In mathematics, Hopf conjecture may refer to one of several conjectural statements from differential geometry and topology attributed to Heinz Hopf.
Positively or negatively curved Riemannian manifolds The Hopf conjecture is an open problem in global Riemannian geometry. It goes back to questions of Heinz Hopf from 1931. A modern formulation is:
A compact, even-dimensional Riemannian manifold with positive sectional curvature has positive Euler characteristic. A compact, (2d)-dimensional Riemannian manifold with negative sectional curvature has Euler characteristic of sign ( − 1 ) d {\displaystyle (-1)^{d}} . For surfaces, these statements follow from the Gauss–Bonnet theorem. For four-dimensional manifolds, this follows from the finiteness of the fundamental group and Poincaré duality and Euler–Poincaré formula equating for 4-manifolds the Euler characteristic with b 0 − b 1 + b 2 − b 3 + b 4 {\displaystyle b_{0}-b_{1}+b_{2}-b_{3}+b_{4}} and Synge's theorem, assuring that the orientation cover is simply connected, so that the odd Betti numbers vanish b 1 = b 3 = 0 {\displaystyle b_{1}=b_{3}=0} . For 4-manifolds, the statement also follows from the Chern–Gauss–Bonnet theorem as noticed by John Milnor in 1955 (written down by Shiing-Shen Chern in 1955.). For manifolds of dimension 6 or higher the conjecture is open. An example of Robert Geroch had shown that the Chern–Gauss–Bonnet integrand can become negative for d > 2 {\displaystyle d>2} . The positive curvature case is known to hold however for hypersurfaces in R 2 d + 1 {\displaystyle \mathbb {R} ^{2d+1}} (Hopf) or codimension two surfaces embedded in R 2 d + 2 {\displaystyle \mathbb {R} ^{2d+2}} . For sufficiently pinched positive curvature manifolds, the Hopf conjecture (in the positive curvature case) follows from the sphere theorem, a theorem which had also been conjectured first by Hopf. One of the lines of attacks is by looking for manifolds with more symmetry. It is particular for example that all known manifolds of positive sectional curvature allow for an isometric circle action. The corresponding vector field is called a killing vector field. The conjecture (for the positive curvature case) has also been proved for manifolds of dimension 4 k + 2 {\displaystyle 4k+2} or 4 k + 4 {\displaystyle 4k+4} admitting an isometric torus action of a k-dimensional torus and for manifolds M admitting an isometric action of a compact Lie group G with principal isotropy subgroup H and cohomogeneity k such that
k − ( rank G − rank H ) ≤ 5. {\displaystyle k-(\operatorname {rank} G-\operatorname {rank} H)\leq 5.} Some references about manifolds with some symmetry are and On the history of the problem: the first written explicit appearance of the conjecture is in the proceedings of the German Mathematical Society, which is a paper based on talks, Heinz Hopf gave in the spring of 1931 in Fribourg, Switzerland and at Bad Elster in the fall of 1931. Marcel Berger discusses the conjecture in his book, and points to the work of Hopf from the 1920s which was influenced by such type of questions. The conjectures are listed as problem 8 (positive curvature case) and 10 (negative curvature case) in "Yau's problems" of 1982.
Non-negatively or non-positively curved Riemannian manifolds There are analogue conjectures if the curvature is allowed to become zero too. The statement should still be attributed to Hopf (for example in a talk given in 1953 in Italy).
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