DIFFERENTIAL GEOMETRY ATTACKS THE TORUS William Schulz Department of Mathematics and Statistics Northern Arizona University, Flagstaff, AZ 86011

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1 DIFFERENTIAL GEOMETRY ATTACKS THE TORUS William Schulz Depatment of Mathematics Statistics Nothen Aizona Univesity, Flagstaff, AZ 86. INTRODUCTION This is a pdf showing computations of Diffeential Geomety quantities using the Tous as example. No advantage is taken of the paticula qualities of the tous; the calculations ae done as they would be fo any suface, but of couse have simple esults because the suface is simple. We make use of the embedding at the beginning to get the metic coefficients g ij then poceed in a Riemannian manne. Fo the tous enthusiast I have added a section at the end that teats the nomal vecto uses it to find the Gaussian Cuvatue. 2. NOTATION AND METRIC COEFFICIENTS We begin by paametizing the tous by longitude latitude as usual. The tous is paametized by θ which is the angle going ound the big sweep of the tous fom to 2π by φ which is the angle going aound the little waist of the tous, also fom to 2π. The paametization is xθ, φ < R + cosφcos θ, R + cosφsin θ, sin φ > Remembe that fo systematic computation puposes we set the paametes u θ u 2 φ. This ode guaantees that the nomal vecto will point outwad. Next we compute the unit tangent vectos: e x u < R + cosφsin θ, R + cosφcos θ, > e 2 x u 2 < sin φcosθ, sin φsin θ, cosφ > Next is to find the matix of metic coefficients g ij e i e j. These ae easily found to be g e e R + cosφ 2 sin 2 θ + R + cosφ 2 cos 2 θ + R + cosφ 2 g 2 e e 2 R + cosφsin θ sin φcosθ cosθ sin φsin θ g 22 e 2 e 2 2 sin 2 φcos 2 θ + sin 2 φsin 2 θ + cos 2 φ 2

2 In matix fom this is g ij R + cosφ 2 2 We will also need the invese matix g ij g ij CONNECTION AND CURVATURE FORMS We fist want to compute the Chistoffel symbols fo which we need the basic fomulas ij k gik 2 u j + g jk u i g ij u k i jk g im jk m Fom these we get, emembeing that i jk i kj, Then 2 g u g2 2 2 u + g 2 u g u 2 R + cosφ sin φ 2 g 2 u 2 + g 2 u g 2 u R + cosφ sin φ 2 2 g2 2 u 2 + g 22 u g 2 u 2 22 g2 2 u 2 + g 2 u 2 g 22 u 22 2 g 22 2 u 2 g m m g 2 g 2m m g 22 2 R + cosφsin φ R + cosφ sin φ 2 2 g m 2 m g 2 sin φ R + cosφ 2 R + cosφ sin φ R + cosφ 2 2 g 2m 2 m g g m 22 m g g 2m 22 m g

3 Fo futue puposes we place these in matices: γ i j 2 sin φ 2 2 sin φ 2 γ 2 i j sin φ Now we want the connection one foms. These ae defined by ω i j i jk duk o in matix fom by ω i j du du 2 which explicitly fo the sphee gives o ω i j sin φ sin φ sin φ sin φ ωj i sin φ dθ + dφ sin φ dθ dθ dφ Next we want to compute the Riemann Cuvatue Tenso Fom which is given by Ω dω + ω ω As the calculations ae goss the intemediate esults of almost no inteest, they will be elegated to an Appendix to this section. The final esult is cos φ Ω dω + ω ω cos φ R + cosφ Recalling that hee the matix enties of Ω is R 2 R2 Ω 2 R 2 2 R2 2 2 we can ead off the values of the Riemann Cuvatue Tenso as R 2 R 2 2 cos φ R 2 2 cos φ R + cosφ R2 2 2 We can now get the Gaussian Cuvatue fom the good old stad fomula of Gauss K g 2mR m 2 detg ij 3

4 which hee gives us K g 22R 2 2 detg ij 2 cos φ R + cosφ 2 R + cosφ 2 cosφ R + cosφ It is amusing to compute the cuvatua intega at this point. 2π 2π cosφ K ds detg ij dθdφ T 2 R + cosφ 2π 2π cosφ R + cosφdθdφ R + cosφ 2π 2π cosφ dθdφ 2π 2π which is the esult we expect fom the Gauss-Bonnet theoem. cosφ dφ APPENDIX: Calculation of the Riemann Cuvatue Tenso Recall that the cuvatue fom ω is given by whee γ i j To compute dω we need sinφ φ R + cosφ so we have ω γ dθ + γ 2 dφ γ 2 i j R + cosφsin φ φ dω sin φ sin φ sin φ R + cosφ cosφ sinφ sinφ R + cosφ 2 R cosφ + R + cosφ 2 R cosφ + cos2 φ sin 2 φ R cos φ+ 2 R cosφ + cos2 φ sin 2 φ 4

5 Note the evesal of signs because the staightfowad application of d to ω esults in dφ dθ which is the wong ode. Next we compute ω ω. Recall that so that ω γ dθ + γ 2 dφ ω ω γ dθ + γ 2 dφ γ dθ + γ 2 dφ γ γ 2 γ 2 γ So we need γ γ 2 γ 2 γ sin φ γ γ 2 sin φ so we have γ 2 γ sin φ ω ω sin 2 φ sin φ sin φ sin φ 2 sin 2 φ 2 To put dω ω ω togethe we need 2 sin 2 φ 2 sin 2 φ R cosφ + R + cosφ 2 2 sin 2 R + cosφ 2 R cosφ sin 2 φ R + cosφ 2 R cosφ + 2 cos 2 φ R + cosφ 2 cosφr + cosφ R + cosφ 2 cosφ R + cosφ so that we finally have Ω dω + ω ω cos φ cos φ R + cosφ 5

6 4. The NORMAL VECTOR AND QUANTI- TIES ASSOCIATED WITH IT Recall that the paametization of the suface is xθ, φ < R + cosφcos θ, R + cosφsin θ, sin φ > with tangent vectos e e 2 given by e x u < R + cosφsin θ, R + cosφcos θ, > e 2 x u 2 < sin φcosθ, sin φsin θ, cosφ > The nomal vecto n is given by n x u x u 2 < R + cosφ cosθ cosφ, R + cosφ sin θ cosφ, R + cosφ sinφ > R + cosφ < cosφcos θ, cosφsin θ, sin φ > its length, which also gives the multiplie in the aea element fo the tous, is n 2 R + cosφ 2 2 cos 2 φcos 2 θ + cos 2 φsin 2 θ + sin 2 φ R + cosφ 2 2 n R + cosφ This gives a unit tangent vecto Recall the stad fomula ˆn < cosφcosθ, cos φsin θ, sin φ > e i u j e k k ij + ˆnb ij We wish to compute the b ij. This is easy fo donuts, because it is so easy to take deivatives of ˆn. We obviously have Since e i ˆn, we can ewite this as We need ˆn θ ˆn φ b ij e i u j ˆn b ij e i ˆn u j < cosφsin θ, cosφcosθ, > < sinφcos θ, sinφsin θ, cosφ > 6

7 Now we have b e ˆn θ R + cosφsin 2 φcosφ + R + cosφcos 2 φcosφ + R + cosφcos φ b 2 b 2 e 2 ˆn θ sin φcosθ cosφsin θ sin φsin θ cosφcos θ + b 22 e 2 ˆn φ sin 2 φcos 2 θ + sin 2 φsin 2 θ cos 2 φ thus b ij R + cosφcos φ With this we can find the Gaussian Cuvatue which elates the infinitesmal aea on the unit sphee to the infinitesmal aea on the suface via the Gauss Map given by ˆn: K detb ij R + cosφcosφ cosφ detg ij 2 R + cosφ 2 R + cosφ 7