3.2. ANALYSIS ON RIEMANNIAN MANIFOLDS 131
and its gradient (i.e. covariant derivatives) read as
(3.2.29)
∇u={Dkui},Dkui=
∂ui
∂xk+Γiklul,and the divergence of∇uis
div·∇ui=DkDkui=gklDl(Dkui) =gkl[
∂
∂xl(Dkui)+Γil jDkuj−ΓkljDjui]
.
Then, by (3.2.29) we obtain that
div·∇u=gkl[
∂
∂xl(
∂ui
∂xk+Γik juj)
+Γil j(
∂uj
∂xk+Γksjus)
−Γklj(
∂ui
∂xj+Γijsus)]
(3.2.30).
6.Laplace-Beltrami operators.The Laplace-Beltrami operators are defined by∆=dδ+δd,wheredis the differential operator, andδthe Hodge operator, which are defined on the spaces
of all differential forms. For a scalar field,∆is the same as div·∇, i.e.
∆u=div·∇u, asu∈W^2 ,p(M⊗pR^1 ).For vector fields and covector fields, we have
(3.2.31)
∆ui=−div(∇ui)−gijRjkuk,
∆ui=−div(∇uj)−gk jRijuk,whereRijis the Ricci curvature tensor defined by (2.3.31), and in (3.2.30) we give the formula
of div(∇uk). The expression of div(∇ui)is written as
div(∇ui) =gkl{
∂
∂xl[
∂ui
∂xk−Γikjuj]
−Γlkj[
∂ui
∂xj−Γrijur]
−Γlij[
∂uj
∂xk−Γrjkur]}
(3.2.32).
Physically, it suffices to introduce the Laplace-Beltrami operators∆only for vector and
covector fields, i.e. the formula in (3.2.31).
Remark 3.10.The Navier-Stokes equations governing a fluid flow on a sphere(e.g. the
surface of a planet) are expressed in the spherical coordinate system, and the viscosity dif-
ferential operators are the Laplace-Beltrami operators given by (3.2.31); see Chapter 7 for
details.
3.2.4 Gauss formula
IfM=Rn, it is known that the Gauss formula is
∫
Rnu·∇fdx=−∫Rnfdivudx ∀u∈H^1 (Rn×Rn),f∈H^1 (Rn),