Differential Geometry and the Quaternions

DIFFERENTIAL GEOMETRY AND THE QUATERNIONS Nigel Hitchin (Oxford) The Chern Lectures Berkeley April 9th-18th 2013 3 26th December 1843 • 16th October 1843 2 3 RIEMANNIAN MANIFOLDS OF FOUR DIMENSIONS 965 We call a frame an ordered set of four mutually perpendicular unit vectors eo, Ci, C2, 63. There exists one and only one rotation carrying 3 one frame to another. The coordinates Xof X\, &2, Xz of a point ï&S with respect to the frame eo, ei, C2, e3 are defined by the equation (2) % = XQCO + Xtfi + X2t2 + Xst3. Let eo*, ei*, e2*, e3* be a frame related to eo, ei, e2, e3 by means of the rela- tions 3 (3) e«* = ]F) daftp, a = 0, 1, 2, 3, OwherN RIEMANNIAe (aap) is a propeN MANIFOLDr orthogonaSl OmatrixF FOU, anR d DIMENSIONSlet #0*, #1*, #2*1 , #3* be the coordinates of the same point x with respect to the frame SHIING-SHEN CHERN eo*, ei*, e2*, e3*. Then we have Introduction. It is well known that3 in three-dimensional elliptic or spherica(3a) l geometry the so-callex*d Clifford'= X) a<*p%p,s parallelis m or parataxot = 0,y 1ha, 2s, 3. locallymany interestinµ = µ ωg properties+ µ ω .+ Aµ group-theoretiω c reason for the most • 1 1 2 2 3 3 importanThe propertiet of thess oef propertiespherical sgeometr is the facy art etha thost the ewhich universa, whel ncoverin expresseg d grouin pterm of ths eo fprope coordinater orthogonas withl grourespecp itn tfouo ar framevariable, remais is thn e invariandirect t ifproducµunde= 0tr chango distinguishedf thee universaof the framel almostcoverin. g complexgroups of structuretwo proper orthogonal • group s in three variables. This last-mentioned property has no ana- Let 1 logue for #oorthogona, #1» ^2, #l3 groupbe th(esµ icoordinate1nI n+ (>4µ2)J svariables+ ofµ a3 Kpoin). Ot n£ thwite hothe respecr handt to, a a framknowledge Co, eie , oe2f , ethree-dimensiona, as defineµ d by (2)l ellipti. To thesc ore coordinatespherical geometrs we associaty is e 3 usefua unil fot rquaternio the studny of orientable Riemannian manifolds of four dimensions, because their tangent spaces possess a geometry of this PROP:(4) This is integrable.X = XQ + xii + X2J + xzk, N(X) = 1, • kind. It is the purpose of this note to give a study of a compact orient- ablwhere Riemanniae N(X) denoten manifols thde onorf foum ro fdimension X. Let s at each point of which F.Battaglia, Circle actions and Morse theory on quaternion- is (4aattache) d a three-dimensionaX* = x* +l sphericax*i+ x*jl space + .x?k. Thi s necessitates a KählermorThee carefun manifolds,the l followinstudy ogJ.Lond.Math.Socf theoresphericam l isgeometr well knowy tha59n n[ (1999)l hithert] : o 345given – in 358. the literature, except, so far as the writer is aware, in a paper by S-S.Chern,E. StudTHEOREy [2].MOn2 0u1. r The Riemannianmai nproper resul t orthogonalconsist manifoldss o f grouptwo formulas of(3a) fourcan, whic be dimensions, expressedh expres s in Bull.twtheo topologica Amer.quaternion Math.l notationinvariant Soc. ins theo51f forma (1945)compac t 964–971.orientable differentiable manifol(5) d of four dimensions asX* integral = AXB,s ove r the manifold of differential invariants constructed from a Riemannian metric previously PROP:givewheren on AMth, eB carriesmanifold are unit a. quaternions.Thes canonicale two topologicaIt complex containsl invariantthe quaternionic two ssubgroups have a linea r • combinatio(6a) n which is the Euler-PoincarX* = AX,é characteristic. 8 1(6b. Three-dimensiona) l sphericaX*l =geometry XB, . We consider an ori- ented Euclidean space of four dimensions E4 with the coordinates called the subgroupsA of left and right translations respectively. A left xo, translationXi, x2, X3. Iisn Ea right let S*translation be the oriente whend anduni tonly hyperspher when it eis defineX* = d± bX.y the equation It is important to 2giv e a2 distinctio2 2n between the left and right (1) Xo + Xl + X2 + #3 = I- Three-dimensional spherical geometry is concerned with properties on 5s which remain invariant under the rotation group (that is, the proper orthogonal group) of E4 leaving the origin fixed. Received by the editors June 22, 1945. 1 The content of this paper was originally intended to be an illustration in the author's article, Some new viewpoints in differential geometry in the large, which is due to appear in this Bulletin. Later it appeared more advisable to publish these results separately, but a comparison with the above-mentioned article, in particular §7, is recommended. 2 Numbers in brackets refer to the references cited at the end of the paper. 964 “Quaternions came from Hamilton after his best work had been done, and though beautifully ingenious, they have been an un- mixed evil to those who have touched them in any way” Lord Kelvin 1890 GEOMETRY OVER THE QUATERNIONS 2 q H quaternions q = x + ix + jx + kx • ∈ 0 1 2 3 algebraic variety? f(q , . , qn) = 0 • 1 q2 + 1 = 0: 2-sphere q = ix + jx + kx , x2 + x2 + x2 = 1 • 1 2 3 1 2 3 5 submanifold M Hn • ⊂ n TxM H • ⊂ m n TxM quaternionic for all x M M = H H • ∈ ⇒ ⊂ 6 INTRINSIC DIFFERENTIAL GEOMETRY 3 quaternionic structure on the tangent bundle T • affine connection Y • ∇X zero torsion Y X = [X, Y ] • ∇X − ∇Y 2 Hn n-dimensional quaternionic vector space • left action by GL(n, H) • commutes with right action of H • GL(n, H) H • · ∗ 3 metric maximal compact subgroup • ⇔ Sp(n) Sp(1) GL(n, H) H • · ⊂ · ∗ Levi-Civita connection : unique torsion-free connection • ∇ preserving metric Quaternionic Kähler preserves quaternionic structure • ⇔ ∇ 4 GL(n, H) preserves action of H on tangent bundle T • I,J,K End(T ) such that I2 = J2 = K2 = IJK = 1 • ∈ − metric Sp(n) GL(n, H) • ⊂ Levi-Civita connection : unique torsion-free connection • ∇ preserving metric Hyperkähler preserves I,J,K • ⇔∇ 3 Kähler form ω Ω1,1 • ∈ SL(n, H) U(1) preserves action of C on tangent bundle T • · dω =0 • if a torsion-free connection preserves this structure, it is • ∇ locallyunique ω = ddcf = dIdf • fcomplexKähler potential quaternionic – complex manifold • volume form U(1) connection on K • ⇒ 2 3 SL(n, H) U(1) • · SL(1, H) U(1) = Sp(1) U(1) = SU(2) U(1) = U(2) • · · · for n = 1 complex quaternionic = Kähler complex surface • with zero scalar curvature n > 1 complex quaternionic is non-metric • 7 Lecture 1 Quaternionic manifolds • Lecture 2 Hyperkähler moduli spaces • Lecture 3 Twistors and holomorphic geometry • Lecture 4 Correspondences and circle actions • 6 THE HYPERKAHLER¨ QUOTIENT 2 hyperkähler manifold M4k • complex structures I, J, K + metric g • Kähler forms ω , ω , ω • ⇒ 1 2 3 1 ω : T T , K = ω− ω etc. • i → ∗ 1 2 3 hyphyperkerkählerähler manifoldmanifold MM44kk •• complexcomplex strstructuresuctures II,,JJ,,KK ++ metricmetric gg •• KKählerähler foformsrms ωω1,,ωω2,,ωω3 •• ⇒⇒ 1 2 3 11 ωωi ::TT TT ,, KK ==ωω−− ωω2 etc.etc. •• i →→ ∗∗ 11 2 33 Lie group G acting on M, fixing ω , ω , ω • 1 2 3 a g vector field Xa • ∈ d(i ω ) + i dω = ω = 0 • Xa i Xa i LXa i moment map i ω = dµa • Xa i i 13 Lie group G acting on M, fixing ω1, ω2, ω3 • Lie group G acting on M, fixing ω1, ω2, ω3 • a g vector field Xa • a ∈ g vector field Xa • ∈ d(i ω ) + i dω = ω = 0 • d(iXaωi) + iXadωi = LXaωi = 0 • Xa i Xa i LXa i moment map i ω = dµa • moment map iXaωi = dµai • Xa i i 13 13 µ : M g R3 • → ∗ ⊗ If G acts properly and freely on µ 1(0) then... • − ... the quotient metric on µ 1(0)/G is hyperkähler... • − ... of dimension dim M 4 dim G • − 5 EXAMPLE EXAMPLE M = Hn = Cn + jCn flat hyperkähler manifold • M EXAMPLE= Hn = Cn + jCn flat hyperkähler manifold • EXAMPLE • M = Hn = Cn + jCn flat hyperkähler manifold i • = n = n + n flat hypω1 =erkähler(dzk manifolddz¯k + dwk dw¯k) •M H C jC i 2 ∧ ∧ • ω = (dz dz¯ + dw dw¯ ) 1 2 k ∧ k k ∧ k • i ω •= (dz dz¯ + dw diw¯ ) ω2 + iω3 = dz dw 1 k k k k k ∧ k 2 ∧ ω1 =∧ ω(dz+kiω dz=¯k +dzdwkdwdw¯k) 22 ∧3 k ∧ ∧ k 1 G = U(1) action u (z, w) = (uz, u− w) • · 1 ωG2 =+ Uiω(1)3 =actiondzk dwu k(z, w) = (uz, u− w) • ∧ ·ω2 + iω3 = dzk dwk ∧ 3 µ(z, w) = (zkz¯k wkw¯k + c, zkwk) R C = R • 1 − ∈ 3× G = U(1) action u (µz(,zw,)w=) =(uz(z,kuz¯−k ww) kw¯k + c, zkwk) R C = R • •·G = U(1) action−u (z, w) = (uz, u∈1w)× • · − 3 3 3 µ(z, w) = (zkz¯k wkw¯k + c, zkwk) R C = R • − µ(z, w) = (z z¯∈ w× w¯ + c, z w ) R C = R3 • k k − k k k k ∈ × 3 3 EXAMPLE EXAMPLE M = Hn = CnEXAMPLE+ jCn flat hyperkähler manifold • M EXAMPLE= Hn = Cn + jCn flat hyperkähler manifold • EXAMPLE M = Hn = Cn + jCn flat hyperkähler manifold • • M = Hn = Cn + jCn flat hyperkähler manifold i • •M = Hn = Cn + jCn flat hypω1 =erkähler(dzk manifolddz¯k + dwk dw¯k) • i 2 ∧ ∧ ω1 = (dzk dz¯k + dwk dw¯k) • 2i ∧ ∧ ω1 = (dzk dz¯k + dwk dw¯k) • i 2 ∧ ∧ ω •= (dz dz¯ + dw diw¯ ) ω2 + iω3 = dz dw 1 k k k k k ∧ k 2 ∧ ω1 =∧ ω(dz+kiω dz=¯k +dzdwkdwdw¯k) 22 ∧3 k ∧ ∧ k ω2 + iω3 = dz dw 1 G = U(1) action u (z, wk)∧= (uzk , u− w) • · 1 ωG2 =+ Uiω(1)3 =actiondzk dwu k(z, w) = (uz, u− w) • ∧ ·ω2 + iω3 = dzk dwk G = U(1) action u (z, w) = (uz, u∧1w) 3 • µ(z, w) = ·(zkz¯k wkw¯k +−c, zkwk) R C = R = (1) action ( ( ) =) =•( ( ¯ 1 ) ¯ +− ) =∈ 3× G U u µz,zw, w uzz,kuz−k wwkwk c, zkwk 1R C R • •·G = U(1) action−u (z, w) = (uz, u∈− w)× • µ(z, w) = (z z¯ w· w¯ , z w ) + const.

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