A differentiable manifold is a topological space which admits differentiable structures. Here we introduce another structure which has relevance in physics. In elementary complex analysis, the partial derivatives are required to satisfy the Cauchy-Riemann relations. We talk not only of the differentiability but also of the analyticity of a function in this case. A complex manifold admits a complex structure in which each coordinate neighbourhood is homeomorphic to and the transition from one coordinate system to the other is analytic.
The reader may consult Chern (1979), Goldberg (1962) or Greene (1987) for further details. Griffiths and Harris (1978), chapter 0 is a concise survey of the present topics. For applications to physics, see Horowitz (1986) and Candelas (1988).
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