitting a Kähler immersion into another. The Sasakian version of this problem had received little attention until recently when it was studied in several papers, see [\textit{G. Bande} et al., Ann. Mat. Pura Appl. (4) 199, No. 6, 2117--2124 (2020; Zbl 1453.53051); \textit{B. Cappelletti-Montano} and \textit{A. Loi}, Ann. Mat. Pura Appl. (4) 198, No. 6, 2195--2205 (2019; Zbl 1427.53053); \textit{G. Placini}, J. Geom. Phys. 166, Article ID 104265, 7 p. (2021; Zbl 1468.53053)]. \N\NFirst, the authors prove a rigidity result, as well as an extension result for local immersions for simply-connected Sasakian manifolds whose proof follows Calabi's ideas. Given a Sasakian manifold, deciding whether it admits a Sasakian embedding or immersion into a Sasakian space form is a highly nontrivial problem. That is why they consider the special case of locally homogeneous Sasakian manifolds. They characterize all homogeneous Sasakian manifolds that admit a local Sasakian immersion into a non-elliptic Sasakian space form. Moreover, they give a characterization of homogeneous Sasakian manifolds which can be embedded into the standard sphere both in the compact and non-compact case.