# Normal coordinate frame

1. Sep 11, 2012

### math6

let x a point on complex manifold X, z_j a coordinate system at x , E a holomorphic bundle and let h_α be a holomorphic frame of E. After replacing h_α by suitable linear combinations with constant coefficients we may assume that h_ α is an orthonormal basis of E_{x}. Then an inner product <h_α, h_β > have an expansion :

<h_α, h_β > = δ_αβ + Ʃ ( a_jαβ z_j + a'_jαβ z_j) + O( |z|^2 ) (1)
for some complex coefficients a_jαβ and a'_jαβ .

I would like to understandd how did he get the expression (1) ?