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In mathematics, Hopf conjecture may refer to one of several conjectural statements from differential geometry and topology attributed to Heinz Hopf.
In mathematics, the Cartan–Hadamard theorem is a statement in Riemannian geometry concerning the structure of complete Riemannian manifolds of non-positive sectional curvature. The theorem states that the universal cover of such a manifold is diffeomorphic to a Euclidean space via the exponential map at any point. It was first proved by Hans Carl Friedrich von Mangoldt for surfaces in 1881, and independently by Jacques Hadamard in 1898. Élie Cartan generalized the theorem to Riemannian manifolds in 1928. The theorem was further generalized to a wide class of metric spaces by Mikhail Gromov in 1987; detailed proofs were published by Ballmann (1990) for metric spaces of non-positive curvature and by Alexander & Bishop (1990) for general locally convex metric spaces.
In mathematics, a Riemannian manifold is said to be flat if its Riemann curvature tensor is everywhere zero. Intuitively, a flat manifold is one that "locally looks like" Euclidean space in terms of distances and angles, e.g. the interior angles of a triangle add up to 180°.
In mathematics, spaces of non-positive curvature occur in many contexts and form a generalization of hyperbolic geometry. In the category of Riemannian manifolds, one can consider the sectional curvature of the manifold and require that this curvature be everywhere less than or equal to zero. The notion of curvature extends to the category of geodesic metric spaces, where one can use comparison triangles to quantify the curvature of a space; in this context, non-positively curved spaces are known as (locally) CAT(0) spaces.
In Riemannian geometry, Cheng's eigenvalue comparison theorem states in general terms that when a domain is large, the first Dirichlet eigenvalue of its Laplace–Beltrami operator is small. This general characterization is not precise, in part because the notion of "size" of the domain must also account for its curvature. The theorem is due to Cheng (1975b) by Shiu-Yuen Cheng. Using geodesic balls, it can be generalized to certain tubular domains.
In mathematics, the Almgren–Pitts min-max theory is an analogue of Morse theory for hypersurfaces.
References
- do Carmo, Manfredo Perdigão (1992), Riemannian geometry, Mathematics: Theory & Applications, Boston, MA: Birkhäuser Boston, ISBN 978-0-8176-3490-2, MR 1138207 CS1 maint: discouraged parameter (link)
- Synge, John Lighton (1936), "On the connectivity of spaces of positive curvature", Quarterly Journal of Mathematics (Oxford Series), 7: 316–320, doi:10.1093/qmath/os-7.1.316 CS1 maint: discouraged parameter (link).
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