Differential Geometry

By Michael Spivak

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Additional resources for A Comprehensive Introduction to Differential Geometry, VOL. 3, 2ND EDITION

Example text

Consider the complex one-dimensional torus T generated from the lattice {1, i}. Let π : C → T be the standard covering map. Then z → −z on the complex plane generates an automorphism, say τ , on T . Now τ has four ﬁxed points, which are π(0) , π(1/2 + i/2) , π(1/2) , π(i/2) . Yet τ does not ﬁx π( 14 ), and so it is not the identity map. 2. We now consider a two-holed torus. This manifold can be generated by a regular octagon centered at the origin of the Poincar´e disk together with its reﬂections.

And a small modiﬁcation of the proof shows that the segment can be arbitrarily short. Picard considered the question of how small a set the image of a nonconstant entire function can omit. Let us pursue the same line of inquiry rather modestly by asking whether a nonconstant entire function can omit one complex value. The answer is “yes,” for f (z) = ez assumes all complex values except zero. It also turns out that it is impossible to construct a metric on the plane less a point that has negative curvature bounded away from zero.

How does this compare with the boundary behavior of the Bergman kernel on the disk? Write an explicit formula for the Bergman kernel on the upper halfplane U = {ζ ∈ C : Im ζ > 0}. Give an example of a planar domain (not the entire plane) for which the Bergman space consists only of the zero function. Can you characterize all such domains? What is the Bergman projection of the function ϕ(ζ) = ζ on the unit j disk? What about ϕj (ζ) = ζ , j = 2, 3, . . Fix a bounded domain Ω ⊆ C. Fix also a point z ∈ Ω.