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In this case the flat connection for is the as the regular exterior differentiation d . If x is the position vector and {} is a local orthonorma frame field with the corresponding local coframe field {} and {} is the connection 1-forms for the frame field , then

dx = (3.30a)

d = (3.30b)

Exterior differentiate Equation (3.30a) gives

0 = d2x = d) .

Substituting in equation (3.30b) and equating coefficients of equal to zero gives

= . (3.30c)

Similarly, exterior differentiating (3.30b) gives

0 = d2= d) .

Substituting in equation (3.30b) and equating coefficients of equal to zero gives

= . (3.30d)

It follows from (3.30d) that the curvature forms for

:= = 0

are identically zero or has constant curvature zero.

We will develop some formulas that can be used to easily compute the total and mean curvature of a surface in E3 defined by a vector function x(u, v) .

Gaussian and Mean Curvatures Using Weingarten Transformation

Let M be a piece surface in E3 with normal vector field N defined on it. For each p M, the Weingarten transform W is linear map of TpM 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 TpM , then

W(Z)Z, W(Z)Y

W(Y)Z, W(Y)Y

K(p) = ----------------------------

( ZZ) (YY) - (ZY )2

W(Z)Z, W(Z)Y + ZZ , ZY

YZ , YY W(Y)Z, W(Y)Y

H(p) = -------------------------------------------------------------------

2 (( ZZ) (YY) - (ZY )2)

Proof: In this proof x denotes the cross product between two vectors in two vectors in E3 . First of all, let us recall the following formulas from vector analysis

2= (ZZ)(YY) – (ZY)2 ,

(Z x Y) (A x B) = (Z A)(Y B) – (Z B)(Y A)

Since {Z, Y} is a basis for TpM , we can write

W(Z) = aZ + bY

W(Y) = cZ + dY

Hence K(p) = det(W) = ad – bc and H(p) = (1/2)trW = (1/2)(a+d) . Using standard properties of cross product, we compute

W(Z) x W(Y) = (aZ + bY) x (cZ + dY)

= (ad – bc)Z x Y

= K(p) Z x Y .

Taking dot products of both sides with Z x Y and solve for K(p) gives us the first identity.

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.

Let the piece of surface be defined parametrically by a vector value function

X(u, v) : U M E3 for a connected open set U in R2 .

Define

Xu = X/u and Xv = X/v

and set

E = XuXu , F = Xv = XvXu , G = Xv Xv .

Then the first fundamental form I of M can be written as

I = dXdX = Fdu2 + 2Fdudv + Gdv2.

Define a normal vector field N by

N = Xu x Xv /|| Xu x Xv || .

We also set

Xuu = 2X/(u )2, Xuv = 2X/uv and Xvv = 2X/vv

l= W(Xu) Xu, m = W(Xu) Xv= W(Xv) Xu, n= W(Xv) Xv

The second fundamental form II of M is then given by

II = d2X N = - dXdN = ldu2 + 2 mdudv + ndv2

Corollary 1. If a piece of surface M is defined parametrically by the vector valued function

X: U , then

K = (ln - m2)/(EG – F2)

H = (G l+ E n - 2F m)/2(EG – F2)

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 Xu N = 0, we have

0 = (N Xu )/ u = N/u Xu + N Xuu .

Therefore

l= W(Xu) Xu = -N/u Xu = N Xuu .

Similarly we have

n = W(Xv) Xv = NXvv and

m = W(Xu) Xv = NXuv

[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

§1 Local Geometry of Submanifolds of a Riemannian manifold in Moving Frames

Let X be a C Riemannian manifold of dimension N = n + p, M be a manifold of dimension n and let

f : M X

differentiable immersion. By imposing on M the induced metric we can suppose M to be Riemannian and f to be an isometric immersion. We agree the following ranges of indices:

(1.1) 1 , , ,, N; 1 i, j, k,, n; n + 1 A, B, C,, n + p;

Let D be the Riemannian connection of X and be a local orthogonal frame define in an open set U of X, with such that (x) TxM, the ordered set

{} defines the orientation of X and satisfies

(1.2) < , > = .

The frame eαdefines, through pulling back of the structure equations in the frame bundle of M, uniquely a dual coframeM. Condition (2) is equivalent to the following expression of the element of arc length:

(1.3) ds2 =

We define a local connection one forms by the equation

D = .

The differential forms satisfy the following

(1.4) + = 0.

Then we have as usual

(1.5) d

We also have

= d -

(1.6)

= - (1/2)

where the coefficients locally defines the Riemannian Christopher tensor field with respect the frame field .

Let M be a manifold of dimension n and let

f : M X

be a differentiable immersion. By imposing on M the induced metric we can suppose M to be Riemannian and f to be an isometric immersion.

If TX denotes the tangent bundle of X, its induced bundle over M splits into a direct sum

(1.7) f*(TX) = TM(TM)┴

where TM and (TM) ┴ are respectively the tangent bundle and normal bundle of M.

We restrict to a neighborhood of M and consider a frame field (m), m M, of the bundle f*(TX) such that (m) are tangent vectors and (m) are normal vectors at m. Let , be the forms previously denoted by , relative to this particular frame field . Then we have

(1.8) = 0 .

Taking exterior derivative of both sides of (8) we have

(1.9) = 0 .

Then by Cartan’s lemma we have

(1.10) = .

where

(1.11) =

(1.12) = ,

where

(1.13) = ,

is called the second fundamental form of M in X. It describes the simplest metrical properties of M as a submanifold of X.

§2 The first variation of the volume of a submanifold and definition of minimal submanifolds

We follow the notations of § 1. If M is compact, possibly with boundary, its totally volume is given by the integral

(2.1) V = .

We apply a variation of M as follows: Let I be the interval - ½ t ½ . Let F: M x I X be a differentiable mapping such that its restriction to M x t, t I, is an immersion and that F(m,0) = f(m) m M. We consider a frame field (m,t) over M x I such that for every t I, (m,t) are tangent vectors to F(M x I) at (m, t) and hence (m,t) are normal to F(M x I). The forms , can then be written as

= + aidt , = dt,

(2.2)

iA= iA+ dt,

We write the operator d on M x I as

(2.3) d = + dt (/t) .

From (1.5) we get

(2.4) (…) =

where

(2.5) = -

Substitute into (2.4) the expression in (2.2), we get its two sides,

LHS = d{… + dt (-1)i-1ai…… },

RHS = ,

(2.6) = -

Equating terms in dt, we get

(2.7) (/t) (…) = { (-1)i-1ai…… }

+,

Integrating over and setting t = 0, we find the first variation of volume

V’(0) = (/t)

(2.8)

= +,

where set t = 0 in the integrands of the last two integrals.

The second terms at the right hand side of (2.8) vanishes if (m, t) = 0,

m M, t = 0, that is, if the deformation vector is orthogonal to M along the boundary M. This condition is a fortiori satisfied if the boundary M remains fixed. The first integral is zero for arbitrary aAif and only if

(2.9) = 0,

that is equivalent to the fact that the mean curvature vector H is zero, that is (following the notations of §1)

H : = (1/n)

(2.10)

= 0 .

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”.

Theorem 1.1. Let Gn-1 be an imbedded (n-1)-dim submanifold of a Riemannian manifold X and P be an n-dim distribution along Gn-1 such that TqGn-1 P(q) for all q Gn-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.Gn-1U N U,

2.TqN = P(q) for all q Gn-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,

Examples of minimal submanifolds.

In the following examples M will be R2 or an open subset of it and X will E3 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:

x(u, v) = (ccosh(v/c)cos(u), ccosh(v/c)sin(u), v)

(c being a positive constant)

The line element is

ds2 = cosh2(v/c) dv2 + c2cosh2 (v/c) du2.

Its mean curvature H = 0 and its Gaussian curvature K is

K = -(1/c2)sech4(v/c),

with principle curvatures (1/c)sech2(v/c)

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.

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

ds2 = dρ2 + (2+ρ2)d2,

and its mean curvature H = 0 Gaussian curvature is given by

K = -2/(2 + ρ2)2,

and the principle curvatures are /(2 + ρ2) .

A physical model of a catenoid can be formed by dipping two circles into a soap solution and slowly drawing the circles apart.

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 system

for, , with deformation parameter ,

where = corresponds to a right-handed helicoid, = /2 corresponds to a catenoid, and = 0 corresponds to a left-handed helicoid.

§2 The reflection principle for a minimal submanifold in an arbitrary Riemannian manifold

It is theorem of Schwartz that every complete minimal submanifolds in E3 that contains a straight line in E3 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 S3 in[L5] .This fact has been used elegantly to construct beautiful and topologically very interesting smooth compact minimal submanifolds of S3 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 pM, v,wTpM and if GM and GN denotes the scalar products of TpM and Th(p)N respectively then

GM (v,w) = GN (h* (v), h* (w)).

The set of all isometries of a Riemannian manifolds M is Lie group G.

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 En is a totally geodesic submanifold of En.

In the n-sphere Sn := {x En+1 : ||x|| =1} considered as Riemannian manifolds of constant curvature, the intersection of Sn with any linear subspace of dimension d +1 n+1, through 0 En+1 , is a d dimensional totally geodesic submanifolds of Sn .

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]).

Definition 1.2. An imbedded submanifold B of a Riemannian manifold X is said to be locally reflective if there exists an involutiveisometryρB (i.e. ρB2 = 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 ρB. Furthermore, if B is globally reflective and N is complete with respect to the induced metric, then N is invariant under ρB.

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 C2 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.

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 ρB is defined in on a neighborhood of W in M. Since ρB is an isometry,

ρB (W) is also a minimal submanifold of M which contains W B. For q W B, TqW considered as a subspace of TqM is spanned by TqB and a nonzero vector v TqWTqB┴, TqB┴ being the orthogonal complement of TqB in TqM. ρB* leaves TqB fixed. Since ρB2 = id, we must have ρB* (v) = -v. In other words we have

TqW = Tq(ρB (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 = ρB (W) U. this proves the first part of the theorem. The second part of the theorem now follows ready from Lemma 2.

§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.

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