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By Luther Pfahler Eisenhart

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2) from Recall 2(QA Q +AAA). = SdS using AS Therefore, d(A -SA, QS = Q) + = 2(A = + Q). (dS A -SQ, = Id(SdS) 2 1 = 2 dS) Q) =2(AAA+AAQ+QAA+QAQ). = But A A Q 0 = by 2S(A following the QA A Similarly stabilizing Notice that QxO 0 QzO independent if = = Proof. which = A* = ff", Q *AQ - 0, because A Let L C H be Lemma5. 7r A S(A + Using and that k)) Q "left have AAQ H Q) type argument: "right A is we + L such that an A(-SQ) = is left K and Q is right surface immersed dSL C L. Then (-AS)Q - QJL and S = 0 is = 0.

Proposition 6. This The = is closed. 3), then >. particular, 1 degS:= 7r is 2(*Q IL7 from Lemma7. ws(X,Y)=, 2. ) The 1. 2 have follows positivity A traceR - a topological f - A M of S. invariant S maps the surface Minto the 8-dimensional Z, deg S cermapping degree of S. But for immersed holomorphic curves of two mapping degrees deg S it is the difference deg N deg R, where N, R : M-4 S2 are the left and right normal vector in affine coordinates, see chapter 7. Remark 7. tainly Since is not the = Proof.

1 4KIdf 14 =< *df dR *df < - (df dR + < N = < < N(df < df df dR + *dNdf, df dfdR, dfR < dNdf, df + < df + < *dNdf, + dfRdR ldf 12 I dyl2(< 21dfI2 (< Kjafter find, we 4K 1 jdfJ2 =< N similar a =< *dR As we use a Proposition to we 10. this The (R = N) df dN, < * < N* dNdf > N * dN > dR, NdNdf *dN, > NdN > dN,N* dN > * and the Ricci X) II(JX, - dN >). 7). 1) K. 5) corollary * dNdf, computation, df (*dR + < On this for the formula proves dR,R > > < > + < < - dNdf > +jdf 12 dR,R*dR *dR, RdR > < This Ndf , dR > > NdNdf + > > dfdR, N dNdf > + < dR > dNdf < > + < *dR, RdR > < *dNdf < - RdR > dNdf, Ndf + < - R * dR > dR, < dfRdR dR > * dR,dfRdR * ldf 12 - *dNdf, < dR + N * * > dNdf) + > *dNdf) dR + * N(-dfdR dNdf, dfR + -df dR + 41 > *dNdf dR + * *dNdf), dNdf, N(-df dR + *dNdf df, -df dR + dNdf * dNdf ), -df dR + * dfdR + < *dN - dR + * dR + * < =- dR * df, -df dN * - Space Euclidean in this f of =< 2-sphere area under R is given *dR, RdR > M yields Kjdf 12 A M is the a version 2 (deg ofthe R+ deg N).

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