Polar foliations and isoparametric maps

A singular Riemannian foliation F on a complete Riemannian manifold M is called a polar foliation if, for each regular point p, there is an immersed submanifold Sigma, called section, that passes through p and that meets all the leaves and always perpendicularly. A typical example of a polar foliation is the partition of M into the orbits of a polar action, i.e., an isometric action with sections. In this article we prove that the leaves of H : M -> Sigma, coincide with the level sets of a smooth map H: M -> Sigma, if M is simply connected. In particular, the orbits of a polar action on a simply connected space are level sets of an isoparametric map. This result extends previous results due to the author and Gorodski, Heintze, Liu and Olmos, Carter and West, and Terng.

Patterns of cell proliferation and apoptosis by topographic region in normal Bos taurus vs. Bos indicus crossbreeds bovine placentae during pregnancy

Background: Placental and fetal growth requires high rates of cellular turnover and differentiation, which contributes to conceptus development. The trophoblast has unique properties and a wide range of metabolic, endocrine and angiogenic functions, but the proliferative profile of the bovine placenta characterized by flow cytometry analysis and its role in fetal development are currently uncharacterized. Complete understanding of placental apoptotic and proliferative rates may be relevant to development, especially if related to the pathogenesis of pregnancy losses and placental abnormalities. Methods: In this study, the proliferation activity and apoptosis in different regions of normal bovine placenta (central and boundary regions of placentomes, placentomal fusion, microplacentomes, and interplacentomal regions), from distinct gestation periods (Days 70 to 290 of pregnancy), were analyzed by flow cytometry. Results: Our results indicated that microplacentomes presented a lower number of apoptotic cells throughout pregnancy, with a higher proliferative activity by the end of gestation, suggesting that such structures do not contribute significantly to normal of placental functions and conceptus development during pregnancy. The placentome edges revealed a higher number of apoptotic cells from Day 170 on, which suggests that placentome detachment may well initiate in this region. Conclusion: Variations involving proliferation and apoptotic rates may influence placental maturation and detachment, compromising placental functions and leading to fetal stress, abnormalities in development and abortion, as frequently seen in bovine pregnancies from in vitro fertilization and cloning procedures. Our findings describing the pattern of cell proliferation and apoptosis in normal bovine pregnancies may be useful for unraveling some of the developmental deviations seen in nature and after in vitro embryo manipulations.