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Lecture 18. The Gauss and Codazzi equations

In this lecture we will prove the fundamental identities which hold for theextrinsic curvature, including the Gauss identity which relates the extrinsiccurvature defined via the second fundamental form to the intrinsic curvaturedefined using the Riemann tensor.

18.1 The fundamental identities

The definitions (from the last lecture) of the connection on the normal andtangent bundles, and the second fundamental form h and the associatedoperator W, can be combined into the following two useful identities: First,for any pair of vector fields U and V on M,

DUDVX=h(U, V) + DX(UV). (18.1)

This tells us how to differentiate an arbitrary tangential vector field DVX,considered as a vector field in RN (i.e. anN-tuple of smooth functions). Thenwe have a corresponding identity which tells us how to differentiate sections ofthe normal bundle, again thinking of them as N-tuples of smooth functions:For any vector field U and section ofN M,

DU= DX(W(U, )) + U. (18.2)

Since we can think of vector fields in this way as N-tuples of smoothfunctions, we can deduce useful identities in the following way: Take a pairof vector fields U and V. Applying the combination U V V U[U, V] toany function gives zero, by definition of the Lie bracket. In particular, we canapplying this to the position vector X:

0 = (U V V U [U, V])X

=h(U, V) + DX(UV) + h(V, U) DX(VU) DX([U, V]).

Since the right-hand side vanishes, both the normal and tangential com-ponents must vanish. The normal component is h(V, U) h(U, V), so thisestablishes the fact we already knew that the second fundamental form issymmetric. The tangential component isDX(UV VU [U, V]), so the

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172 Lecture 18. The Gauss and Codazzi equations

vanishing of this tells us that the connection is symmetric (as we provedbefore).

18.2 The Gauss and Codazzi equations

We will use the same method as above to deduce further important identities,by applying U V V U [U, V] to an arbitrary tangential vector field.

Let Wbe a smooth vector field on M. Then we have

0 = (U V V U [U, V])W X

=U(V W X) V(U W X) [U, V]W x

(18.1)= U(h(V, W) + (VW)X) V(h(U, W) + (UW)X)

h([U, V], W) + ([U,V]W)X

(18.2)

= DX(W(U, h(V, W))) (h(U,V ,W ) + h(UV, W) + h(V, UW))

+ DX(W(V, h(U, W))) (h(V ,U,W ) + h(VU, W) + h(U,VW))

+ U(VW)X V(UW)X+ h([U, V], W) ([U,V]W)X(18.1)

= DX(W(U, h(V, W))) (h(U,V ,W ) + h(UV, W) + h(V, UW))

+ DX(W(V, h(U, W))) (h(V ,U,W ) + h(VU, W) + h(U,VW))

h(U,VW) + (UVW)X+ h(V, UW) (VUW)X

+ h([U, V], W) ([U,V]W)X

=DX(R(V, U)W W(U, h(V, W)) + W(V, h(U, W)))

+ h(U,V ,W ) h(V ,U,W ).

In deriving this we used the definition of curvature in the last step, andused the symmetry of the connection to note that several of the terms cancelout. The tangential and normal components of the resulting identity are the

following: R(U, V)W=W(U, h(V, W)) W(V, h(U, W)) (18.3)

andh(U,V ,W ) = h(V ,U,W ). (18.4)

Note that the tensor h appearing here is the covariant derivative of thetensor h, defined by

h(U,V ,W ) = U(h(V, W)) h(UV, W) h(V, UW)

for any vector fields U, V andW. Here the appearing in the first term onthe right-hand side is the connection on the normal bundle, and the othertwo terms involve the connection on T M.

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18.3 The Ricci equations 173

Equation (18.3) is called the Gauss equation, and Equation (18.4) theCodazzi equation. Note that the Gauss equation gives us a formula for theintrinsic curvature ofMin terms of the extrinsic curvatureh.

It is sometimes convenient to write these identities in local coordinates:Given a local chart with coordinate tangent vectors 1, . . . , n, choose alsoa collection of smooth sections e of the normal bundle, = 1, . . . , N n,

which are linearly independent at each point. Let g be the metric onT Mandg the metric onN M, and write gij =g(i, j) and g = g(e, e). Then wecan write

h(i, j) = hije

andW(i, e) = Wi

jj.

The relation between h and Wthen tells us that Wij =gjk ghik

.The Gauss identity then becomes

Rijkl =

hjkhil

hjkhil

g (18.5)

If we write h= ihjkdxi dxj dxk e, then the Codazzi identity

becomes ihjk =jhik

. (18.6)

Since we already know that hij is symmetric in j and k, this implies thatkhij is totally symmetric ini, j and k .

18.3 The Ricci equations

We will complete our suite of identities by applying U V V U [U, V] to anarbitrary section of the normal bundle:

0 = (U V V U [U, V])

(18.2)

= U(W(V, )X+ V) V(W(U, )X+ U) (W([U, V], )X+ [U,V])

(18.1)= h(U,W(V, )) + (W(U,V ,) + W(UV, ) + W(V, U))X

+ h(V, W(U, )) (W(V ,U,) + W(VU, ) + W(U,V))X

+ UV VU W([U, V], )X [U,V]

(18.2)= h(U,W(V, )) + (W(U,V ,) + W(V, U))X

+ h(V, W(U, )) (W(V ,U,) + W(U,V))X

+ W(U,V)X+ UV W(V, U)X VU [U,V]

=R(V, U) h(U,W(V, )) + h(V, W(U, ))

+ (W(U,V ,) W(V ,U,)) X.

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174 Lecture 18. The Gauss and Codazzi equations

As before, this gives us two sets of identities, one from the tangentialcomponent and one from the normal component. In fact the tangential com-ponent is just the Codazzi identities again, but the normal component givesa new identity, called the Ricci identity, which expresses the curvature of thenormal bundle in terms of the second fundamental form:

R(U, V)= h(V, W(U, )) h(U,W(V, )). (18.7)

In local coordinates, with a local basis for the normal bundle as above, thisidentity can be written as follows: If we write Rij = g(R

(i, j)e, e)and hij = g(h(i, j), e), then we have

Rij =gkl(hikhjl hjkhil). (18.8)

18.4 Hypersurfaces

In the case of hypersurfaces the identities we have proved simplify somewhat:First, since the normal bundle is one-dimensional, the normal curvature van-

ishes and the Ricci equations become vacuous.Also, the basis {e} for N Mcan be taken to consist of the single unitnormal vector n, and the Gauss and Codazzi equations become

Rijkl= hikhjl hjkhil

andihjk = jhik.

The curvature tensor becomes rather simple in this setting: At any pointx Mwe can choose local coordinates such that 1, . . . , n are orthonormalat x and diagonalize the second fundamental form, so that

hij = i, i= j

0, i=j

Then we have an orthonormal basis for the space of 2-planes 2TxM, givenby{ei ej : i < j}. The Gauss equation gives for all i < j and k < l

Rm(ei ej , ek el) =

ij , i= k, j = l

0, otherwise.

In particular, this basis diagonalizes the curvature operator, and the eigen-values of the curvature operator are precisely ij for i < j. Note that all ofthe eigenvectors of the curvature operator are simple planes in 2T M. It fol-lows that ifMis any Riemannian manifold which has a non-simples 2-planeas a eigenvector of the curvature operator at any point, then M cannot beimmersed (even locally) as a hypersurface in Rn+1.

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18.4 Hypersurfaces 175

Example 18.4.1 (Curvature of the unit sphere). Consider the unit sphere Sn,which is a hypersurface of Euclidean space Rn+1 (so we take the immersionX to be the inclusion). With a suitable choice of orientation, we find thatthe Gauss map is the identity map on Sn that is, we have n(z) = X(z)for all z Sn. Differentiating this gives

DX(W(u)) = Du

n= DX(u)

so that W is the identity map on T Sn, and hij = gij . It follows that all ofthe principal curvatures are equal to 1 at every point, and that all of thesectional curvatures are equal to 1. Therefore the unit sphere has constantsectional curvatures equal to 1.

Example 18.4.2(Totally umbillic hypersurfaces). A hypersurface Mn in Eu-clidean space is called totally umbillic if for every x Mthe principal cur-vatures 1(x), . . . , n(x) are equal that is, the second fundamental formhas the form hij = (x)gij. The Codazzi identity implies that a connectedtotally umbillic hypersurface in fact has constant principal curvatures (hencealso constant sectional curvatures by the Gauss identity):

(k)gij =khij =ihkj = (i)gkj

for any i, j and k. Fix k, and choose j = i = k (we assume n 2, sinceotherwise the totally umbillic condition is vacuous). This gives k = 0.Since k is arbitrary, this implies= 0, hence is constant. There are twopossibilities: = 0 (in which case M is a subset of a plane), or = 0 (inwhich case Mis a subset of a sphere).

Example 18.4.3 Spacelike hypersurfaces in Minkowski space The definitionswe have made for the second fundamental form were given for submanifolds ofEuclidean space. However the same definitions work with very minor modifi-cations for certain hypersurfaces in the Minkowski space Rn,1: A hypersurfaceMn in Rn,1 is calledspacelikeif the metric induced onMfrom the Minkowski

metric is Riemannian equivalently, if every non-zero tangent vectoru ofMhasu, uRn,1 >0. This is equivalent to the statement thatMis given as thegraph of a smooth function with slope less than 1 over the plane Rn {0}.

Given a spacelike hypersurface, we can choose at each x M a unitnormal by taking the unique future-pointing vector n which is orthogonal toTxMwith respect to the Minkowski metric, normalized so that n, n= 1.

The definitions are now identical to those for hypersurfaces in Euclideanspace, except that the metric on the normal bundle is now negative defi-nite. The Codazzi identity is unchanged, and the Gauss identity is almostunchanged: The one difference arises from the presence of the g term inEquation (18.5), which is now negative instead of positive. This gives

Rijkl = (hikhjl hjkhil) . (18.9)

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176 Lecture 18. The Gauss and Codazzi equations

We can now compute the second fundamental form for a very specialexample, namely Hyperbolic space Hn, which is the set of unit future timelikevectors in Minkowski space. In this case we have n(z) =z for every z Hn,since 0 = Duz, z= 2Duz, z= 2u, z. Differentiating, we find (exactly asin the calculation for the sphere in Example 18.4.1 above) that hij =gij, sothat the principal curvatures are all equal to 1. The Gauss equation therefore

gives that the sectional curvatures are identically equal to 1.More generally, a hypersurface with all principal curvatures of the same

sign (i.e. a convex hypersurface) in Minkowski space has negative curvatureoperator, while a hypersurface with all principal curvatures of the same signin Euclidean space has positive curvature operator.