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

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

The 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...

thus the traces 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...

the Euler characteristic of the manifold. This theorem is now called the Poincaré–Hopf theorem. Hopf spent the year after his doctorate at the University...

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

In the mathematics of dynamical systems and differential equations, a Hopf bifurcation is said to occur when varying a parameter of the system causes the...

of Lagrange's theorem for subgroups. As a corollary of this and integral theory, a Hopf subalgebra of a semisimple finite-dimensional Hopf algebra is automatically...

Famous theorems in differential topology include the Whitney embedding theorem, the hairy ball theorem, the Hopf theorem, the Poincaré–Hopf theorem, Donaldson's...

there must be at least one zero. This is a consequence of the Poincaré–Hopf theorem. In the case of the torus, the Euler characteristic is 0; and it is possible...

the Hopf invariant is a homotopy invariant of certain maps between n-spheres. In 1931 Heinz Hopf used Clifford parallels to construct the Hopf map η...

mathematics, especially algebraic topology and homotopy theory, the Hopf–Whitney theorem is a result relating the homotopy classes between a CW complex and...

Myers) and cyclic groups of order 3 (J. Hyam Rubinstein). Killing–Hopf theorem Hopf, Heinz (1926), "Zum Clifford-Kleinschen Raumproblem", Mathematische...

when f has no fixed points, the Lefschetz–Hopf theorem trivially implies the Lefschetz fixed-point theorem. A. Katok and B. Hasselblatt(1995), Introduction...

R {\displaystyle \mathbb {R} } . The Hopf–Rinow theorem characterizes geodesically complete manifolds. Theorem: Let ( M , g ) {\displaystyle (M,g)} be...

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...

compactness theorem (symplectic topology) Gromov–Ruh theorem (differential geometry) Hilbert's theorem (differential geometry) Hopf–Rinow theorem (differential...

The theorem is also sometimes known as the Carathéodory–Fréchet extension theorem, the Carathéodory–Hopf extension theorem, the Hopf extension theorem and...

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

Poincaré–Bendixson theorem: a statement about the long-term behaviour of orbits of continuous dynamical systems on the plane, cylinder, or two-sphere. Poincaré–Hopf theorem:...

is known. The fundamental theorem of algebra is a corollary of Hopf's theorem. Dropping the requirement of commutativity, Hopf generalized his result: Any...

geodesic flow on Riemannian manifolds, starting with the results of Eberhard Hopf for Riemann surfaces of negative curvature. Markov chains form a common context...

version of the theorem has a much shorter proof, relying on a key theorem from differential topology known as the Poincaré–Hopf theorem. It can be summarized...

global pinching. This result is known as the differentiable sphere theorem. Heinz Hopf conjectured that a simply connected manifold with pinched sectional...

Wiener–Hopf method is a mathematical technique widely used in applied mathematics. It was initially developed by Norbert Wiener and Eberhard Hopf as a method...

embedding theorem Critical value Sard's theorem Saddle point Morse theory Lie derivative Hairy ball theorem Poincaré–Hopf theorem Stokes' theorem De Rham...

properties are isometric to each other. It is a consequence of the Killing–Hopf theorem. To prove the existence of such a space as described above one can explicitly...

algebra, the Milnor–Moore theorem, introduced by John W. Milnor and John C. Moore (1965) classifies an important class of Hopf algebras, of the sort that...

path-connected as well, providing a homotopy to the identity map. The Lefschetz–Hopf theorem states that the sum of the indices (in this context, multiplicity) of...

{\displaystyle T_{p}M} , the entire tangent space at p {\displaystyle p} . The Hopf–Rinow theorem gives alternative characterizations of completeness. Let ( M , g )...