Poincaré–Hopf theorem

The set of Poincaré - Hopf is an important mathematical theorem of differential topology. He is also known as the Poincaré -Hopf index formula, Poincaré -Hopf index theorem or Hopf index theorem. The set is named after Henri Poincaré and Heinz Hopf. For two dimensions, the statement of Poincaré was proved and later generalized by Hopf in higher dimensions. Often the special case of the theorem is used by hedgehogs as an illustration of the statement.

Index of a vector field

Is a vector field and a sphere with a small radius to a singular point, so that the following applies. The index of the vector field at the point is the mapping degree of the mapping

And is quoted at. This definition can be generalized to manifolds as follows. Is one -dimensional differentiable manifold and a vector field, then choose a card, so that is true. Then the above definition of the indicator can be transferred to the in -lying area maps, and which proves to be independent of the choice of the map.

Set of Poincaré

The sake of completeness is presented first found the statement by Henri Poincaré in 1881. Let be a compact surface with induced metric. It should also be a smooth vector field with a finite number of isolated singular points. Then we have

The Euler characteristic denoted by. That is to say, the Euler characteristic of being equal to the sum of the indices of all isolated from singular points.

Set of Poincaré - Hopf

The set of Poincaré - Hopf was proved in 1926 by Hopf as a generalization of the theorem of Poincaré. Let be a compact differentiable manifold and let a vector field which has only finitely many isolated zeros. Then we have

Has an edge, it must point to the edge in the direction of the outward normal.