Hopf conjecture [SEP] named after
Hopf conjecture
In mathematics, Hopf conjecture may refer to one of several conjectural statements from differential geometry and topology attributed to either Eberhard Hopf or Heinz Hopf.
Positively curved Riemannian manifolds.
For surfaces, this follows from the Gauss–Bonnet theorem. For four-dimensional manifolds, this follows from the finiteness of the fundamental group and Poincaré duality. The conjecture has been proved for manifolds of dimension 4"k"+2 or 4"k"+4 admitting an isometric torus action of a "k"-dimensional torus and for manifolds "M" admitting an isometric action
Seifert conjecture
In mathematics, the Seifert conjecture states that every nonsingular, continuous vector field on the 3-sphere has a closed orbit. It is named after Herbert Seifert. In a 1950 paper, Seifert asked if such a vector field exists, but did not phrase non-existence as a conjecture. He also established the conjecture for perturbations of the Hopf fibration.
The conjecture was disproven in 1974 by Paul Schweitzer, who exhibited a formula_1 counterexample. Schweitzer's construction was then modified by Jenny Harrison in 1988 to make a