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In this article we show how the classical theory of Reeb and others extends to the case of codimension one singular foliations on closed oriented manifolds, provided that at the singular set sing(F), the foliation is locally defined by Bott-Morse functions which are transversely centers. We prove, in this setting, the equivalent of the local and the complete stability theorems of Reeb. We show that if F has a compact leaf with finite fundamental group, or if a component of sing(F) has codimension ≥ 3 and finite fundamental group, then all leaves of F are compact and diffeomorphic, sing(F) consists of two connected components, and there is a Bott-Morse function f : M → [0, 1] such that f : M \ sing(F) → (0, 1) is a fiber bundle defining F and sing(F) = f.1({0, 1}). This yields a topological description of the type of leaves that appear in these foliations, and also the type of manifolds admitting such foliations. These results unify and generalize well known results for cohomogeneity one isometric actions, and a theorem of Reeb for foliations with Morse singularities of center type.

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In this paper we study the singularities of holomorphic functions of bicomplex variables introduced by G. B. Price ($An Introduction to Multicomplex Spaces and Functions$, Dekker, New York,1991). In particular, we use computational algebra techniques to show that even in the case of one bicomplex variable, there cannot be compact singularities. The same techniques allow us to prove a duality theorem for such functions.

There have been many attempts to define the notion of quasilocal mass for a spacelike 2-surface in spacetime by the Hamilton-Jacobi analysis. The essential difficulty in this approach is to identify the right choice of the background configuration to be subtracted from the physical Hamiltonian. Quasilocal mass should be nonnegative for surfaces in general spacetime and zero for surfaces in flat spacetime. In this letter, we propose a new definition of gauge-independent quasilocal mass and prove that it has the desired properties.