S1-Bundles Over Surfaces

  • José María Montesinos-Amilibia
Part of the Universitext book series (UTX)


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.


Tangent Bundle Euler Number Tubular Neighbourhood Lens Space Klein Bottle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. HOPF, H.: Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche. Math. Ann. 104, 637–665 (1931)MathSciNetCrossRefGoogle Scholar
  2. SEIFERT, H.: [S]Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1987

Authors and Affiliations

  • José María Montesinos-Amilibia
    • 1
  1. 1.Facultad de MatemáticasUniversidad ComplutenseMadridSpain

Personalised recommendations