Poincaré-Hopf theorem

The Poincaré – Hopf theorem is an important mathematical theorem of differential topology . It 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 . The statement by Poincaré was proven for two dimensions and later generalized by Hopf for higher dimensions. Often the special case of the sentence about the hedgehog is used as an illustration of the statement.

Index of a vector field

Let be a vector field and an isolated zero, that is, there is a closed ball around with . The index of the vector field at the point is the degree of mapping of the image ${\ displaystyle X \ colon M \ subset \ mathbb {R} ^ {n} \ to \ mathbb {R} ^ {n}}$ ${\ displaystyle z}$ ${\ displaystyle Q}$ ${\ displaystyle z}$ ${\ displaystyle X | _ {Q \ setminus \ {z \}} \ neq 0}$ ${\ displaystyle z}$

{\ displaystyle f \ colon \ partial Q \ to \ mathbb {S} ^ {n-1}, \ qquad x \ mapsto {\ frac {X (x)} {\ | X (x) \ |}}.}

and is noted with . This definition can be generalized to manifolds as follows . Is one -dimensional differentiable manifold and a vector field, then choose a card to , so true. Then the above definition of the index can be transferred to the map area in the area, and that turns out to be independent of the choice of map. ${\ displaystyle \ operatorname {ind} (X, z)}$ ${\ displaystyle M}$ ${\ displaystyle n}$ ${\ displaystyle X \ in \ Gamma (TM)}$ ${\ displaystyle (U, \ phi)}$ ${\ displaystyle z}$ ${\ displaystyle Q \ subset U}$ ${\ displaystyle \ mathbb {R} ^ {n}}$

Poincaré's theorem

For the sake of completeness, the statement found by Henri Poincaré in 1881 is presented first. Let be a compact surface with an induced metric. In addition, let it be a smooth vector field with a finite number of isolated singular points . Then applies ${\ displaystyle S \ subset \ mathbb {R} ^ {3}}$ ${\ displaystyle X \ in \ Gamma (TS)}$ ${\ displaystyle A_ {1}, \ ldots, A_ {k}}$

{\ displaystyle \ sum _ {j = 1} ^ {k} \ operatorname {ind} (X, A_ {j}) = \ chi (S).}

Here called the Euler characteristic of . That means: The Euler characteristic of is equal to the sum over the indices of all isolated singular points of . ${\ displaystyle \ chi (S)}$ ${\ displaystyle S}$ ${\ displaystyle S}$ ${\ displaystyle X}$

Poincaré-Hopf theorem

The Poincaré-Hopf theorem was proved by Hopf in 1926 as a generalization of the Poincaré theorem. Let be a compact differentiable manifold and let be a vector field that has only finitely many isolated zeros . Then applies ${\ displaystyle M}$ ${\ displaystyle X}$ ${\ displaystyle M}$ ${\ displaystyle A_ {1}, \ ldots, A_ {k}}$

{\ displaystyle \ sum _ {i = 1} ^ {k} \ mathrm {ind} (X, A_ {i}) = \ chi (M).}

If there is an edge , the edge must point in the direction of the outer normal. ${\ displaystyle M}$ ${\ displaystyle X}$

literature

M. Hazewinkel: Poincaré – Hopf theorem . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
Paul Alexandroff , Heinz Hopf: Topology. Volume 1: Basic concepts of set-theoretical topology, topology of complexes, topological invariance theorems and subsequent concept formation, entanglements in the n-dimensional Euclidean space, continuous mapping of polyhedra . Springer, Berlin 1935, p. 549 (corrected reprint. Ibid. 1974, ISBN 3-540-06296-3 ) (= The Basic Teachings of Mathematical Sciences in Individual Representations 45, ISSN 0072-7830 )