HARMONIC READING COURSE

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1 HARONIC READING COURSE BRIAN COLLIER 1. Harmonic maps between Riemannian anifolds 1.1. Basic Definitions. This section comes from [1], however we will try to do our computations globally first and then locally. Let (,g) and (N,h) be Riemannian manifolds with compact, denote their Levi Civita connections by r g and r h. They are given by Kozul s formula 2g(r g XY,) =X(g(Y,)+Y (g(, X)) (g(x, Y )) +g([x, Y ],) g([y,],x) g([x, ],Y) With respect to coordinates {x 1,...,x m } on and {y 1,...,y n } on N, we have and where r g x j r h y x i = k ij x k = y y k ij = 1 2 gkl ( g jl + g il x i x j = 1 2 h ( y h + y h x l g ij ) y h ) Let f :!N be a smooth map between them (we really only need to assume f is C 1,) the di erential df 2 0 (,T f TN) and the bundle T f TN has metric g f h and connection r g f r h. The Christo el symbols for r g are given by r g x j dx i = k ij dx k = j ik dx k We can view df is a slightly manor, namely df 2 1 (,f TN). The covariant derivative f r h induces an exterior di erential operator d f r h : (,f TN)! +1 (,f TN) The distinction between (r g f r h )(df ) and d f r h (df ) will be important, d f r h is the skew symmeterization of r g f r h. Definition 1.1. Locally we have df (x) 2 = 1 2 gij h (f(x)) f (x) f (x) x i x j. Let (,g) and (N,h) be Riemannian manifolds with compact, the energy of a C 1 map f :(,g)!(n,h) is given by E(f) = 1 hdf, df i 2 T f TN dv ol The objects we will be interested in are the critical points of the energy functional E : C 1 (,N)!R. Definition 1.2. A C 1 map f :(,g)!(n,h) is called harmonic if it is a critical point of the energy function E. 1

2 2 BRIAN COLLIER 1.2. Euler Lagrange Equations for E. We want to find the Euler Lagrange Equations for harmonic maps. To do this, we start with a map f :!N and consider a variation of f. By a variation we mean a vector field along f, that is, a section 2 (,f TN), for the moment we won t be too concerned with smoothness of. Given 2 (,f TN) we consider the one parameter family F (x, t) =exp f(x) ( (x)t) : [ and calculate t (E(F )). The critical points of E are then all maps f so that t (E(F )) = 0 for all., ]!TN Theorem 1.3. The Euler Lagrange Equations for the energy functional are or equivalently (d f r h ) (df )=0 Tr g (r g f r h )(df )=0 Before proving the Theorem we need two auxiliary calculations. Lemma 1.4. Let 2 (,f TN), locally = f Proof. We just compute Thus then f r h = x i f dx i + f x i r f TN x i = x i = x i = x i = r f TN f x f + r f x i f + r f TN f + f TN f x i f r f TN x i = f + x i f x i f f r h = x i f dx i + f x i ( f ) f ( f ) ( f ) ( f ) f dx i dx i Lemma 1.5. Let 2 (,f TN) and let F = exp f (t ) : [, ]!N, then Proof. Locally For the rest of the proof denote r T [ r t df r T [, ] f TN t (df ) =(f r h ) df = F dx i + F dt x i F t F t 0, ] f TN (df ) by r t df. We have = r t ( F dx i x i F + F t dt F )

3 F x i r t (dx i ) F HARONIC READING COURSE 3 =( 2 F dx i tx i F + 2 F t 2 dt F ) + F x i r t (dx i =(d F t F ) + F t r t (dt) + F t r t (dt F + 2 F t 2 dt F ) + F F ) +( F x i dx i +(dt) F t ) f r h t The second and third terms above are 0 and we can evaluate at everything except r t get = d f + f (r F TN x i t = x i dx i f + f (r F TN x i = dx i x i f + f F x i t F t F ) F TN (rf F F ) F ) = dx i x i f + f (r F TN x i F ) F = dx i x i f + f ( x i F ) F F at t =0to = dx i x i f + f ( x i f ) Finally, since the Levi Civita connection is Torsion free, the Christo el symbols have the symmetry =, thus and we are done by Lemma 1.4. = x i f dx i + f x i = f r h f dx i We are now ready to prove the theorem that f :!N is harmonic if and only if (d f r h )(df )=0 Proof. (Of Theorem 1.3) Consider variation 2 We have = 1 2 (,f TN), we need to calculate t E(F ). t E(F ) = 1 hdf, df idvol t 2 t hdf, df i dvol = hdf, r t df i = hdf, f r h idvol dvol Thus f is harmonic if and only if (d f r h ) (df )= 1 2 Tr g(r g f r h )(df )=0.

4 4 BRIAN COLLIER Locally the Euler Lagrange equations for a map f :(,g)!(n,h) are given by trace(rdf )=g ij 2 f g ij i f jk + g x i x j x k The first part is the Laplace Beltrami operator f = So the Euler Lagrange equations become or equivalently 1 p ( p ij f det(g)g )=g ij det(g) x i x j ij f f f + g =0 x i x j 1 p ( p ij f det(g)g )+g ij det(g) x i x j ij f f x i x j 2 f g ij i f jk x i x j x k (f(x)) f x i f x j =0 Remark 1.6. Since the Cristo el symbols for R with the Euclidean metric are zero, we see that a function f :(,g)!r is harmonic if and only if f = 0; recovering the standard notion Harmonic maps from a Riemann Surface. For maps with a Riemann surface as domain, the harmonic map equations simplify, this part also comes mostly from [1]. Definition 1.7. Let be a Riemann surface, a Riemannian metric g on is called conformal if in local coordinates it can be written as 2 (z)dz d z for a positive real valued function. and In real coordinates z = x + iy this means and g( z, z )=g( z, z )=0 g( z, z )= 2 (z). g( x, x )=g( y, y )= 2 (z) g( x, y )=0. For the rest of part 2 (,g) will be a Riemann surface with a conformal metric. Definition 1.8. Let (N,h) be a Riemannian manifold. A C 1 map f :!N is called conformal if h( f z, f z )=0. We will now look at the Harmonic map equations for maps with domain. Proposition 1.9. The energy of a map f :!(N,h) is conformally invariant. Proof. The energy of f is given by E(f) = 1 2 hdf, df i T f TN dv ol writing the integrand locally and remembering the metric on T has conformal factor 1 hdf, df i T f TN dv ol = 4 2 (x) h ( f z p f 1 z ) 2 2 (z)dz ^ d z we have

5 HARONIC READING COURSE 5 Thus the energy is given by which is conformally invariant. E(f) = p 1 h f z f z Lemma The Laplace Beltrami operator for (,g) is given by Proof. We compute, the metric is given by = 1 p ( p det(g)g ij det(g) x i = z z! )= 1 x j 2 ( x dz ^ d z = 1 2 ( 2 x y 2 )= z z With this we have the following form of the harmonic map equations Lemma The harmonic map equations for f :(,g)!(n,h) are! y ( ) x y 1 A) locally the harmonic map equations are (d rf h ( f z dz))( z )= Proof. The harmonic map equations are (d rf h) 01 d 10 f =0 2 f z z dz ^ d z f z ij f f f + g =0. x i x j f z dz ^ d z =0 By the previous lemma and after converting everything to complex coordinates we may rewrite them as: proving the Lemma. 4 f 2 (2 z z + f z f z )=0 Corollary The harmonic map equations for maps (,g)!(n,h) only depend on the conformal class of and not on the actual metric g. In particular holomorphic and anti holomorphic maps between Riemann surfaces are harmonic. Proof. By the previous lemma we see that the conformal factor does not appear in the equations. Finally we define the Hopf di erential for a map f :(,g)!n. To do this we use the metric on to define a contraction defined by Definition The Hopf di erential Note that : 0 (,T f TN) 0 (,T f TN)! 0 (,T T ) f = 0 if and only if f is conformal. (! s, t) =hs, ti f h!. f of a map f :(,g)!(n,h) is the quadratic di erential f = (df 1,0,df 1,0 ) 2 0 (,K 2 ) Proposition If f :(,g)!(n,h) is harmonic then the Hopf di erential harmonic implies f 2 H 0 (,K 2 ). f is holomorphic, that is, f

6 6 BRIAN COLLIER Proof. We calculate f, locally df 10 = f z by Lemma 1.11 f = f h z, f z idz2 =2h(r f TN 01 f ) z, f z idz2 =0

7 o / " o o 7 HARONIC READING COURSE 7 2. Reductions of Structure and Energy 2.1. Reductions of structure group. Let (,g) be a compact Riemannian manifold, let G be a semisimple Lie group and let i : H!G be a subgroup so that g = h m is Ad H invariant. An important example of this is when (,g) is a Riemann surface and g is a conformal metric, G is a noncompact semisimple Lie group (for example SL(n, R) or SL(n, C)) and H is the maximal compact subgroup of G (for example SO(n, R) SL(n, R) or SU(n) SL(n, C)). Let P! be a principal G bundle, a reduction of structure of P from G to H is a subbundle P H,! P G so that P H i G = P G. Reductions of structure are in one to one correspondence with sections (we will always work in the smooth category) of the associated G/H fiber bundle P G G G/H!. To see this, the following diagram is very helpful: G H P G P G P G /H = / P G G G/H G/H Given a section 2 C 1 (; P G G G/H) we can pullback P G!P G /H to a H bundle P G over which naturally includes, H-equivariantly, in P G! H G H P G P G P G P G G G/H G/H Remark 2.1. We will always use tilde s to denote lifted objects or maps (i.e. objects and maps. ) and hats for descended It will be useful to sometimes think of a section of P G G G/H as G-equivariant maps P G!G/H. / 2.2. Reductions and Connections. We will now start with a connection on P G and see how to measure the compatibility of a reduction with a connection. Given a principal bundle G P G we get a exact sequence of tangent bundles 0 / ker(d ) / TP G d / T / 0 the bundle ker(d ) is the vertical bundle V G!P G. Recall that a connection on a principal bundle is given by a B 2 1 (P g, g) satisfying: 1. (Vertical) For all X 2 g let X P be the vector field determined by the G action, then B(X P )=X. 2. (Equivariance) If R g : P G!P G is the di eomorphism of P G given by the right action of G then we require (R gb)(y )=Ad g 1B(Y ) for all g 2 G and Y 2 C 1 (P G ; TP G ). Such a B defines an equivariant projection TP!V G and thus gives an equivariant splitting TP G = V G ker(b), with ker(b) = T. The subbundle ker(b) is called the horizontal distribution associated to B and will be denoted H B. Recall that a section of (P G, g) is called horizontal if it vanishes on vertical vector fields. Equivariant horizontal sections of a principal bundle are called basic and there is a 1 to 1 correspondence between (P G, g) basic and (,P G G g).

8 o ' % o 7 8 BRIAN COLLIER Now fix a connection B 2 1 (P G, g), we have the following diagram to consider (P G,B) P G /H If pr h : g!h denotes the projection then set B h = pr h B. Since B h 2 1 (P G, h) ish equivariant and B h (X P )=X for all X 2 h we conclude B h defines connection on P G!P G /H. We will write B = B h + µ where µ 2 1 (P G, m). It is straight forward to check that µ is a basic form and hence descends to ˆµ 2 1 (P G /H, P G H m). Note that the vertical bundle ker(d ) TP G /H is isomorphic to P G H m so ˆµ defines a projection ˆµ : TP G /H!V G/H. Given a reduction of structure :!P G /H we get a principal H bundle ( P G, B) / (P G,B) % P G /H Definition 2.2. Given a reduction of structure :!P G /H we define the vertical derivative of, with respect to the connection B, to be D B =ˆµ d that is T d / TP G /H ˆµ / V 7 G/H = PG H m D B Here D B 2 C 1 (; T V G/H )= 1 (, P G H m). We can pull back B by to B 2 1 ( P G, g), since the map is H equivariant, we get the decomposition We will use the following notation B = B h + µ. A = B h 2 1 ( P G, h) = µ 2 1 ( P G, m) As before, A defines a connection on P G and is a basic 1-form valued in m which we identify with a section ˆ 2 1 (, P G H m). Proposition 2.3. With the set up above, ˆ can be identified with vertical derivative D B. Proof. With the above set up, the proof is straight forward. We defined by = µ and saw that both µ and were basic forms so descend to sections ˆµ and ˆ of the appropriate bundles. We have that ˆ = ˆµ,

9 HARONIC READING COURSE 9 thus for a vector field X 2 C 1 (; T) by definition of the vertical derivative. ˆ (X) =( ˆµ)(X) =ˆµ(d (X)) = D B ( )(X) 2.3. Reduction of structure thought of as an equivariant map. Reductions of structure are sections of P G G G/H! and sections of an associated bundle are in one to one correspondence with equivariant maps from the total space to the fiber space. This means reductions of structure can also be thought of as maps % : P G!G/H with the property that %(p g) =g 1 %(p) for all p 2 P G and all g 2 G. The exterior derivative of % is an equivariant 1 form valued in the pull back of the tangent bundle of G/H, that is d% 2 1 (P G,% T G/H) = 1 (P G,% G H m) and is G equivariant. (ORE TO COE ON THIS) want to compare d% with the vertical derivative of and things like this

10 10 BRIAN COLLIER 3. Harmonic Reductions In this section we need to fix a Riemannian metric g on and a splitting g = h m with an Ad H invariant inner product h, i on m. If H is the maximal compact of G, the Killing form of g is nondegenerate (since G is semisimple) and maximally negative definite on h g. We define m to be the orthogonal compliment of h giving g = h m, the Killing form is then positive definite on m. Definition 3.1. Let G be a semisimple Lie group and H be a subgroup with g = h m is Ad H invariant and so that m admits a Ad H invariant inner product h, i. Let (P G,B)!(,g) be principal G bundle with connection B over a compact Riemannian manifold (,g), Then the energy of a reduction of structure :!P G /H is defined by E( ) = 1 2 ˆ 2 dvol here the norm is defined with respect to the metric on T and the metric on V G/H = PG H m induced by h, i. To derive the harmonic section equations we follow [3]. The connection ˆµ gives an isomorphism TP G /H = T P G H m thus we have a g h, i defines a Riemannian metric on TP G /H. Let r LC denote the Levi Civitta connection of g h, i, with respect to this connections define covariant derivative r v on P G H m!p G /H by r v (V )=ˆµ(rLC X X V ). Theorem 3.2. With the notation of the previous section, the Euler Lagrange equations for the energy functional E are (d r v ) ( ˆ ) =0 or equivalently Tr g ((r g ˆµr LC )(ˆµd )) = Tr g ((r g r v )( ˆ )) = 0 Proof. First note that the variations appropriate for this problem are in the vertical direction since we want to consider a 1 parameter family of sections. Let 2 0 (, V G/H ) be such a variation of, consider the smooth one parameter family of sections!p G /H given by We need to compute First, as in the proof of 1.5 we observe that t r v t ( ˆ t ) t (x) =exp (x) (t (x)) d dt E( t) = r v ( d dt t )= r v ( ) Note that in the proof of Lemma 1.5 we used that the connection was torsion free, thus it is important that we are using a connection coming from the Levi Civita connnection on P G /H. We are now ready to compute: d dt E( t) = 1 2 d dt h ˆ t, ˆ t idvol Since the metric is symmetric and the connection is metric this becomes = h ˆ t, r v ˆ t idvol t = h ˆ, r v idvol

11 HARONIC READING COURSE 11 = h(d r v ) ˆ, idvol So the critical points of E correspond to reductions with or equivalently 0=(d r v ) ˆ Tr g ((r g ˆµr LC )(ˆµd )) = Tr g ((r g r v )( ˆ )) = 0 proving the proposition Harmonic sections equations with respect to canonical connection. Right now we have the Harmonic section equations in terms of the Levi Civitta connection on P G /H, however there is a canonical metric connection on P G H m induced by the connection 1-form B h on P G!P G /H, denote this canonical connection by r c, note that r c is a metric connection. Define a symmetric bilinear form U on m by hu(a, b),ci = h[c, a] m,bi + ha, [c, b] m i a, b, c 2 m. By Ad H invariance of h, i, U defines a bilinear form on P G H m, and ˆµ U defines a bilinear form on TP G /H. Note that if G/H is a symmetric space then U = 0, a homogeneous space G/H is called naturally reductive if there is a Ad H -invariant inner product of m with U =0. Recall that B h = A and ˆµ = ˆ. We will prove the following theorem Theorem 3.3. Let be a section P G /H!, then is harmonic if and only if (d r A ) ( ˆ ) 1 2 Tr g(r g ˆ U)=0 We will need to introduce some auxillary tensors to prove the theorem, however before we do this observe that we have the following corollary of Theorem 3.3. Corollary 3.4. If G/H is a symmetric space then is harmonic if and only if (d ra ) ˆ =0, in particular the harmonic section equations are independent of the Ad H -invariant inner product on m. In fact, the dependence the harmonic map equations on the Ad H -invariant inner product of m is measured by ˆ U. We begin by defining the torsion of the canonical connection r c. Definition 3.5. Let N! be a homogeneous fiber bundle with fiber G/H and connection ˆµ 2 1 (N,V G/H ), let r be a connection on the vertical bundle V G/H!N. The ˆµ torision of r is defined by T rˆµ = d r (ˆµ). When the fiber bundle N! is P G /H and ˆµ comes from a connection B on P G!, we have Note that the curvature of B is given by This gives the following lemma T rc ˆµ (X, Y )=T c (X, Y )=r c X(ˆµY ) r Y (ˆµX) ˆµ[X, Y ]. 8 < F B = : db h [Bh,B h ]+ 1 2 [ˆµ, ˆµ]h +dˆµ +[B h, ˆµ]+ 1 2 [ˆµ, ˆµ]m 8 < F B h [ˆµ, ˆµ]h = : +d rbh (ˆµ)+ 1 2 [ˆµ, = F h ˆµ]m B + F B m Lemma 3.6. The canonical torsion is given by T c = FB m fields and W 1,W 2 are horizontal vector fields then: 1 2 [ˆµ, ˆµ], furthermore if V 1,V 2 are vertical vector

12 12 BRIAN COLLIER (1) T c (V 1,V 2 )= [V,W] m (2) T c (W 1.W 2 )=F m B (W 1,W 2 ) (3) T c (V 1,W 1 )=0 We now define the following di erence tensor Lemma S =ˆµ U T c S(X, Y )=ˆµ(r L C X Y ) r c X(ˆµY ) Proof. Let X, Y 2 0 (R G /H, T P G /H) be vector fields and V 2 0 (P G /H, P G H m) be a vertical vector field. Using the Koszul formula for the Levi Civita connection we have 2hS(X, Y ),Vi = XhˆµY, V i + Y hˆµx, V i V hˆµx, ˆµY i hˆµx, ˆµ[Y,V ]i hˆµy, ˆµ[X, V ]i + hˆµv, ˆµ[X, Y ]i 2hr c X(ˆµY ),Vi Vg(d X, d Y )+Xg(d V, d Y )+Yg(d V, d X) g(d X, d [Y,V ]) g(d Y, d [X, V ]) + g(d V, d [X, Y ]) Since d (V ) = 0 two terms in the third line and one in the fourth are automatically zero, in fact all terms involving the metric g vanish. Now using the fact that r c is a metric connection we may rewrite the first line in terms of the the canonical connection, for instance the first term can be rewritten as After collection terms we have now using lemma 3.6 we have Proving the Lemma. XhˆµY, V i = hr X (ˆµY ),Vi + hˆµy, r c XV i 2hS(X, Y ),Vi = ht c (X, V ), ˆµY i + ht c (Y,V ), ˆµXi 2hS(X, Y ),Vi = hu(ˆµx, ˆµY ) T c (X, Y ),Vi Corollary 3.8. For all vertical vector fields V 2 0 (P G /H.P G H m) we have ht c (X, Y ),Vi ˆµ(r LC X V )=r v XV = r c XV U(ˆµX, V )+1 [ˆµX, V ]m 2 Proof. By lemma 3.7 we have 2S =ˆµ (U +[, ] m ) FB m, now using the definition of S we have r v X(V )=r c XV U(ˆµX, V )+1 [ˆµX, V ]m 2 proving the Lemma. We are now ready to prove Theorem 3.3. Proof. (Of Theorem 3.3) By Theorem 3.2 is harmonic if and only if (d r v ) ( ˆ ) = Tr g ((r g r v )( ˆ )) = 0 By corollary 3.8 we have So r v = r c + (ˆµ 1 2 U +[, ]m )=r A + ˆ ( 1 2 U)+ [, ] m Tr g ((r g r v )( ˆ )) = Tr g (r g (r A + ˆ ( 1 2 U)+ [, ] m ))( ˆ ) = Tr g (r g r A )( ˆ )+Tr g (r g ( ˆ ( 1 2 U)+ [, ] m ))( ˆ ) Since U is the symmetric part of U +[, ] m, we can simplify to = (d ra ) ( ˆ )+ 1 2 Tr g(r g ˆ U)( ˆ ). Thus is a harmonic section if and only if (d ra ) ( ˆ ) 1 2 Tr g(r g ˆ U)( ˆ ) =0

13 HARONIC READING COURSE 13 Remark 3.9. A homogeneous space G/H is called natually reductive if the tensor U vanishes. For naturally reduction homogeneous fiber bundles, the harmonic reduction equations with respect to the canonical connection are the same as the harmonic reduction equations with respect to the connection coming from the Levi Civitta connection. 4. Hamonicity of equivariant map We want show that the harmonicity of as a section of the homogeneous fiber bundle P G /H!(,g) is equivalent to the corresponding G-equivariant map % : P G!G/H being harmonic, however, this is not true in general. To remedy this, we look at a more specific class of homogeneous fiber bundles. Before specifying, we need to introduce the notion of a Kaluza Klein metric. Definition 4.1. A homogeneous space G/H is called normal if the Ad H invariant innerproduct on m is the restriction of an Ad G invariant innerproduct on g. We will prove the following theorem, which is Theorem 3 in [2]. Theorem 4.2. Let P G!(,g) be a principal G bundle and let G/H be a normal homogeneous space. Then a section :!P G /H is a harmonic reduction if and only if the corresponding equivariant map % : P G!G/H is harmonic.

14 14 BRIAN COLLIER 5. Harmonic bundles and Corlette s Theorem Definition 5.1. Let G be a semisimple Lie group, let H G be its maximally compact subgroup and g = h m be the splitting of the Lie algebra of G as the Lie algebra of h and its orthogonal compliment with respect to the Killing form. A Harmonic G-bundle over a Riemann manifold is a triple (P H, A, ) where P H! is principal H bundle, A 2 1 (P H, h) is a connection 2 1 (,P H H m) that satisfy Part 1. TO DO F A [, ] =0 r A =0 (r A ) =0 (1) Reword Wood s proof of relation between harmonicity of equivariant map and section in language and generality of our set up (2) Reword Corlette s set up and prove general set up of harmonic map to ariski closure of representation. (3) Prove things about harmonic reductions (i.e. commuting diagrams of reductions) include proof of how Corlette for SL n (C) together with our results prove analogous theorem for G a real reductive group. References [1] Jürgen Jost. Riemannian geometry and geometric analysis. Universitext.Springer-Verlag,Berlin,fifthedition,2008. [2] C.. Wood. Harmonic sections and equivariant harmonic maps. anuscripta ath., 94(1):1 13,1997. [3] C.. Wood. Harmonic sections of homogeneous fibre bundles. Di erential Geom. Appl., 19(2): ,2003.