Hermitian Geometry and Kählerian Geometry

  • Shiing-shen Chern
Part of the Universitext book series (UTX)


Let M be a complex manifold of dimension m. M is called hermitian if an hermitian structure H is given in its tangent bundle T(M). With the local coordinates z1,…, zm a natural frame field is given by
$$ {S_i} = \frac{\partial }{{\partial {z^i}}},\;l \mathbin{\lower.3ex\hbox{$\buildrel<\over{\smash{\scriptstyle=}\vphantom{_x}}$}} i,j,k,l \mathbin{\lower.3ex\hbox{$\buildrel<\over{\smash{\scriptstyle=}\vphantom{_x}}$}} m $$
and this frame is holomorphic. Let
$$ {h_{ik}} = H\left( {\frac{\partial }{{\partial {z^i}}},\frac{\partial }{{\partial {z^k}}}} \right) = {\bar h_{ki}} $$
Then the matrix
$$ H = {}^t\bar H = \left( {{h_{ik}}} \right) $$
is positive definite hermitian.


Meromorphic Function Complex Manifold Hermitian Structure Hermitian Manifold Restricted Type 
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Copyright information

© S.-s. Chern 1979

Authors and Affiliations

  • Shiing-shen Chern
    • 1
  1. 1.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

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