A NOTE ON OPEN 3-MANIFOLDS SUPPORTING FOLIATIONS BY PLANES

We show that if N, an open connected n-manifold with finitely generated fundamental group, is C-2 foliated by closed planes, then pi(1)(N) is a free group. This implies that if pi(1)(N) has an abelian subgroup of rank greater than one, then F has at least a nonclosed leaf. Next, we show that if N is three dimensional with fundamental group abelian of rank greater than one, then N is homeomorphic to T-2 x R. Furthermore, in this case we give a complete description of the foliation.

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.

Magnetic fabrics and their relationship with the emplacement of the Piracaia pluton, SE Brazil

Magnetic fabric and rock-magnetism studies were performed on the four units of the 578 +/- 3-Ma-old Piracaia pluton (NW of Sao Paulo State, southern Brazil). This intrusion is roughly elliptical (similar to 32 km(2)), composed of (i) coarse-grained monzodiorite (MZD-c), (ii) fine-grained monzodiorite (MZD-f), which is predominant in the pluton, (iii) monzonite heterogeneous (MZN-het), and (iv) quartz syenite (Qz-Sy). Magnetic fabrics were determined by applying both anisotropy of low-field magnetic susceptibility (AMS) and anisotropy of anhysteretic remanent magnetization (AARM). The two fabrics are coaxial. The parallelism between AMS and AARM tensors excludes the presence of a single domain (SD) effect on the AMS fabric of the units. Several rock-magnetism experiments performed in one specimen from each sampled units show that for all of them, the magnetic susceptibility and magnetic fabrics are carried by magnetite grains, which was also observed in the thin sections. Foliations and lineations in the units were successfully determined by applying magnetic methods. Most of the magnetic foliations are steeply dipping or vertical in all units and are roughly parallel to the foliation measured in the field and in the country rocks. In contrast, the magnetic lineations present mostly low plunges for the whole pluton. However, for eight sites, they are steep up to vertical. Thin-section analyses show that rocks from the Piracaia pluton were affected by the regional strain during and after emplacement since magmatic foliation evolves to solid-state fabric in the north of the pluton, indicating that magnetic fabrics in this area of the pluton are related to this strain. Otherwise, the lack of solid-state deformation at outcrop scale and in thin sections precludes deformation in the SW of the pluton. This evidence allows us to interpret the observed magnetic fabrics as primary in origin (magmatic) acquired when the rocks were solidified as a result of magma flow, in which steeply plunging magnetic lineation suggests that a feeder zone could underlie this area.

ENERGY AND VOLUME OF VECTOR FIELDS ON SPHERICAL DOMAINS

We present a "boundary version" for theorems about minimality of volume and energy functionals on a spherical domain of an odd-dimensional Euclidean sphere.

Using AMS combined with mineral shape preferred orientation analysis to understand the emplacement fabrics of the Apiai gabbro-norite (Ribeira Belt, SE Brazil)

The Apiai gabbro-norite is a massive fine-grained Neoproterozoic intrusion emplaced in a core of synformal structure that deforms low-grade marine metasedimentary rocks of the Ribeira Belt of south-eastern Brazil. The lack of visible magmatic layering or any internal fabric has been a major limitation in deciding whether the emplacement occurred before or after the regional folding. To assist in the tectonic interpretations, we combine low-field anisotropy of magnetic susceptibility (AMS) and silicate shape preferred orientation (SPO) to reveal the internal structure of the mafic intrusion. Magnetic data indicate a mean susceptibility of about 10(-2) SI and a mean anisotropy degree (P) of about 1.08, essentially yielded by titanomagnetite. The magnetic and silicate foliations for P >= 1.10 are parallel to each other, while the lineations tend to scatter on the foliation plane, in agreement with the dominant oblate symmetry of the AMS and SPO ellipsoids. For lower P values, the magnetic and silicate fabrics vary from coaxial to oblique, and for P <= 1.05, their shapes and orientations can be quite distinct. The crystal size distribution (CSD) of plagioclase for P > 1.05 is log linear, in agreement with a bulk simple crystallisation history. These results combined show that for a strong SPO, corresponding to a magnetic anisotropy above 1.10, AMS is a reliable indicator of the magmatic fabric. They indicate that the Apiai gabbro-norite consists of sill-like body that was inclined gently to the north by the regional folding.