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A Simple Proof of Frobenius Theorem

  • Shiing-shen Chern
  • Jon G. Wolfson
Chapter
Part of the Progress in Mathematics book series (PM, volume 14)

Abstract

Frobenius Theorem, as stated in Y. Matsushima, Differential Manifolds, Marcel Dekker, N.Y., 1972, p. 167, is the following:
Let D be an r-dimensional differential system on an n-dimensional manifold M. Then D is completely integrabte if and only if for every local basis {X1,...,Xr} of D on any open set V of M , there are C -functvons c ij k on V such that we have
$$ [{x_i},{x_j}] = \sum\limits_k {c_i^k} {j^{{x_k}}},1 \mathbin{\lower.3ex\hbox{$\buildrel<\over{\smash{\scriptstyle=}\vphantom{_x}}$}} i,j,k \mathbin{\lower.3ex\hbox{$\buildrel<\over{\smash{\scriptstyle=}\vphantom{_x}}$}} r $$
(1)
.

Keywords

Induction Hypothesis Differential System Simple Proof Algebraic System Local Coordinate System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1981

Authors and Affiliations

  • Shiing-shen Chern
    • 1
  • Jon G. Wolfson
    • 1
  1. 1.University of CaliforniaBerkeleyUSA

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