By Terrence Napier, Mohan Ramachandran

ISBN-10: 0817646922

ISBN-13: 9780817646929

ISBN-10: 0817646930

ISBN-13: 9780817646936

This textbook offers a unified method of compact and noncompact Riemann surfaces from the perspective of the so-called L2 $\bar{\delta}$-method. this technique is a strong strategy from the speculation of numerous complicated variables, and gives for a different method of the essentially various features of compact and noncompact Riemann surfaces.

The inclusion of constant workouts working through the e-book, which result in generalizations of the most theorems, in addition to the workouts integrated in every one bankruptcy make this article excellent for a one- or two-semester graduate path. the must haves are a operating wisdom of ordinary themes in graduate point actual and complicated research, and a few familiarity of manifolds and differential forms.

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**Additional info for An Introduction to Riemann Surfaces**

**Example text**

Thus X is a compact Riemann surface. The Riemann surface X is called a complex torus because topologically, X is the torus T2 = S1 × S1 . In fact, we have a commutative diagram of C ∞ maps R2 ϒ0 α C ϒ β T2 X where α is the real linear isomorphism (and therefore diffeomorphism) given by (t1 , t2 ) → t1 ξ1 + t2 ξ2 , ϒ0 is the C ∞ mapping onto the real torus T2 = S1 × S1 given by (t1 , t2 ) → (e2πit1 , e2πit2 ), and the induced mapping β is a well-defined diffeomorphism. For it is easy to verify that β is a well-defined bijection.

For any element ξ of (Tp X)C or (Tp∗ X)C , we call P r,s ξ the (r, s) part of ξ . 3. 4). The (0, 1) tangent bundle (T X)0,1 and cotangent bundle (T ∗ X)0,1 are examples of C ∞ line bundles. 4. A reader familiar with vector bundles will recognize the decomposition (T X)C = (T X)1,0 ⊕ (T X)0,1 as a decomposition of the C ∞ vector bundle (T X)C into a sum of C ∞ subbundles (as is the decomposition of (T ∗ X)C ). 4 Holomorphic Tangent Bundle 43 5. For (r, s) ∈ {(1, 0), (0, 1)} and for any open subset of a complex 1-manifold X with inclusion mapping ι : → X, we identify (T )r,s and (T ∗ )r,s with the −1 r,s ∗ r,s sets (T X)r,s ( ) ⊂ (T X) and −1 (T ∗ X)r,s ( ) ⊂ (T X) , respectively, under the bijections ι∗ : (T )r,s → tively.

4 Holomorphic Tangent Bundle 41 (d) A C 1 function f on an open set ⊂ X is holomorphic if and only if for each point p ∈ , ∂f/∂ z¯ ≡ 0 on U ∩ for some (equivalently, for every) local holomorphic coordinate neighborhood (U, z) of p. If : X → Y is a C 1 mapping of X into a complex 1-manifold Y and (r, s) ∈ {(1, 0), (0, 1)}, then the following are equivalent: (i) is holomorphic. (ii) ∗ (T X)r,s ⊂ (T Y )r,s . (iii) ∗ (T ∗ Y )r,s ⊂ (T ∗ X)r,s . Proof Let (U, = z, U ) be a local holomorphic chart in X and let p ∈ U .

### An Introduction to Riemann Surfaces by Terrence Napier, Mohan Ramachandran

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