Parallel Locally Strictly Convex Surfaces in Four-Dimensional Affine Space Contained in Hyperquadrics

Locally strictly convex surfaces in four-dimensional affine space are studied from a perspective of the affine structure invented by Nuño-Ballesteros and Sánchez, which is especially suitable in convex geometry. The surfaces that are embedded in locally strictly convex hyperquadrics are classified under assumptions that the second fundamental form is parallel with respect to the induced connection and the normal connection is compatible with a metric on the transversal bundle. Both connections are induced by a canonical transversal plane bundle, which is defined by certain symmetry conditions. The obtained surfaces are always products of an ellipse and a conical planar curve.

Mediastinal Veins and Anomalies

The chapter titled abnormalities of systemic veins discusses a series of mediastinal venous anomalies and tributaries that may produce confusing imaging findings. Such findings may be seen on a radiograph that shows an abnormal course of a central line or catheter in one of these anomalous vessels. The more frequently encountered systemic vein variants are presented, namely left superior vena cava (SVC) and azygous continuation of an interrupted inferior vena cava (IVC). In these conditions the normal connection of a right-sided vena cava to right atrium may be altered. Commonly encountered venous tributaries, including the left superior intercostal vein (which produces the aortic nipple on radiography), the left pericardiophrenic vein, and systemic to pulmonary vein collaterals will also be included. The latter may result in right-to-left shunting and cyanosis.

ANASTOMOSIS ENTRE LA RAMA PROFUNDA DEL NERVIO CUBITAL Y EL NERVIO MEDIANO EN LA MANO. Anastomosis between the deep branch of the ulnar nerve and the median nerve in the hand

Introducción: La anastomosis de Riche-Cannieu (ARC) es una variación anatómica formada entre la rama tenar del nervio mediano (NM) y la rama profunda del nervio cubital (NC). Debido a la importancia clínica y electromiográfica su descripción anatómica es de gran interés, ya que debido a esta variación anatómica existen distintas formas de inervación motora a nivel de la mano. Materiales y Métodos: Se realizaron disecciones cadavéricas en 38 manos (19 cadáveres) de ambos sexos formolizados en solución al 5 %, de entre 50 y 70 años de edad. Se utilizó instrumental y técnicas convencionales de disección. Resultados: En la rama profunda del NC no se evidenciaron variaciones y finalizaba su recorrido en el músculo aductor del pulgar. En el 86,84% de los casos emerge una rama que se anastomosa con el NM de diferentes formas. Esta rama anastomótica, en el 50% de las manos, era una arcada nerviosa de considerable calibre entre el NC y NM, que daba ramas motoras a los músculos de la eminencia tenar. Discusión: El conocimiento de esta anastomosis es muy importante ya que, en casos de lesión del nervio mediano o cubital, puede causar confusión clínica, quirúrgica y en los hallazgos electromiográficos. Debido a su alta frecuencia fue considerada un rasgo anatómico normal. Introduction: The Riche-Cannieu anastomosis (RCA) is an anatomic variation formed between the thenar branch of the median nerve and the deep branch of the ulnar nerve. Its anatomical description is of great interest because of its clinical and electromyographic relevance. Due to the RCA, there are various types of hand motor innervation. Materials and Methods: Thirty eight hands from 19 corpses (formolized in a 5% solution) whose ages ranged from 50 to 70 years old were dissected. Conventional instruments and techn-iques were used. Results: The pathway of the deep branch of the ulnar nerve did not show variations and ended at the adductor pollicis muscle. In 86.84% of the cases a branch emerged there and anastomosed the median nerve by different modalities. A nervous arcade of considerable important size was observed between the ulnar and median nerve in 50 % of the anastomosis and it gave motor branches for the thenar muscles. Discussion: It is of great importance to learn about this anastomosis because of the possibility of median or ulnar nerve injuries; it may cause clinical, surgical or electromyographic confusion. Due to its high frequency, it was regarded as a normal connection.

PINCHING THEOREMS FOR A COMPACT MINIMAL SUBMANIFOLD IN A COMPLEX PROJECTIVE SPACE

We give a formula for the Laplacian of the second fundamental form of an n-dimensional compact minimal submanifold M in a complex projective space CPm. As an application of this formula, we prove that M is a geodesic minimal hypersphere in CPm if the sectional curvature satisfies K≥1/n, if the normal connection is flat, and if M satisfies an additional condition which is automatically satisfied when M is a CR submanifold. We also prove that M is the complex projective space CPn/2 if K≥3/n, and if the normal connection of M is semi-flat.

A characterization of harmonic foliations by the volume preserving property of the normal geodesic flow

We prove that a Riemannian foliation with the flat normal connection on a Riemannian manifold is harmonic if and only if the geodesic flow on the normal bundle preserves the Riemannian volume form of the canonical metric defined by the adapted connection.

Submanifolds with Nonparallel First Normal Bundle

We provide a complete local geometric description of submanifolds of spaces with constant sectional curvature where the first normal spaces, that is, the subspaces spanned by the second fundamental form, form a vector subbundle of the normal bundle of low rank which is nonparallel in the normal connection. We also characterize flat submanifolds with flat normal bundle in Euclidean space satisfying the helix property.