Chapter 1 DIFFERENTIAL GEOMETRY OF REAL MANIFOLDS

294 CHAPTER 1. DIFFERENTIAL GEOMETRY ...
Exercise 1.3.27 Show that for an R k -valued function f on a Riemannian manifold the two
definitions of ∆f given above are identical. (R k is endowed with the standard flat Euclidean
metric.)
Exercise 1.3.28 Assume M has dimension 1, so that M is either an open interval in R or
the circle. Show that f : M → N is harmonic if and only if f is a geodesic.
Example 1.3.9 Let j : M → R N be an isometric immersion so that M maybe locally
regarded as a submanifold of R N . We want to see when the mapping j is harmonic. We
choose moving frames on R N such that e1, · · · , em are tangent to j(M) and also use e1, · · · , em
as a moving frame on M. Denoting the coframes forms on M and R N by ωi and θA, and the
connection forms by ωij and θAB, we obtain
j ⋆ (θi) = ωi, j ⋆ (θp) = 0, j ⋆ (θij) = ωij. (1.3.35)
It follows that jk i = δk i and j p
i = 0. (Recall the index convention 1 ≤ i, j, · · · ≤ m, m + 1 ≤
p, q, · · · ≤ N.) Therefore (1.3.32) becomes
j ⋆ (θpj) ∧ ωj = 0,
j
which, by means of Cartan’s lemma, determines j p
ij . Comparing with the definition of second
fundamental form it follows that the symmetric matrix (j p
ij ) is the matrix of the second
fundamental form of j(M) in the normal direction ep, and
j
j p
jj = mHp, (1.3.36)
where Hp is mean curvature in the direction ep. The same calculation carried out for a
tangential direction as well. In fact going through the calculation of the Laplacian ji jk we see
that, for each i ≤ m, the matrix (ji jk ) is determined by Cartan’s lemma and the equation
In other words,
(ωik − j ⋆ (θik)) ∧ ωk = 0.
k
ωik − j ⋆ (θik) =
m
l=1
j i klωl.