History of The Sphere Theorem
Hopf conjectured that a simply connected manifold with pinched sectional curvature is a sphere. In 1951, Harry Rauch showed that a simply connected manifold with curvature in is homeomorphic to a sphere. In 1960, Berger and Klingenberg proved the topological version of the sphere theorem with the optimal pinching constant.
Read more about this topic: Sphere Theorem
Famous quotes containing the words history of the, history of, history, sphere and/or theorem:
“Three million of such stones would be needed before the work was done. Three million stones of an average weight of 5,000 pounds, every stone cut precisely to fit into its destined place in the great pyramid. From the quarries they pulled the stones across the desert to the banks of the Nile. Never in the history of the world had so great a task been performed. Their faith gave them strength, and their joy gave them song.”
—William Faulkner (1897–1962)
“He wrote in prison, not a History of the World, like Raleigh, but an American book which I think will live longer than that. I do not know of such words, uttered under such circumstances, and so copiously withal, in Roman or English or any history.”
—Henry David Thoreau (1817–1862)
“While the Republic has already acquired a history world-wide, America is still unsettled and unexplored. Like the English in New Holland, we live only on the shores of a continent even yet, and hardly know where the rivers come from which float our navy.”
—Henry David Thoreau (1817–1862)
“Wherever the State touches the personal life of the infant, the child, the youth, or the aged, helpless, defective in mind, body or moral nature, there the State enters “woman’s peculiar sphere,” her sphere of motherly succor and training, her sphere of sympathetic and self-sacrificing ministration to individual lives.”
—Anna Garlin Spencer (1851–1931)
“To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
—Albert Camus (1913–1960)