S¹-Bundles Over Surfaces

Abstract

A closed surface is a compact, connected 2-manifold without boundary. From the tangent bundle TX of a closed surface X we can construct the spherical (or unit) tangent bundle of X, denoted ST(X), as the subbundle of TX consisting of vectors of norm 1 (see [GP], page 55). The fiber of ST(X) is the 1-sphere S¹, and thus ST(X) is a closed 3-manifold, which always has a canonical orientation, even when the base X of the bundle is non-orientable (see [GP], pp. 76 and 106). These S¹-bundles were known by the suggestive name of “bundles of oriented line elements” (see [ST]). The “bundles of unoriented line elements” of X, denoted by PT(X), are obtained from ST(X) by identifying the vectors (x, v) and (x, −v), for every (x, v) E ST(X). The S¹-bundle PT(X), which is also called projective tangent bundle, is a canonically oriented, closed 3-manifold and the natural map ST(X)→PT(X) is a 2-fold covering.

“Cap. LXXVII, Donde se cuentan mil zarandajas, tan impertinentes, como necesarias al verdadero entendimiento de esta grande Historia.”

“Chapter LXXVII. In which a Thousand Trifles are recounted, as nonsensical as they are necessary to the True Understanding of this great History.” Cervantes, Don Quixote