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HARMONIC MAPS Contents 1. Introduction 2 1.1. Notational HARMONIC MAPS ANDREW SANDERS Contents 1. Introduction 2 1.1. Notational conventions 2 2. Calculus on vector bundles 2 3. Basic properties of harmonic maps 7 3.1. First variation formula 7 References 10 1 2 ANDREW SANDERS 1. Introduction 1.1. Notational conventions. By a smooth manifold M we mean a second- countable Hausdorff topological space with a smooth maximal atlas. We denote the tangent bundle of M by TM and the cotangent bundle of M by T ∗M: 2. Calculus on vector bundles Given a pair of manifolds M; N and a smooth map f : M ! N; it is advantageous to consider the differential df : TM ! TN as a section df 2 Ω0(M; T ∗M ⊗ f ∗TN) ' Ω1(M; f ∗TN): There is a general for- malism for studying the calculus of differential forms with values in vector bundles equipped with a connection. This formalism allows a fairly efficient, and more coordinate-free, treatment of many calculations in the theory of harmonic maps. While this approach is somewhat abstract and obfuscates the analytic content of many expressions, it takes full advantage of the algebraic symmetries available and therefore simplifies many expressions. We will develop some of this theory now and use it freely throughout the text. The following exposition will closely fol- low [Xin96]. Let M be a smooth manifold and π : E ! M a real vector bundle on M or rank r: Throughout, we denote the space of smooth sections of E by Ω0(M; E): More generally, the space of differential p-forms with values in E is given by Ωp(M; E) := Ω0(M; ΛpT ∗M ⊗ E): Definition 2.1. A connection on E is an R-linear map r :Ω0(M; E) ! Ω1(M; E) such that for all X; Y 2 Ω0(M; T M); u; v 2 Ω0(M; E) and f 2 C1(M); (1) rX+Y s = rX s + rY s; (2) rfX s = frX s; (3) rX (u + v) = rX u + rY v (4) rX fu = X(f) ⊗ u + frX u: i m Given a local coordinate system fx gi=1 on M and a basis of local sections r k fejgj=1 of E; then the connection coefficients fAijg of r relative to these choices are the locally defined quantities k r@i ej = Aijek: A section u 2 Ω0(M; E) is parallel for r (or covariant constant) if ru = 0: Given a connection r on E; the curvature Rr of r is given by the expression R(X; Y )u = rX rY u − rY rX u − r[X;Y ]u: HARMONIC MAPS 3 A direct calculation exploiting the properties of the connection r shows that R 2 Ω2(M; End(E)): Given two vector bundles with connections (E; rE) and (F; rF ); there is a functorial way to assign a connection to each of the bundles which we construct out of E and F: Again, let X 2 Ω0(M; T M) and u 2 Ω0(M; E) and v 2 Ω0(M; F ): • On the direct sum E ⊕ F; E F rX (u ⊕ v) = rX u ⊕ rX v: • On the tensor product E ⊗ F; E F rX (u ⊗ v) = (rX u) ⊗ v + u ⊗ rX v: • On the dual bundle E∗; let φ 2 Ω0(M; E∗); E rX (φ)(u) = X(φ(u)) − φ(rX u): • On the bundle of bundle homomorphisms Hom(E; F ); let Φ 2 Ω0(M; Hom(E; F )); F E rX (Φ)(u) = rX Φ(u) − Φ(rX u): The reader should check that these rules are compatible with the isomorphism Hom(E; F ) ' E∗ ⊗ F: Furthermore, the exterior powers ΛpE and symmetric powers SpE can be viewed as a subbundles of the p-th tensor power E⊗p whose sections are cor- responding antisymmetric and symmetric sections of the the tensor power. Hence, these bundles are equipped with the restriction of the connection on E⊗p: Finally, we describe how to pull-back bundles with connections. Let f : M ! N be a smooth map and (E; r) a vector bundle with connection over N: Then we can form the pullback vector bundle f ∗E over M: There is a linear map f ∗ :Ω0(N; E) ! Ω0(M; f ∗E) s 7! s ◦ f: Given a section of this type and a tangent vector X; we can define the pull-back covariant derivative via f ∗E ∗ ∗ rX f s = f (rdf(X)s): Given any local section u of f ∗E; there exists a finite number of locally defined smooth functions hi on M and locally defined sections si of E such that X ∗ u = hif si: i The general definition of the pull-back connection is then f ∗E X f ∗E ∗ X ∗ ∗ rX u := rX hif si = (X(hi) ⊗ f si + hif (rdf(X)si)): i i Now we move to a discussion of Riemannian vector bundles. Given a vector bundle E over M; a smooth section of S2E∗ is a smoothly varying symmetric 4 ANDREW SANDERS bilinear form on the fibers of E: A section a 2 Ω0(M; S2E∗) is a metric on E if the restriction to each fiber is a positive definite inner product. A vector bundle E with a connection r and a metric a is a Riemannian vector bundle (E; r; a) if the metric is parallel: ra = 0: Explicitly, this condition leads to the following Leibniz rule 1 d(a(u; v)) = a(ru; v) + a(u; rv) 2 Ω (M; R): It is straightforward to check that given Riemannian vector bundles (E; rE; a) and (F; rF ; b); then all associated bundles equipped with their induced connections and metrics become Riemannian vector bundles. The tangent bundle TM of a Riemannian manifold (M; g) equipped with the Levi-Civita connection rg is the prototypical example of a Riemannian vector bundle. Here g is a metric on the tangent bundle T M: Given a local coordinate i k g system fx g on M; the connection coefficients fΓijg of r with respect to this coordinate system are known as the Christoffel symbols. A short calculation reveals the explicit formula for the Christoffel symbols gkm Γk = (@ g + @ g − @ g ): ij 2 i jm j im m ij Here, gkm is the (km)-th entry of the inverse of the Gram matrix fgijg: The entries of the Gram matrix of g are defined by gij = g(@i;@j): By inspection, the Christoffel symbols satisfy the symmetric property k k Γij = Γji which is a consequence of the Levi-Civita connection being torsion free: g g rX Y − rY X = [X; Y ] for every pair of vector fields X; Y 2 Ω0(M; T M): We also record the action of the Levi-Civita connection on coordinate differential forms rg dxj = −Γj dxk: @i ik We now explain some exterior calculus for bundles with connections. The exte- rior covariant derivative associated to (E; r) is the first order differential operator which is defined on a simple tensor ! ⊗ u 2 Ωp(M; E) via dr :Ωp(M; E) ! Ωp+1(M; E) ! ⊗ u 7! d! ⊗ u + (−1)p! ^ ru and extends to all tensors by linearity. We will leave the following lemma as an exercise for the reader. HARMONIC MAPS 5 Lemma 2.2. The exterior covariant derivative dr is the skew symmetrization of the covariant derivative rg ⊗ r :Ωp(M; E) ! Ω0(M; T ∗M ⊗ ΛpT ∗M ⊗ E): Furthermore, dr ◦ r :Ω0(M; E) ! Ω2(M; E) u 7! Rr(u) Here, Rr 2 Ω2(M; End(E)) is the curvature of r: Now choose a Riemannian vector bundle (E; r; a) over an oriented Riemannian p manifold (M; g): Combining the metrics g and a yields a metric h; ig⊗a on Ω (M; E): Combining this with integration gives rise to the L2-pairing defined on every α; β 2 p Ω0(M; E) via Z hα; βi = hα; βig⊗a dVg: M p Here, Ω0(M; E) denotes compactly supported forms. Now, let p m−p ? :Ω (M; R) ! Ω (M; R) be the Hodge star operator derived from g which is uniquely characterized by the formula 2 ! ^ ?! = j!jg dVg: This may be comined with the metric duality a : E ! E∗ u 7! (a(u): v 7! a(u; v)) for u; v 2 Ω0(M; E). The Hodge star ? and the duality a combine to yield an extension of the Hodge star operator to E-valued forms defined by the formula p m−p ∗ ?E :Ω (M; E) ! Ω (M; E ) ! ⊗ u 7! ?! ⊗ a(u): p 2 Then, given α; β 2 Ω0(M; E); we can express the L -pairing as Z hα; βi = α ^ ?E β M where here ^ refers to the natural pairing p q ∗ p+q ^ :Ω (M; E) ⊗ Ω (M; E ) ! Ω (M; R) which combines the wedge product of forms with the evaluation pairing of vectors on covectors. The utility of this algebraic gadget is that the exterior covariant derivative now satisfies a graded Leibniz rule: Lemma 2.3. Let (E; r; a) be a Riemannian vector bundle over (M; g): Then for every α 2 Ωp(M; E) and β 2 Ωq(M; E∗); E E∗ d(α ^ β) = dr α ^ β + (−1)pα ^ dr β: 6 ANDREW SANDERS p−1 p Now, if α 2 Ω0 (M; E) and β 2 Ω0(M; E); then by Stokes theorem Z 0 = d(α ^ ?E β)(2.1) M Z ∗ rE (p−1) rE = d α ^ ?E β + (−1) α ^ d ?E β: M Now, we use the fact that 2 p(m−p) ?E = (−1) on p-forms.
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