Classical di®erential geometry of two-dimensional surfaces 1 Basic de¯nitions This section gives an overview of the basic notions of di®erential geometry for two- dimensional surfaces. It follows mainly Kreyszig [Kre91] in its discussion. De¯nition of a surface Let us consider the vector function X(»1; »2) R3 with 2 X : R2 ¥ (»1; »2) X(»1; »2) U R3 ; (1) 3 7! 2 ½ where ¥ is an open subset of R2. Let X(»1; »2) be of class r 1 in ¥, which means ¸ that one of its component functions Xi (i x; y; z ) is of class r and the other 1 2 f g @(Xx;Xy;Xz) ones are at least of this class. Let furthermore the Jacobian matrix @(»1;»2) be of rank 2 in ¥ which implies that the vectors @X ea := = @aX ; a 1; 2 ; (2) @»a 2 f g are linearly independent. The mapping (1) then de¯nes a smooth two-dimensional surface patch U embedded in three-dimensional Euclidean space R3 with coordi- nates »1 and »2 (see Fig. 1). A union § of surface patches is called a surface if two arbitrary patches U and U 0 of § can be joined by ¯nitely many patches 0 U = U1; U2; : : : ; Un¡1; Un = U in such a way that the intersection of two subsequent patches is again a surface patch [Kre91, p. 76]. To simplify the following let us restrict ourselves to a surface that can be covered by one patch U only. The vectors ea, de¯ned in Eqn. (2), are the tangent vectors of the surface. They are not normalized in general. Together with the unit normal e e n := 1 £ 2 ; (3) e e j 1 £ 2j they form a local basis (local frame) in R3 (see Fig. 2): ea n = 0 ; and n n = 1: (4) ¢ ¢ 1 A function of one or several variables is called a function of class r if it possesses continous partial derivatives up to order r. 1 z Σ U y X ξ 2 Ξ x ξ1 Figure 1: Parametrization of a surface n e2 γ e1 z X y x Figure 2: Local frame on the surface 2 The metric tensor (¯rst fundamental form) With the tangent vectors ea, one can de¯ne the metric tensor (also called the ¯rst fundamental form) gab := ea eb : (5) ¢ This covariant second rank tensor is symmetric (gab = gba) and positive de¯nite [Kre91, p. 86]. It helps to determine the in¯nitesimal Euclidean distance in terms of the coordinate di®erentials [Kre91, p. 82] 2 1 1 2 2 1 2 2 1 2 2 ds = [X(» + d» ; » + d» ) X(» ; » )] = (e1 d» + e2 d» ) a 2 ¡a b = (ea d» ) = (ea eb) d» d» a b ¢ = gab d» d» ; (6) where the sum convention is used in the last two lines (see App. ??). The con- travariant dual tensor of the metric may be de¯ned via cb b 1; if a = b gac g := ± := ; (7) a 0; if a = b ( 6 b where ±a is the Kronecker symbol. The metric and its inverse can be used to raise and lower indices in tensor equations. Consider for instance the second rank tensor tab: cb b c Raising: tac g = ta ; and lowering: ta gcb = tab : (8) The determinant of the metric2 g := det g = gab = g g g g (9) j j 11 22 ¡ 12 21 can be exploited to calculate the in¯nitesimal area element dA: let be the angle between e1 and e2 (see Fig. 2). Then e e 2 = e 2 e 2 sin2 = g g (1 cos2 ) = g g (e e )2 j 1 £ 2j j 1j j 2j 11 22 ¡ 11 22 ¡ 1 ¢ 2 = g g g g = g ; (10) 11 22 ¡ 12 12 and thus dA = e e d»1 d»2 = pg d2» : (11) j 1 £ 2j The covariant derivative The partial derivative @a is itself not a tensor. One therefore de¯nes the covariant a1a2:::an derivative a on a tensor t r b1b2:::bm a1a2:::an a1a2:::an ct = @ct r b1b2:::bm b1b2:::bm d a2:::an a1 a1d:::an a2 a1a2:::d an + tb1b2:::bm ¡dc + tb1b2:::bm ¡dc + : : : + tb1b2:::bm ¡dc ta1a2:::an ¡ d ta1a2:::an ¡ d : : : ta1a2:::an ¡ d ; (12) ¡ d b2:::bm b1c ¡ b1d:::bm b2c ¡ ¡ b1b2:::d bmc 2 Note that g is the matrix consisting of the metric tensor components gab. 3 c where the ¡ab are the Christo®el symbols of the second kind with c c ¡ = (@aeb) e ; (13) ab ¢ and a is now a tensor. For the covariant di®erentiation of sums and products of tensorsr the usual rules of di®erential calculus hold. The metric-compatible Laplacian a ¢ can be de¯ned as ¢ := a . r r Note in particular that c aeb = @aeb ¡ab ec ; and (14) r bc¡ agbc = ag = ag = 0 : (15) r r r Equation (15) is also called the Lemma of Ricci. It implies that raising and lowering of indices commutes with the process of covariant di®erentiation. Orientable surfaces The orientation of the normal vector n in one point S of the surface depends on the choice of the coordinate system [Kre91, p. 108]: exchanging, for instance, »1 and »2 also ips n by 180 degrees. A surface is called orientable if no closed curve through any point S of the surface exists which causes the sense of n to change whenC displacing n continuously from S along back to S. An example of a surface that is not orientable is the MÄobius strip. C The extrinsic curvature tensor (second fundamental form) Two surfaces may have the same metric tensor gab but di®erent curvature properties in R3. In order to describe such properties let us consider a surface § of class3 r 2 and a curve of the same class on § with the parametrization X(»1(s); »2(s))¸on §, where s isCthe arc length of the curve (see Fig. 3). At every point of the curve where its curvature k > 0, one may de¯ne a moving trihedron t; p; b where t = X_ is the unit tangent vector, p = t_= t_ = t_=k is the unit principalf normalg vector, and b = t p is the unit binormal vectorj jof the curve.4 Furthermore, let ´ be the angle between£the unit normal vector n of the surface and the unit principal normal vector p of the curve with cos ´ = p n (see again Fig. 3). The curvature k of the curve can then be decomposed into a¢part which is due to the fact that the surface is curved in R3 and a part due to the fact that the curve itself is curved. The former will be called the normal curvature Kn, the latter the geodesic curvature Kg. One de¯nes: K := t_ n = k (p n) = k cos ´ ; and (16) n ¡ ¢ ¡ ¢ ¡ K := t (t_ n) = k t (p n) = k sin ´ sign (n b) : (17) g ¢ £ ¢ £ ¢ 3 This means that its parametrization X(»1; »2) is of class r 2. 4 The dot denotes the derivative with respect to the arc length¸ s. 4 Σ n η S t p C b Figure 3: Curve on a surface Here, we are interested in the curvature properties of the surface. Therefore, the 5 normal curvature Kn is the relevant quantity that has to be studied a bit further. The vector t_ may be written as d t_ = XÄ = (e »_a) = (@ e ) »_a»_b + e »Äa : (18) ds a b a a Thus, Eqn. (16) turns into a b K = k cos ´ = ( n @aeb) »_ »_ ; (19) n ¡ ¡ ¢ where it has been exploited that @aeb = @bea. The expression in brackets is the extrinsic curvature tensor or second fundamental form Kab := n @aeb = ea @bn : (20) ¡ ¢ ¢ It is a symmetric covariant second rank tensor such as the metric. The second relation in Eqn. (20) follows if one di®erentiates the ¯rst equation of (4) with respect to »a. The extrinsic curvature can be written covariantly: Kab := n aeb : (21) ¡ ¢ r This is possible because @aeb di®ers from aeb only by terms proportional to the r tangent vectors ec, which vanish when multiplied by n (see Eqn. (14)). 5 The minus sign in the de¯nition of Kn, Eqn. (16), is unfortunately a matter of convention and is here chosen in accordance to the literature where the surface stress tensor for uid membranes has been introduced [CG02, Guv04]. A sphere with outward pointing unit normal has a positive normal curvature then. Note that this di®ers from Ref. [Kre91]. 5 One can easily see from Eqn. (20) that Kab has got something to do with curvature: at every point of the surface it measures the change of the normal vector in R3 for an in¯nitesimal displacement in the direction of a coordinate curve. To learn more about the normal curvature let us consider a reparametrization of the curve with the new parameter t. One gets C d»a dt »a0 »_a = = ; (22) dt ds s0 where 0 denotes the derivative with respect to t. Equation (19) thus takes the form a0 b0 a0 b0 a b K » » (6) K » » K d» d» K = K »_a»_b = ab = ab = ab : (23) n ab 0 2 a0 b0 a b (s ) gab » » gab d» d» For a ¯xed point S, Kab and gab are ¯xed as well. The value of Kn then only depends on the direction of the tangent vector t of the curve. One may search for extremal values of Kn at S by rewriting Eqn.