Homology of Fibrations
In 1935, at the instigation of Hopf, his student Stiefel undertook in his dissertation  to extend Hopf’s work on vector fields (Part 2, chap. Ill, §3). Given an n-dimensional compact C∞ manifold M, the problem was to investigate whether there exists on M, not only one nowhere vanishing vector field, but a system of m vector fieldsXj (1≤j≤m) for some m≤n, subject to the condition that at each point x ∈ M, the m tangent vectors Xj(x) are linearly independent (hence≠0). The case m = n is particularly interesting in differential geometry, because the existence of such systems of n vector fields is equivalent to the existence of a parallelism on M.
KeywordsVector Field Differential Geometry Tangent Vector Algebraic Topology Differential Topology
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