Differential forms. Proposition 3 Let X be a Riemann surface, a X and (U, z = x + iy) a coordinate neighborhood of a.

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1 Differential forms Proposition 3 Let X be a Riemann surface, a X and (U, z = x + iy) a coordinate neighborhood of a. 1. The elements d a x and d a y form a basis of the cotangent space T (1) a. 2. If f is a function that is differentiable in a neighborhood of a, then d a f = f x (a) d ax + f y (a) d ay = f z (a) d az + f z (a) d az. Let (U, z) and (U, z ) be two holomorphic coordinate neighborhoods of a point a X. Then the holomorphic compatibility of the charts implies z z z (a) C (a) C z. z z (a) = 0 z z (a) = 0 The vector subspaces Ta 1,0 := Cd a z and Ta 0,1 := Cd a z of T a (1) independent of the choice of coordinate. This motivates the following definition: are thus Christoph Schweigert, Differentialformen p.1/12

2 Definition 4 Differential forms II 1. Elements of Ta 1,0 are called cotangent vectors of type (1, 0), elements of Ta 0,1 type (0, 1). 2. For a differentiable function f, we define: by d a f = d af + d af mit d af T a (1,0) and d af T a (0,1) d af = f z (a)d az and d af = f z (a)d az. 3. Let Y be an open subset of a Riemann surface X. A differential form of order one or a one-form on Y is a map ω : Y a Y T (1) a such that ω(a) T a (1) for all a Y. If we have ω(a) T a (1,0) or ω(a) T a (0,1) for all a Y, the differential form ω is said to be of type (1, 0) or (0, 1), respectively. of Christoph Schweigert, Differentialformen p.2/12

3 Sheaves of differential forms 4. A one-form ω is called differentiable or smooth, if ω can be written in any holomorphic chart (U, z) as ω = fdz + gdz in U Y with f, g E(U Y ). (The notion of a smooth differential form can be introduced for any smooth manifold.) 5. A one-form ω on a Riemann surface is called holomorphic, if ω can be written with respect to any complex chart (U, z) as ω = fdz in U Y with f O(U Y ). 6. For any open subset U X denote by E (1) (U) the complex vector space of smooth one-forms on U. By E 1,0 (U) and E 0,1 (U) denote the vector subspace of differential forms of type (1, 0)and (0, 1) respectively. By Ω(U) denote the vector subspace of holomorphic differential forms. Note that holomorphic differential forms are by definition of type (1, 0). Together with the standard restriction maps, we obtain the sheaves E (1), E 1,0, E 0,1 and Ω of complex vector spaces on the Riemann surface X. Christoph Schweigert, Differentialformen p.3/12

4 Observation 5 The residue 1. Let Y be an open subset of a Riemann surface X. Let a Y and ω Ω(Y \ {a}). Let (U, z) be a coordinate neighborhood with U Y and z(a) = 0. Then we have on U that ω = fdz with f O(U \ {a}). Let f = n= c n z n be the Laurent expansion of the holomorphic function f. 2. If c n = 0 for all n < 0, then ω can be continued holomorphically to all of Y. The differential form ω then has a removable singularity in a. Similarly, we define when the 1-form ω has in a a pole of order k or an essential singularity. 3. The coefficient c 1 is called the residue of the one-form of ω in a, c 1 =: res a (ω). Lemma 6 The residue of a one-form does not depend on the chart (U, z). Christoph Schweigert, Differentialformen p.4/12

5 Definition 7 Abelian differentials 1. A meromorphic differential form ω on an open subset Y of a Riemann surface X is a holomorphic one-form on an open subset Y Y such that (a) Y \ Y consists of isolated points only. (b) ω has a pole in all points a Y \ Y. 2. The set of all meromorphic differential forms on Y is denoted by M (1) (Y ). With the natural operations and the restriction maps, we obtain a sheaf M (1) of vector spaces on X. 3. Meromorphic differential forms on X are called abelian differentials. An abelian differential is said to be (a) of first kind if it is holomorphic everywhere. (b) of second kind, if the residue in any pole vanishes. (c) of third kind, else. Christoph Schweigert, Differentialformen p.5/12

6 2-forms Observation 8 1. Let X be a Riemann surface and a X. The exterior product of complex vector spaces yields T (2) a := 2 T (1) a. For a coordinate neighborhood (U, z) of a and z = x + iy, the element d a x d a y is a basis of T a (2). Another basis is dz a dz a. The complex dimension of T a (2) equals one. In analogy to definition 4, we introduce 2-forms and denote by E (2) the sheaf of smooth 2-forms on X. 2. For any open subset U X, we define derivatives d, d, d : E (1) (U) E (2) (U). Write the differential form ω locally as a finite sum ω = f k dg k with differentiable functions f k and g k, e.g. ω = f 1 dz + f 2 dz. Let dω := df k dg k d ω := d f k dg k d ω := d f k dg k. An explicit calculation shows that this is well-defined. Christoph Schweigert, Differentialformen p.6/12

7 Rules for derivatives Lemma 9 Let U be an open subset of a Riemann surface and f E(U) and ω E (1) (U). Then 1. ddf = d d f = d d f = dω = d ω + d ω. 3. d(fω) = df ω + fdω and similar rules for the differential operators d and d. Remarks The rules are direct consequences of the definitions, e.g. ddf = d(1df) = d1 df = Moreover, and thus 0 = ddf = (d + d )(d + d ) f = d d f + d d f d d f = d d f 3. With respect to a local chart (U, z = x + iy), one has d d f = 2 f z z dz dz = 1 2i ( 2 ) f x f y 2 dx dy. Therefore, a smooth function on an open subset of a Riemann surface is called harmonic, if d d f = 0. Christoph Schweigert, Differentialformen p.7/12

8 Closed and holomorphic differential forms Definition 11 Let Y be an open subset of a Riemann surface X. A one-form ω E (1) (Y ) is called closed, if dω = 0. A one-form is called exact, if there is a smooth function f E(Y ) with ω = df. Remarks Because of ddf = 0, exact forms are always closed. The converse, however, fails in general. More on that in section Connection between holomorphic and closed differential forms: For Y X open: (a) Every holomorphic form ω Ω(Y ) is closed. (By definition 4, it is of type (1, 0).) (b) Every closed form ω E 1,0 (Y ) is holomorphic. A smooth differential form ω of type (1, 0) can be written locally as ω = fdz with a smooth function f. We compute dω = df dz = ( f z ) f dz + z dz dz = f dz dz. z Thus dω = 0 is equivalent to f z = If u is a harmonic function, then d u is a holomorphic one-form, because we have dd u = (d + d )d u = d d u = 0. Christoph Schweigert, Differentialformen p.8/12

9 Pullback of differential forms Observation 13 Let F : X Y be a holomorphic map of Riemann surfaces. For any open subset U Y the map F induces a homomorphism, called the pullback F : E(U) E(F 1 (U)) f f F. In the same way, we introduce F : E (k) (U) E (k) (F 1 (U)) für k = 1, 2 fi dg i F (f j )d(f g j ) fi dg i dh i F (f j )d(f g j ) d(f h j ) These maps are well-defined and yield linear pullback maps which commutes with the differential, F : E (k) (U) E (k) (F 1 U) F (df) = d(f f) F (dω) = df ω and analogous rules for the differential operators d and d. Christoph Schweigert, Differentialformen p.9/12

10 Integration of one-forms Observation 1 Let X be a Riemann surface and c : [0, 1] X a piecewise smooth curve. Let ω E (1) (X) be a smooth one-form. Define the integral c ω by choosing a subdivision of the interval [0, 1] 0 = t 0 < t 1 <... < t n = 1 and charts (U k, z k ) of X with c([t k 1, t k ]) U k. Write ω U k = f k dx k + g k dy k with smooth functions f k, g k and let c ω := n k=1 ( f k (c(t)) dx k(c(t)) dt + g k (c(t)) dy ) k(c(t)) dt dt One checks that the right hand side is independent of the subdivision of the interval and the choice of charts of X. Christoph Schweigert, Differentialformen p.10/12

11 Holomorphic primitives The main theorem of calculus relating integration and differentiation can be applied patch by patch: Proposition 2 Let X be a Riemann surface and c : [0, 1] X a piecewise smooth curve and F E(X) a smooth function. Then c df = F (c(1)) F (c(0)). Definition 3 Let X be a Riemann surface and ω E (1) (X) a smooth differential form. A function F E(X) is called a primitive of ω, if df = ω. Remarks 4 1. Any differential form that has a primitive is closed. This follows at once from dω = d 2 F = The primitive is only unique up to an additive constant. 3. Proposition 2 provides an easy way to compute curve integrals over differential forms with known primitive. The integral of an exact differential form only depends on the initial and final point of the curve. Christoph Schweigert, Differentialformen p.11/12

12 Local existence of primitives Let U an open disc around zero in C. Let ω E (1) (U). We write Then ω is closed, dω = 0, if and only if ( ) Consider the integral F (x, y) := It is a primitive: 1 0 ω = fdx + gdy mit f, g E(U). g x = f y. dt (f(tx, ty)x + g(tx, ty)y) F (x, y) x = 1 0 ( f dt x ) g (tx, ty)tx + (tx, ty)ty + f(tx, ty) x. Using ( ) and d dt f f f(tx, ty) = (tx, ty)x + (tx, ty)y, we find x y F (x, y) x = 1 0 dt (t ddt ) f(tx, ty) + f(tx, ty) = 1 0 ( ) d dt (t f(tx, ty)) dt = f(x, y). Globally, primitives of closed differential forms only exist as multivalued functions on X. Christoph Schweigert, Differentialformen p.12/12