By Leonard Lovering Barrett

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**Extra resources for An introduction to tensor analysis **

**Example text**

9). , the second fundamental form of X' vanishes at p, provided either dim X' = 1 or X is Kahler. 12) Corollary Let (X, dsi) be a Hermitian man(fold whose holomorphic sectional curvature is bounded above by - 1. Then for f E Hol(D, X), where ds 2 is the Poincare metric of D. 13) to higher dimensional spaces X. 13) Theorem. Let X be a complex space with an upper semicontinuous pseudolength function F with the property that, at each v E f X with F(v) > 0, F has negative holomorphic curvature. Let ~ be a holomorphic vector field on X.

A ::: o. We recall the notation dC = i (d" - d') so that dd c = 2id'd". 1 ) Ric(w) = -dd c log A = 2Kw, where . 2) is the (Gaussian) curvature of da 2 • Both Ric(w) and K are defined wherever A is positive. Let Da denote the open disc of radius a in the Gaussian plane C, Da = {z E C; Izl < a}. 3) 2 dSa 4a 2dzdz = A(a 2 _ IzI2)2' (A> 0) 20 Chapter 2. Schwarz Lemma and Negative Curvature on Da is complete and has curvature -A. In the special case where a = 1, the unit disc D j will be denoted D and the Poincare metric ds~ with A = 1 will be denoted ds 2 .

Using a local coordinate system z in Vi with Z(Pi) = 0, let r be a positive number such that B = {z; Izl < r} C V;. Let B' = {z; Izl < r/2}. Choose a Coo functiona(z,Z) on V; such that (i) 0 :::: a(z, z) :::: 1 on Vi, (ii) a(z, z) = 1 on B', and (iii) a(z, z) = 0 on V; - B. Let c be a constant, 0 < c < 1, and set 30 Chapter 2. Schwarz Lemma and Negative Curvature where 1 = II iI + Id2 and h = (l + zz)a(z, z). Then the metric dsi = 2gdzdz coincides with da 2 on Vi - B. Since the curvature of da 2 is bounded above by a negative constant on the compact set B - Bf, the curvature of the metric ds~ is strictly negative on B - B' if c is sufficiently small.