In complex geometry, a Hopf manifold (Hopf 1948) is obtained as a quotient of the complex vector space (with zero deleted) ( C n ∖ 0 ) {\displaystyle...

The concept is due to Charles Ehresmann and Heinz Hopf in the 1940s. Let M be a smooth manifold. An almost complex structure J on M is a linear complex...

mathematics, the Poincaré–Hopf theorem (also known as the Poincaré–Hopf index formula, Poincaré–Hopf index theorem, or Hopf index theorem) is an important...

Heinz Hopf (19 November 1894 – 3 June 1971) was a German mathematician who worked on the fields of dynamical systems, topology and geometry. Hopf was born...

of Kähler manifolds include smooth projective varieties and more generally any complex submanifold of a Kähler manifold. The Hopf manifolds are examples...

The Hopf–Rinow theorem characterizes geodesically complete manifolds. Theorem: Let ( M , g ) {\displaystyle (M,g)} be a connected Riemannian manifold. The...

mathematical physics, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere...

with no Kähler metric. Higher-dimensional analogues of Hopf surfaces are called Hopf manifolds. Hopf surfaces are surfaces of class VII and in particular...

The Hopf–Rinow theorem gives alternative characterizations of completeness. Let ( M , g ) {\displaystyle (M,g)} be a connected Riemannian manifold and...

mathematical field of differential topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional...

Hopf–Rinow theorem is a set of statements about the geodesic completeness of Riemannian manifolds. It is named after Heinz Hopf and his student Willi...

complex manifolds Enriques–Kodaira classification GAGA Hartogs' extension theorem Hermitian symmetric space Hodge decomposition Hopf manifold Imaginary...

compact Kähler manifold is even, by Hodge symmetry. This is not true for compact complex manifolds in general, as shown by the example of the Hopf surface,...

goes back to questions of Heinz Hopf from 1931. A modern formulation is: A compact, even-dimensional Riemannian manifold with positive sectional curvature...

In geometry, the Killing–Hopf theorem states that complete connected Riemannian manifolds of constant curvature are isometric to a quotient of a sphere...

mathematics, particularly differential geometry, a Finsler manifold is a differentiable manifold M where a (possibly asymmetric) Minkowski norm F(x, −) is...

In mathematical knot theory, the Hopf link is the simplest nontrivial link with more than one component. It consists of two circles linked together exactly...

Cartan–Hadamard manifolds are diffeomorphic to the Euclidean space R n . {\displaystyle \mathbb {R} ^{n}.} Furthermore it follows from the Hopf–Rinow theorem...

as in classic Hopf bifurcation. Invariant manifold Stable manifold Lagrangian coherent structure Normally hyperbolic invariant manifold Roberts, A.J....

The Hopf theorem (named after Heinz Hopf) is a statement in differential topology, saying that the topological degree is the only homotopy invariant of...

combinatorial topology, appeared in the book of Pavel Alexandrov and Heinz Hopf, Topologie I (1935). Rosenfeld et al. proposed digital connectivity such...

are all zero. A stronger form of the theorem, also known as the Lefschetz–Hopf theorem, states that, if f {\displaystyle f} has only finitely many fixed...

structure of a Z2-graded Hopf algebra. Likewise the representations of this Hopf algebra turn out to be Z2-graded comodules. This Hopf algebra gives the global...

bundles, topological fibrations and fibered manifolds are a special case, is attributed to Seifert, Heinz Hopf, Jacques Feldbau, Whitney, Norman Steenrod...

decomposition Schouten tensor Weyl curvature Ricci flow Einstein manifold Holonomy Gauss–Bonnet theorem Hopf–Rinow theorem Cartan–Hadamard theorem Myers theorem Rauch...

field on a compact manifold with finitely many zeroes, the Poincaré-Hopf theorem states that the vector field’s index is the manifold’s Euler characteristic...

spinC bundle. By a theorem of Hopf and Hirzebruch, closed orientable 4-manifolds always admit a spinC structure. When a manifold carries a spinC structure...

Riemannian manifolds. The sectional curvature K(σp) depends on a two-dimensional linear subspace σp of the tangent space at a point p of the manifold. It can...

dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related...

3-dimensional "surface"). Topologically, a 3-sphere is an example of a 3-manifold, and it is also an n-sphere. In coordinates, a 3-sphere with center (C0...