We prove some structure results for transversely reducible Sasaki manifolds. In particular, we show a Sasaki manifold with positive Ricci curvature is transversely irreducible, and so join (product) construction cannot be generalized to irregular SasakiEinstein manifolds, as opposed to the quasi-regular case done by WangZiller and BoyerGalicki. As an application, we classify compact Sasaki manifolds with nonnegative transverse bisectional curvature, which can be viewed as the generalized Frankel conjecture in Sasaki geometry.

We use adiabatic limits to study foliated manifolds. The Bott connection naturally shows up as the adiabatic limit of Levi-Civita connections. As an application, we then construct certain natural elliptic operators associated to the foliation and present a direct geometric proof of a vanshing theorem of Connes [Co], which extends the Lichnerowicz vanishing theorem [L] to foliated manifolds with spin leaves, for what we call almost Riemannian foliations. Several new vanishing theorems are also proved by using our method.

In this companion piece to [9], some variations on the main results there are sketched. In particular, • the recursions in [9], which we interpreted as the quantum Lefschetz, is re- formulated in terms of Givental’s quantization formalism, or equivalently, a summation of finitely many graphs; • varietiesofmodificationoftheauxilliaryspaces(masterspaces)forthefixed point localization are given, leading to different (looking) recursions; • applications of this circle of ideas to derive (apparently) new relations of Gromov–Witten invariants.

Motivated by the study of coupled Kähler-Einstein metrics by Hultgren and Witt Nyström (2018) and coupled Kähler-Ricci solitons by Hultgren (2017), we study in this paper coupled Sasaki-Einstein metrics and coupled Sasaki-Ricci solitons. We first show an isomorphism between the Lie algebra of all transverse holomorphic vector fields and certain space of coupled basic functions related to coupled twisted Laplacians for basic functions, and obtain extensions of the well-known obstructions to the existence of Kähler-Einstein metrics to this coupled case. These results are reduced to coupled Kähler-Einstein metrics when the Sasaki structure is regular. Secondly we show the existence of toric coupled Sasaki-Einstein metrics when the basic first Chern class is positive extending the work of Hultgren (2017).

We prove several vanishing theorems for a class of generalized elliptic genera on foliated manifolds, by using classical equivariant index theory. The main techniques are the use of the Jacobi theta-functions and the construction of a new class of elliptic operators associated to foliations.