1. Differential Geometry Dominic Leung 梁树培 Lecture 17 Geometry of Submanifolds in R n

19. We will develop some formulas that can be used to easily compute the total and mean curvature of a surface in E 3 defined by a vector function x(u, v). Gaussian and Mean Curvatures Using Weingarten Transformation Let M be a piece surface in E 3 with normal vector field N defined on it. For each p  M, the Weingarten transform W is linear map of T p M defined by W(X) = XN  p. The Gaussian K and mean curvature H of M can be computed as follow, K(p) = det(W) and H(p) = tr(W). Proposition 1. If Z and Y are two independent vectors of T p M, then  W(Z)  Z, W(Z)  Y   W(Y)  Z, W(Y)  Y  K(p) = ---------------------------- ( Z  Z) (Y  Y) - (Z  Y ) 2  W(Z)  Z, W(Z)  Y  +  Z  Z, Z  Y   Y  Z, Y  Y   W(Y)  Z, W(Y)  Y  H(p) = ------------------------------------------------------------------- 2 (( Z  Z) (Y  Y) - (Z  Y ) 2 )

21. Similarly, W(Z) x Y = a Z x Y Z x W(Y) = d Z x Y. Therefore we have W(Z) x Y + Z x W(Y) = (a+d)(Z x Y) = tr(W)(Z x Y) = 2H(p)(Z x Y). Taking dot product on both sides with Z x Y and solve for H(p) gives the second identity. If Z and Y are vector fields independent at each point p  M, the above formulas will give the curvature function and mean curvature function respectively.

22. Let the piece of surface be defined parametrically by a vector value function X(u, v) : U  M  E 3 for a connected open set U in R 2. Define X u =  X/  u and X v =  X/  v and set E = X u  X u, F =  X v = X v  X u, G = X v  X v. Then the first fundamental form I of M can be written as I = dX  dX = Fdu 2 + 2Fdudv + Gdv 2. Define a normal vector field N by N = X u x X v /|| X u x X v ||. We also set X uu =  2 X/(  u ) 2, X uv =  2 X/  u  v and X vv =  2 X/  v  v l = W(X u )  X u, m = W(X u )  X v = W(X v )  X u, n = W(X v )  X v The second fundamental form II of M is then given by II = d 2 X  N = - dX  dN = l du 2 + 2 m dudv + n dv 2

23. Corollary 1. If a piece of surface M is defined parametrically by the vector valued function X: U , then K = ( l n - m 2 )/(EG – F 2 ) H = (G l + E n - 2F m )/2(EG – F 2 ) Proof: This is a direct consequence of the two formulas of Proposition 1. Q.E.D. E, F and G can be computed fairly easily. We can also find simpler ways to compute l, m and n. From X u  N = 0, we have 0 =  (N  X u )/  u =  N/  u  X u + N  X uu. Therefore l = W(X u )  X u = -  N/  u  X u = N  X uu. Similarly we have n = W(X v )  X v = N  X vv and m = W(X u )  X v = N  X uv

24. Differential Geometry Dominic Leung 梁树培 Lecture 18

25. References [G] C. Gorodski, Enlarging Totally Geodesic Submanifolds of Symmetric Spaces to Minimal Submanifolds of One Dimension Higher, Proceedings of the American Mathematical Society Volume 132, Number 8 (2004) 2441-2447 [K] S. Kobayashi, Transformation Groups in Differential Geometry (Classics in Mathematics) (Feb 15, 1995) [L1] D.S.P. Leung, Deformation of integrals of exterior differential systems, Trans. Amer. Math, Soc. 170 (1972) 334-358 [L2] D.S.P. Leung, The reflection principle for minimal submanifolds of Riemannian symmetric spaces, J. Differential Geometry 8 (1973)153- 160 [L3] D.S.P. Leung, On the Classification of Reflective Submanifolds of Riemannian Symmetric Spaces, Indiana University Mathematics Journal, Vol. 24, No. 4 (1974) 327-339 [L4] D.S.P. Leung, Reflective Submanifolds. III. Congruency of Isometric Reflective Submanifolds and Corrigenda to the Classification of Reflective Submanifolds, J. Differential Geometry 14 (1979) 167-177 [L5]H.B. Lawson, Jr., Complete Minimal Surfaces in S3. Annals of Mathematics, Second Series, Vol. 92, No. 3 (1970) 335-374

33. The submanifold M is called minimal, if it satisfies either (9) or (10). Most people would prefer (10) since it is stated in a seeming more ‘geometric vector field’ formulation. On the other hand, (10) is the preferred way, or rather the only way, to define the condition of a minimal submanifold if we want to use Cartan-Kahler theory of exterior differential systems to prove a theorem on the initial value problem for minimal submanifold. The following theorem is used in establishing “the reflection principle for a minimal submanifold in an arbitrary Riemannian manifold”.

34. Theorem 1.1. Let G n-1 be an imbedded (n-1)-dim submanifold of a Riemannian manifold X and P be an n-dim distribution along G n-1 such that T q G n-1  P(q) for all q  G n-1. Then assuming the data are real analytic, for each there exists in every sufficiently small neighborhood U of q a unique imbedded analytic minimal submanifold N of dimension such that 1.G n-1  U N U, 2.T q N = P(q) for all q  G n-1 ∩U. A proof of Theorem 1.1 is given in [L1], it uses Cartan-Kahler theory by considering an appropriate exterior differential system I in the bundle of orthonormal frames F(X) of the Riemannian manifold X. The local solutions or the so called integral manifolds of I, satisfying the appropriate initial conditions actually project on to submanifolds giving rise to solutions required by Theorem 1.1,

35. Examples of minimal submanifolds. In the following examples M will be R 2 or an open subset of it and X will E 3 with usual flat Euclidean metric. Catenoid Not counting the plane, it is the first minimal surface to be discovered. It was found and proved to be minimal by Leonhard Euler in 1744. One parametrization of the cantenoid is as follows:minimal surfaceLeonhard Euler x(u, v) = (ccosh(v/c)cos(u), ccosh(v/c)sin(u), v) (c being a positive constant) The line element is ds 2 = cosh 2 (v/c) dv 2 + c 2 cosh 2 (v/c) du 2. Its mean curvature H = 0 and its Gaussian curvature K is K = -(1/c 2 )sech 4 (v/c), with principle curvatures  (1/c)sech 2 (v/c)

36. Helicoid The helicoid, after the plane and the catenoid, is the third minimal surface to be known. It was first discovered by Jean Baptiste Meusnier in 1776. Its name derives from its similarity to the helix: for every point on the helicoid there is a helix contained in the helicoid which passes through that point.planecatenoidminimal surfaceJean Baptiste Meusniernamehelixpoint The Helicoid can be described by the following parametric equations: x(ρ,  ) = (ρcos(  ), ρsin(  ),  ) where -   ρ   and -     , while α is a nonzero constant. If α is positive then the helicoid is right-handed as shown in the figure; if negative then left-handed. The line element is ds 2 = dρ 2 + (  2 +ρ 2 )d  2, and its mean curvature H = 0 Gaussian curvature is given by K = -  2 /(  2 + ρ 2 ) 2, and the principle curvatures are   /(  2 + ρ 2 ).

37. A physical model of a catenoid can be formed by dipping two circles into a soap solution and slowly drawing the circles apart.circles One can bend a catenoid into the shape of a helicoid without stretching. In other words, one can make a continuous and isometric deformation of a catenoid to a helicoid such that every member of the deformation family is minimal (having a mean curvature of zero). A parametrization of such a deformation is given by the systemhelicoid continuousisometric helicoidminimalmean curvature parametrization for,, with deformation parameter, where  =  corresponds to a right-handed helicoid,  =  /2 corresponds to a catenoid, and  = 0 corresponds to a left-handed helicoid.

38. §2 The reflection principle for a minimal submanifold in an arbitrary Riemannian manifold It is theorem of Schwartz that every complete minimal submanifolds in E 3 that contains a straight line in E 3 is symmetric with respect to the perpendicular reflection about that line (check this out for the helicoids). It has been observed that this is also true for minimal submanifolds of three sphere S 3 in[L5].This fact has been used elegantly to construct beautiful and topologically very interesting smooth compact minimal submanifolds of S 3 in [L5]. We will prove a theorem that generalizes this reflection principle for any totally geodesic submanifolds B of a Riemannian manifolds X, provided that the geodesic reflection with respect to B is an isometry of X. We will first of all recall some basic facts about isometric of a Riemannian manifolds. A diffeomorphism h: M  N between Riemannian manifolds is an isometry if it preserves the Riemannian metrics. Thus for each p  M, v,w  T p M and if G M and G N denotes the scalar products of T p M and T h(p) N respectively then G M (v,w) = G N (h * (v), h * (w)). The set of all isometries of a Riemannian manifolds M is Lie group G.

39. Definition 1.1. Let M be a Riemannian manifold and S a connected submanifolds of M, Let p  S. The submanifold S is called to be geodesic at if each M- geodesic which is tangent to S at p is an S-geodesic. The submanifold S is called totally geodesic if it is geodesic at each of its points. Examples: Each linear subspace of dimension d  n of E n is a totally geodesic submanifold of E n. In the n-sphere S n := {x  E n+1 : ||x|| =1} considered as Riemannian manifolds of constant curvature, the intersection of S n with any linear subspace of dimension d +1  n+1, through 0  E n+1, is a d dimensional totally geodesic submanifolds of S n. Theorem 1.2. Let M be a Riemannian manifold and B be any set of isometries of X. Let F be the set of points of X which are left fixed by all elements of B, Then each connected component of F is a closed totally geodesic submanifolds of X. A proof of this fact can be found in [K] (Theorem5.1 of [K]).

40. Definition 1.2. An imbedded submanifold B of a Riemannian manifold X is said to be locally reflective if there exists an involutive isometry ρ B (i.e. ρ B 2 = id), call, defined at least in an open neighborhood U of B in X such that B is precisely the fixed poi t ser of when restricted to U. B is said to be globally reflective or simply relective if it is complete and the isometry is defined every ere exists on X with B contains in its fixed point set. Theorem 1.3. Let M be an analytic Riemannian manifolds, and B a locally reflective (imbedded) submanifold of codimension greater than one. If N is a minimal submanifold of M which contains B as a hypersurface, then every point p  B has an open neighborhood W in N such that W is invariant under the reflection map ρ B. Furthermore, if B is globally reflective and N is complete with respect to the induced metric, then N is invariant under ρ B. We will assume from now on unless otherwise specified that M is analytic. We need two technical lemmas for the proof of Theorem 1.3. Lemma 1. Any C 2 minimal submanifold of M is real analytic. Lemma 1 follows ready from a well known theorem in quasi-linear elliptic partial differential equations. The next lemma asserts the uniqueness of the analytic continuation of real analytic submanifolds.

41. Lemma 2. Let N and N’ be two l-dimensional submanifolds of an analytic manifold M which are both maximal (i.e. not a proper subset of any other connected submanifolds). If there is an open set W in M such that W  N = W N’ and W N contains a coordinate neighborhood of N, then N = N’. Proof of Theorem1.3.: By Lemma 1, we can assume that N and B to be real analytic. Let p  B. Then there exists an open neighborhood W of p in N such that W and W  B are imbedded submanifolds of M and ρ B is defined in on a neighborhood of W in M. Since ρ B is an isometry, ρ B (W) is also a minimal submanifold of M which contains W  B. For q  W  B, T q W considered as a subspace of T q M is spanned by T q B and a nonzero vector v  T q W  T q B ┴, T q B ┴ being the orthogonal complement of T q B in T q M. ρ B* leaves T q B fixed. Since ρ B 2 = id, we must have ρ B* (v) = -v. In other words we have T q W = T q (ρ B (W)) for all q  W  B. Applying Theorem 1.1, we can conclude that there exists an open neighborhood U of W in M such that W  U = ρ B (W)  U. this proves the first part of the theorem. The second part of the theorem now follows ready from Lemma 2.

42. §3 Reflective submanifolds of a Riemannian symmetric spacese For arbitrary Riemannian manifold, there may not exit any totally geodesic submanifold of dimension greater than one. If we will restrict ourselves to Riemannian symmetric spaces, there are many reflective submanifolds in every Riemannian symmetric space. See [L3 and L4] for details. In the next few classes, we will briefly study symmetric spaces, a theory that was initiated by Cartan.