**43**(2007), 471–504

**Modified Elastic Wave Equations** **on Riemannian and K¨** **ahler Manifolds**

By

YoshiyasuYasutomi^{∗}

**Abstract**

We introduce some geometrically invariant systems of diﬀerential equations on
any Riemannian manifolds and also on any K¨ahler manifolds, which are natural exten-
sions of the elastic wave equations onR^{3}. Further we prove the local decomposition
theorems of distribution solutions for those systems. In particular, the solutions of
our systems on K¨ahler manifolds are decomposed into 4 solutions with diﬀerent prop-
agation speeds.

**§0 Introduction and Results**

**Introduction**

The elastic wave equation onR^{3} is written as follows:

*P u*:=*ρ∂*^{2}

*∂t*^{2}*u−*(λ+*µ)grad div* *u−µ∆u*

=*ρ∂*^{2}

*∂t*^{2}*u−*(λ+ 2µ)grad div*u*+*µ*rot rot*u*=*f,*

where*u*is a 3-dimensional vector ﬁeld of the displacement of an elastic body,*ρ*
is the density constant and*λ, µ*are the Lam´e constants. It is well-known that
any distribution solution*u*of*P u*= 0 is decomposed into a sum*u*=*u*1+*u*2of
solutions *u*1*, u*2 satisfying the following additional equations:

rot*u*_{1}= 0, div*u*_{2}= 0.

Communicated by H. Okamoto. Received May 9, 2003. Revised August 17, 2005, Febru- ary 1, 2006.

2000 Mathematics Subject Classiﬁcation(s): 58A10, 35L10, 32C38.

*∗*Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba,
Meguro, Tokyo 153-0041, Japan.

c 2007 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.

We call *u*_{1}*, u*_{2} a longitudinal wave solution and a transverse wave solution,
respectively.

The elastic wave equations on Euclidean space are well studied in the Scattering theorem and propagation problems by Kawashita [6], Shimizu [7], and so on.

Real elastic waves (earthquakes) propagate through many layers which do not necessarily lie in good order. Therefore, more general study, we extend the elastic wave equations on Euclidean space to ones on Riemannian manifolds. In a physical generalization of this system of equations to Riemannian manifolds we replace div, rot with some covariant diﬀerentiations. However covariant diﬀerentiations do not commute with each other in general. Hence we consider the new elastic wave equations which do not depend on the choice of coordi- nates, and ﬁnd the natural model of hyperbolic equations with multi-values and multi-modes extended from the theory of the elastic wave equations as the model to apply the polarization theory.

To begin with, in Chapter 2, we consider the physical generalization*P*^{org}*u*

=0 of the elastic wave equation on a Riemannian manifold. In Chapter 3, we
introduce a new diﬀerential equation*P*_{R} *u*= 0 which is a modiﬁcation on the
lower order term of the original equation*P*org*u*= 0. Then we show that; the
new diﬀerential equation admits a decomposition of any solutions into longi-
tudinal wave solutions and transverse wave solutions. However, the original
equation does not admit any similar decompositions in general. Moreover we
generalize *P*_{R} and *P*_{org} to operators on *p-diﬀerential forms. In Chapter 4, we*
deal with the diﬀerential equations *P*_{C}*u*= 0,*P*_{C}^{∗}*u*= 0 on complex manifolds
and *P*_{K} *u*= 0 on K¨ahler manifolds. We show any distribution solutions of the
diﬀerential equations *P*_{C} *u*= 0 and*P*_{C}^{∗}*u*= 0 admit some decompositions into
2 solutions with diﬀerent propagation speeds. In the same way, we also show
that any distribution solution of the equation*P*_{K} *u*= 0 admits a decomposition
into 4 solutions with 4 diﬀerent propagation speeds.

**Results**

**Definition 0.1.** Let *M* be an *n-dimensional Riemannian manifold and*
*M* = R*t**×M*. Let *u* =

*u*^{i}*∂** _{i}* be a contravariant vector ﬁeld on

*M*with parameter

*t; precisely, a contravariant vector ﬁeld on*

*M*with

*dt, u*= 0.

We assume the density constant *ρ* and the L`ame constants *λ,* *µ*are positive.

Because the Riemannian metric tensor*g** _{ij}* (and the inverse metric tensor

*g*

*of*

^{ij}*g*

*) and the covariant diﬀerentiation*

_{ij}*∇*

*are commutative on the Riemannian manifolds (cf. [2] Section 15), we deﬁne the original elastic wave equation as*

_{j}follows:

*P*^{org} *u** ^{i}*:=

*ρ∂*

^{2}

*∂t*^{2}*u*^{i}*−λg*^{ij}*∇*_{j}*∇*_{k}*u*^{k}*−µg*^{ik}*∇*_{j}*∇*_{k}*u*^{j}*−µg*^{jk}*∇*_{j}*∇*_{k}*u*^{i}

=*ρ∂*^{2}

*∂t*^{2}*u*^{i}*−λ∇*^{i}*∇**k**u*^{k}*−µ∇**j**∇*^{i}*u*^{j}*−µ∇**j**∇*^{j}*u*^{i}

=*ρ∂*^{2}

*∂t*^{2}*u*^{i}*−λ∇*^{i}*∇*_{k}*u*^{k}*−µ∇*_{k}*∇*^{i}*u*^{k}*−µ∇*_{k}*∇*^{k}*u** ^{i}*=

*f*

^{i}*,*where we denote

*∇*

*=*

^{i}*g*

^{ij}*∇*

*j*according to the custom (cf. [1] Section 26).

In this paper, we often omit

by Einstein’s convention.

Because of the duality between contravariant vector ﬁelds and covariant forms, and the fact that diﬀerential operators attach covariant vector (tensor), we consider a new diﬀerential equation

*P*_{org}*u** _{i}*: =

*ρ∂*

^{2}

*∂t*^{2}*u*_{i}*−λ∇*_{i}*∇*^{k}*u*_{k}*−µ∇*^{k}*∇*_{i}*u*_{k}*−µ∇*^{k}*∇*_{k}*u*_{i}

=*ρ∂*^{2}

*∂t*^{2}*u*_{i}*−*(λ+ 2µ)*∇**i**∇*^{k}*u** _{k}*+

*µ∇*

^{k}*∇*

*i*

*u*

_{k}*−µ∇*

^{k}*∇*

*k*

*u*

_{i}*−*2µR

^{l}

_{i}*u*

_{l}=*f*_{i}*,*

where *R*^{l}* _{i}* is the Ricci tensor (cf. [1] Section 26).

When we put
*P*_{R} *u** _{i}* :=

*ρ∂*

^{2}

*∂t*^{2}*u*_{i}*−*(λ+ 2µ)*∇*_{i}*∇*^{k}*u** _{k}*+

*µ∇*

^{k}*∇*

_{i}*u*

_{k}*−µ∇*

^{k}*∇*

_{k}*u*

_{i}*,*

the diﬀerential equation *P*_{R} *u** _{i}* =

*f*

*on*

_{i}*M*is a modiﬁcation on the part of order 0 of the diﬀerential equation

*P*

_{org}

*u*

*=*

_{i}*f*

*, and we can rewrite*

_{i}*P*

_{R}

*u*=

*ρ*

_{∂t}

^{∂}^{2}

_{2}

*u*+ (λ+ 2µ)dδu+

*µδdu*=

*f*for a 1-diﬀerential form

*u*=

*u*

_{i}*dx*

*(cf. [1]*

^{i}Section 26). Here, *d, δ*are the exterior diﬀerential operator and the associated
exterior diﬀerential operator on *M*, respectively.

The diﬀerential operators*d,δ*operate on*p-diﬀerential forms for allp, then*
we extend the equations naturally to equations for*p-diﬀerential forms.*

Let(*p*)

*T*^{∗}*M* be a vector bundle of*p-diﬀerential forms onM*. Let*E*_{M}^{(}^{p}^{)}be
a sheaf of *p-forms onM* with*C** ^{∞}* coeﬃcients, and

*Db*

^{(}

_{M}

^{p}^{)}a sheaf of

*p-currents*on

*M*; that is,

*p-forms with distribution coeﬃcients. In this article, we do*not mean distributions the dual space of

*C*

_{0}

*(M). Our distributions behave as*

^{∞}“functions” for coordinate transformations. Further we deﬁne*E*_{M}^{(}^{p}^{)} and*Db*^{(}_{M}^{p}^{)}.
**Definition 0.2.** We denote by*E*_{M}^{(}^{p}^{)}, *Db*^{(}_{M}^{p}^{)}the sheaves of sections of*E*_{M}^{(}_{f}^{p}^{)},
*Db*^{(}_{f}^{p}^{)}

*M* which do not include the covariant vector*dt. That is, setting the projec-*

tion*π*:R*t**×M* *→M*, we deﬁne
*E*_{M}^{(}^{p}^{)}:=*E*_{M}^{(0)}_{f} *⊗*

*π*^{−1}*E*_{M}^{(0)}

*π*^{−}^{1}*E*_{M}^{(}^{p}^{)}*,* *Db*^{(}_{M}^{p}^{)}:=*Db*^{(0)}_{f}

*M* *⊗*

*π*^{−1}*E*_{M}^{(0)}

*π*^{−}^{1}*E*_{M}^{(}^{p}^{)}*.*

For

*u*=

1*≤i*1*<···<i**p**≤n*

*u*_{i}_{1}_{···i}* _{p}*(t, x)dx

^{i}^{1}

*∧ · · · ∧dx*

^{i}

^{p}*∈Db*

^{(}

_{M}

^{p}^{)}

*,*

we deﬁne an operator*P*_{R}for*Db*^{(}_{M}^{p}^{)}on*M* (1*≤p≤n−*1), where the coeﬃcients
*{u*_{i}_{1}_{···i}_{p}*}* are supposed to be alternating with respect to (i1*· · ·i** _{p}*).

**Definition 0.3.** We deﬁne sheaf-morphisms*P*_{R}:*Db*^{(}_{M}^{p}^{)}*−→Db*^{(}_{M}^{p}^{)}by
*P*_{R} *u*:=*ρ∂*^{2}

*∂t*^{2}*u*+ (λ+ 2µ)dδu+*µδdu.*

For*p*= 1, this equation is the covariant form of*P*_{R} *u** ^{i}*.

When *p* = 0 or *n,* *P*_{R} *u*= 0 reduces to a wave equation. Therefore we
suppose 1*≤p≤n−*1.

For*u∈Db*^{(}_{M}^{p}^{)}, we deﬁne equationsM^{R}, M^{R}1,M^{R}2,M^{R}0 as follows:

M^{R} : *P*_{R} *u*= 0,
M^{R}1 :

*P*_{R} *u*= 0,
*du*= 0, *⇐⇒*

(∂_{t}^{2}+*α∆)u*= 0,
*du*= 0,

M^{R}2 :

*P*_{R} *u*= 0,
*δu*= 0, *⇐⇒*

(∂_{t}^{2}+*β∆)u*= 0,
*δu*= 0,

M^{R}0 :

*P*_{R} *u*= 0,
*du*= 0,
*δu*= 0,

*⇐⇒*

*∂*_{t}^{2}*u*= 0,
*du*= 0,
*δu*= 0.

Here, *α* = (λ+ 2µ)/ρ, *β* = *µ/ρ* and *∆* = *dδ*+*δd* : *Db*^{(}_{M}^{p}^{)} *→* *Db*^{(}_{M}^{p}^{)} is the
Laplacian on*M*.

Further we deﬁne subsheaves*Sol(*M^{R};*p),Sol(*M^{R}* _{j}*;

*p), (j*= 0,1,2) of

*Db*

^{(}

_{M}

^{p}^{)}as follows: For N

^{R}=M

^{R}

*,*M

^{R}

*,*

_{j}*Sol(*N^{R};*p) :=*

*u∈Db*^{(}_{M}^{p}^{)}*u*satisﬁes N^{R}
*.*

Then, we have the following theorem.

**Theorem A**(Theorem 3.1)**.** *For any germu∈ Sol(*M^{R};*p)*

(^{◦}*t,**x** ^{◦}*)

*, there*

*exist some germs*

*u*

_{j}*∈Sol(*M

^{R}

*;*

_{j}*p)*

(^{◦}*t,*^{◦}*x*)(j= 1,2) *such thatu*=*u*_{1}+*u*_{2}*.*
*Furthermore, the equationu*=*u*_{1}+u_{2}= 0*impliesu*_{1}*, u*_{2}*∈ Sol(*M^{R}0;*p)*

(^{◦}*t,*^{◦}*x*)*.*
*Equivalently, we have the following exact sequence:*

0*−→ Sol(*M^{R}0;*p)−→ S*^{F}*ol(*M^{R}1;*p)⊕ Sol(*M^{R}2;*p)−→ S*^{G}*ol(*M^{R};*p)−→*0,

*where* *F(U*) =*U⊕*(*−U*),*G(U*_{1}*⊕U*_{2}) =*U*_{1}+*U*_{2}*.*

Let*X*be an*n-dimensional complex manifold with a Hermitian metric, and*
(*q,r*)

*T*^{∗}*X* a vector bundle of (q, r)-type diﬀerential forms on*X*. Let*E*_{X}^{(}^{q,r}^{)}be
a sheaf of (q, r)-forms on*X* with*C** ^{∞}*coeﬃcients, and

*Db*

^{(}

_{X}

^{q,r}^{)}a sheaf of (q, r)- currents on

*X. SettingX*=R

_{t}*×X*, we also deﬁne

*E*

_{X}^{(}

^{q,r}^{)},

*Db*

^{(}

_{X}

^{q,r}^{)}similarly to

*E*

_{M}^{(}

^{p}^{)},

*Db*

^{(}

_{M}

^{p}^{)}.

**Definition 0.4.** We deﬁne sheaf-morphisms *P*_{C},*P*_{C}* ^{∗}* :

*Db*

^{(}

_{X}

^{q,r}^{)}

*−→Db*

^{(}

_{X}

^{q,r}^{)}on

*X*which are similar to

*P*

_{R}:

*P*_{C}= *∂*^{2}

*∂t*^{2} +*α*_{1}*∂ϑ*+*α*_{2}*ϑ∂,* *P*_{C}* ^{∗}*=

*∂*

^{2}

*∂t*^{2}+*α*_{3}*∂ϑ*+*α*_{4}*ϑ∂,*

where *α*_{1}*, α*_{2}*, α*_{3} and *α*_{4} are positive constants. Here, *∂,* *∂* are the exterior
diﬀerential operator, the conjugate exterior diﬀerential operator on*X*, and*ϑ,*
*ϑ*are the associated operators of*∂,∂, respectively.*

For *u* *∈* *Db*^{(}_{X}^{q,r}^{)}, we deﬁne equations M^{C}, M^{C}1, M^{C}2, M^{C}* ^{∗}*, M

^{C}3

*, M*

^{∗}^{C}4

*as follows:*

^{∗}M^{C} : *P*_{C} *u*= 0,
M^{C}1 :

*P*_{C} *u*= 0,

*∂u*= 0, *⇐⇒*

(∂_{t}^{2}+*α*1)u= 0,

*∂u*= 0,
M^{C}2 :

*P*_{C} *u*= 0,
*ϑu*= 0, *⇐⇒*

(∂_{t}^{2}+*α*_{2})u= 0,
*ϑu*= 0,

M^{C}* ^{∗}* :

*P*

_{C}

^{∗}*u*= 0, M

^{C}3

*:*

^{∗}*P*_{C}^{∗}*u*= 0,

*∂u*= 0, *⇐⇒*

(∂_{t}^{2}+*α*_{3})u= 0,

*∂u*= 0,
M^{C}4* ^{∗}*:

*P*_{C}^{∗}*u*= 0,
*ϑu*= 0, *⇐⇒*

(∂_{t}^{2}+*α*4)u= 0,
*ϑu*= 0.

Here, = *∂ϑ*+*ϑ∂* and = *∂ϑ*+*ϑ∂* are the complex Laplace-Beltrami
operators.

Further we deﬁne subsheaves *Sol(*M^{C};*q, r),* *Sol(*M^{C}* _{j}*;

*q, r) (j*= 1,2),

*Sol(*M

^{C}

*;*

^{∗}*q, r),Sol(*M

^{C}

_{k}*;*

^{∗}*q, r) (k*= 3,4) of

*Db*

^{(}

_{X}

^{q,r}^{)}as follows: ForN

^{C}=M

^{C}

*,*M

^{C}

*, M*

_{j}^{C}

^{∗}*,*M

^{C}

_{k}*,*

^{∗}*Sol(*N^{C};*q, r) :=*

*u∈Db*^{(}_{M}^{q,r}^{)}*u*satisﬁesN^{C}
*.*
Then, we get the following theorems.

**Theorem B**(Theorem 4.1)**.** *For any germu∈ Sol(*M^{C};*q, r)*

(^{◦}*t,*^{◦}*z*)*, there*
*exist some germs* *u*_{j}*∈ Sol(*M^{C}* _{j}*;

*q, r)*

(^{◦}*t,*^{◦}*z*) (j= 1,2) *such thatu*=*u*1+*u*2*.*
**Theorem B*** ^{}* (Theorem 4.2)

**.**

*For any germ*

*u*

*∈ Sol(*M

^{C}

*;*

^{∗}*q, r)*

(^{◦}*t,**z** ^{◦}*)

*,*

*there exist some germs*

*u*

_{k}*∈ Sol(*M

^{C}

_{k}*;*

^{∗}*q, r)*

_{(}

*◦*

*t,*^{◦}*z*) (k = 3,4) *such that* *u* =
*u*_{3}+*u*_{4}*.*

Now we assume that *X* is a K¨ahler manifold; that is, for the Hermitian
metric*h, we have the equationd*

*h** _{jk}*(z)dz

^{j}*∧dz*

^{k}= 0, and we know that
*h** _{jk}* can be described as

*h*

*=*

_{jk}*∂*

_{j}*∂*

_{k}*φ*with a smooth real function

*φ*locally (cf.

[3] Chapter 1, Section 7). Then the following equations for operators on*Db*^{(}_{X}^{q,r}^{)}
are well-known (cf. [4] Chapter 3, Section 2):

== ^{1}_{2}*∆,*

*∂ϑ*+*ϑ∂*= 0, *∂ ϑ*+*ϑ ∂*= 0,

*∂∂*+*∂∂*= 0, *ϑϑ*+*ϑϑ*= 0.

(0.1)

As for the relationship between the conditions (0.1) and the K¨ahler condition, we give a brief introduction and a proof of the equivalency in Appendix.

**Definition 0.5.** We deﬁne sheaf-morphisms *P*_{K} : *Db*^{(}_{X}^{q,r}^{)} *−→* *Db*^{(}_{X}^{q,r}^{)} on
*X* by

*P*_{K}= *∂*^{2}

*∂t*^{2} +*α*_{1}*∂ϑ*+*α*_{2}*ϑ∂*+*α*_{3}*∂ϑ*+*α*_{4}*ϑ∂.*

Here,*α*1*, α*2*, α*3 and*α*4are positive constants.

When*q, r*= 0 or*n,* *P*_{K} *u*= 0 reduces to a wave equation. When *q*= 0, n
or *r* = 0, n, *P*_{K} stands for *P*_{C}* ^{∗}* or

*P*

_{C}, respectively. Therefore, we suppose 1

*≤q, r≤n−*1.

For *u∈Db*^{(}_{X}^{q,r}^{)}, we deﬁne equationsM^{K}, M^{K}* _{i}* (i = 1,2,3,4), M

^{K}

*, M*

_{jk}^{K}

_{jk}_{0}(jk) = (13),(23),(14),(24)

as follows:

M^{K} : *P*_{K} *u*= 0,
M^{K}1 :

*P*_{K}*u*= 0,

*∂u*= 0, M^{K}2 :

*P*_{K}*u*= 0,
*ϑu*= 0,
M^{K}3 :

*P*_{K}*u*= 0,

*∂u*= 0, M^{K}4 :

*P*_{K}*u*= 0,
*ϑu*= 0,

M^{K}13 :

*P*_{K}*u*= 0,

*∂u*= 0,

*∂u*= 0,

*⇐⇒*

*∂*_{t}^{2}+*α*_{1}+*α*_{3}

2 *∆*

*u*= 0,

*∂u*= 0,

*∂u*= 0,

M^{K}23 :

*P*_{K}*u*= 0,
*ϑu*= 0,

*∂u*= 0,

*⇐⇒*

*∂*_{t}^{2}+*α*2+*α*3

2 *∆*

*u*= 0,
*ϑu*= 0,

*∂u*= 0,

M^{K}14 :

*P*_{K}*u*= 0,

*∂u*= 0,
*ϑu*= 0,

*⇐⇒*

*∂*_{t}^{2}+*α*_{1}+*α*_{4}

2 *∆*

*u*= 0,

*∂u*= 0,
*ϑu*= 0,

M^{K}24 :

*P*_{K}*u*= 0,
*ϑu*= 0,
*ϑu*= 0,

*⇐⇒*

*∂*_{t}^{2}+*α*2+*α*4

2 *∆*

*u*= 0,
*ϑu*= 0,

*ϑu*= 0,

M^{K}130:

*P*_{K}*u*= 0,

*∂u*= 0,

*∂u*= 0,

*∆*^{2}*u*= 0,

*⇐⇒*

*∂*_{t}^{4}*u*=*∆*^{2}*u*= 0,

*∂*_{t}^{2}+*α*_{1}+*α*_{3}

2 *∆*

*u*= 0,

*∂u*= 0,

*∂u*= 0,

M^{K}230:

*P*_{K}*u*= 0,
*ϑu*= 0,

*∂u*= 0,

*∆*^{2}*u*= 0,

*⇐⇒*

*∂*_{t}^{4}*u*=*∆*^{2}*u*= 0,

*∂*_{t}^{2}+*α*_{2}+*α*_{3}

2 *∆*

*u*= 0,
*ϑu*= 0,

*∂u*= 0,

M^{K}140:

*P*_{K}*u*= 0,

*∂u*= 0,
*ϑu*= 0,

*∆*^{2}*u*= 0,

*⇐⇒*

*∂*_{t}^{4}*u*=*∆*^{2}*u*= 0,

*∂*_{t}^{2}+*α*1+*α*4

2 *∆*

*u*= 0,

*∂u*= 0,
*ϑu*= 0,

M^{K}240:

*P*_{K}*u*= 0,
*ϑu*= 0,
*ϑu*= 0,

*∆*^{2}*u*= 0,

*⇐⇒*

*∂*_{t}^{4}*u*=*∆*^{2}*u*= 0,

*∂*_{t}^{2}+*α*_{2}+*α*_{4}

2 *∆*

*u*= 0,
*ϑu*= 0,

*ϑu*= 0.

Further we deﬁne subsheaves *Sol(*M^{K};*q, r),* *Sol(*M^{K}* _{i}*;

*q, r)*

*i*= 1,2,3,4
,
*Sol(*M^{K}* _{jk}*;

*q, r),*

*Sol(*M

^{K}

*0;*

_{jk}*q, r)*

(jk) = (13),(23),(14),(24)

of *Db*^{(}_{X}^{q,r}^{)} as the
sheaves of*Db*^{(}_{X}^{q,r}^{)}-solutions, respectively.

Then, we have the following theorem.

**Theorem C**(Theorem 4.3)**.** *For any germu∈ Sol(*M^{K};*q, r)*

(^{◦}*t,*^{◦}*z*)*, there*
*exist some germs* *u*_{ij}*∈ Sol(*M^{K}* _{ij}*;

*q, r)*

(^{◦}*t,**z** ^{◦}*) ((ij) = (13),(23), (14),(24))

*such*

*that*

*u*=

*u*

_{13}+

*u*

_{23}+

*u*

_{14}+

*u*

_{24}

*.*

*Further, we ﬁnd thatu*=*u*_{13}+*u*_{23}+*u*_{14}+*u*_{24}= 0*implies*
*u*_{jk}*∈ Sol(*M^{K}* _{jk}*0;

*q, r)*

(jk) = (13),(23),(14),(24)
*.*
*Equivalently, we have the following exact sequence:*

0*−→*

(*ij*)

*Sol(*M^{K}* _{ij}*0;

*q, r)−→*

^{G}(*ij*)

*Sol(*M^{K}* _{ij}*;

*q, r)−→ S*

^{H}*ol(*M

^{K};

*q, r)−→*0.

*Here,*

(*ij*)

*Sol(*M^{K}* _{ij}*0;

*q, r) :=*

(u* _{ij}*)

*∈*

(*ij*)

*Sol(*M^{K}* _{ij}*0;

*q, r)*

(*ij*)

*u** _{ij}*= 0

*,*

*G(U*13*⊕U*23*⊕U*14*⊕U*24) =*U*13*⊕U*23*⊕U*14*⊕U*24*,H*(U13*⊕U*23*⊕U*14*⊕U*24) =
*U*13+*U*23+*U*14+*U*24*.*

**§1.****Preparation from Riemannian Geometry**

In this section, we recall some notations and terminologies in Riemannian geometry used in this paper according to [1] (Chapter 2,5), [2] (Chapter 3), and [5] (Chapter 1,4).

We assume that *M* is oriented. Then, there is a global section Ω of *E*_{M}^{(}^{n}^{)}
on*M*, which never vanishes on *M*.

**Definition 1.1.** The inner products*·,·*:(1)

*T*_{x}^{∗}*M×*(1)

*T*_{x}*M* *→*R,

*·,·** ^{∗}* :(

*p*)

*T*_{x}^{∗}*M×*(*p*)

*T*_{x}^{∗}*M* *→*R, are deﬁned as follows. We choose a local
positive orthonormal system (ω^{1}*, . . . , ω** ^{n}*) of

*C*

*sections of*

^{∞}*T*

^{∗}*M*concerning the Riemannian metric; that is, there is a positive valued

*C*

*function*

^{∞}*α*such that

*ω*

^{1}

*∧ · · · ∧ω*

*=*

^{n}*αΩ>*0, and for

*ω*

*=*

^{i}

_{n}*j*=1*a*_{ij}*dx** ^{j}*(i= 1,2, . . . , n) with a
local coordinate system (x

^{1}

*, . . . , x*

*), we have*

^{n}*g*

*=*

_{ij}

_{n}*k*=1*a*_{ki}*a** _{kj}*. Then for

*σ*=

1*≤i≤n*

*σ*_{i}*dx*^{i}*,* *τ* =

1*≤i≤n*

*τ*^{i}*∂*_{i}*,*

we deﬁne

*σ, τ*:=

1*≤i≤n*

*σ*_{i}*τ*^{i}*,*
and for

*φ*=

1*≤i*1*<···<i**p**≤n*

*φ*_{i}_{1}_{···i}_{p}*dx*^{i}^{1}*∧ · · · ∧dx*^{i}^{p}*,*

*ψ*=

1*≤i*1*<···<i**p**≤n*

*ψ*_{i}_{1}_{···i}_{p}*dx*^{i}^{1}*∧ · · · ∧dx*^{i}^{p}*,*
we deﬁne

*φ, ψ** ^{∗}*:=

1*≤i*1*<···<i**p**≤n*

*φ*_{i}_{1}_{···i}_{p}*ψ*^{i}^{1}^{···i}^{p}

:=

1*≤i*1*<···<i**p**≤n*
1*≤j*1*<···<j**p**≤n*

*φ*_{i}_{1}_{···i}_{p}*g*^{i}^{1}^{j}^{1}*· · ·g*^{i}^{p}^{j}^{p}*ψ*_{j}_{1}_{···j}_{p}*.*

**Definition 1.2.** We denote by*d*:*Db*^{(}_{M}^{p}^{)}*→ Db*^{(}_{M}^{p}^{+1)} the exterior diﬀer-
ential operator which acts on *Db*^{(}_{M}^{p}^{)} as a sheaf-morphism. Then the following
formulas are well-known:

*d(φ±ψ) =dφ±dψ* (φ, ψ*∈ Db*^{(}_{M}^{p}^{)}),

*d(φ∧ψ) =dφ∧ψ*+ (*−*1)^{p}*φ∧dψ* (φ*∈ Db*^{(}_{M}^{p}^{)}*, ψ∈ Db*^{(}_{M}^{q}^{)}),

*d(dφ) = 0* (φ*∈ Db*^{(}_{M}^{p}^{)}),

for*f* *∈ Db*^{(0)}* _{M}*,

*df*:=

*∂f*

*∂x*_{j}*dx*^{j}*∈ Db*^{(1)}* _{M}*.
Here 0

*≤p≤n. Ifp*=

*n,dφ*= 0 holds.

**Definition 1.3.** The isomorphism*∗*:

*T*^{∗}*M* *→*

*T*^{∗}*M* of vector bun-
dle is deﬁned as follows:

*∗*:(*p*)

*T*_{x}^{∗}*M* *→*(*n−p*)

*T*_{x}^{∗}*M* is a linear map,

*∗*(ω^{i}^{1}*∧ · · · ∧ω*^{i}* ^{p}*) =

*δ*

_{1}

*···n*

*i*1

*···i*

*p*

*j*1

*···j*

*n−p*

*ω*^{j}^{1}*∧ · · · ∧ω*^{j}^{n−p}

= (*−*1)^{(}^{i}^{1}^{−}^{1)+}^{···}^{+(}^{i}^{p}^{−p}^{)}*ω*^{j}^{1}*∧ · · · ∧ω*^{j}^{n−p}*,*
for any permutation (i_{1}*, . . . , i*_{p}*, j*_{1}*, . . . , j** _{n−p}*) of (1, . . . , n).

Here (i_{1}*· · ·i** _{p}*) and (j

_{1}

*· · ·j*

*) are indices satisfying*

_{n−p}(i_{1}*· · ·i*_{p}*j*_{1}*· · ·j** _{n−p}*) is a permutation of (1

*· · ·n),*1

*≤i*

_{1}

*<· · ·< i*

_{p}*≤n,*1

*≤j*

_{1}

*<· · ·< j*

_{n−p}*≤n.*

*Remark.* The deﬁnition above does not depend on the choice of the
positive orthonormal system*{ω*^{1}*, . . . , ω*^{n}*}*.

**Proposition 1.1.** *We setφ, ψ∈*(*p*)

*T*_{x}^{∗}*M. Then we obtain*

*φ∧ ∗ψ*= (*∗φ)∧ψ*=*φ, ψ*^{∗}*ω*^{1}*∧ · · · ∧ω*^{n}*,*

*∗*1 =*ω*^{1}*∧ · · · ∧ω** ^{n}*=

*√*

*g dx*^{1}*∧ · · · ∧dx*^{n}*,*

*∗φ*= (*−*1)^{(}^{i}^{1}^{−}^{1)+}^{···}^{+(}^{i}^{p}^{−p}^{)}*√g g*^{i}^{1}^{j}^{1}*· · ·g*^{i}^{p}^{j}^{p}*φ*_{i}_{1}_{···i}_{p}*dx*^{j}^{1}*∧ · · · ∧dx*^{j}^{n−p}

*∈*(*n−p*)

*T*_{x}^{∗}*M .*
*Here* *g*= det(g* _{kl}*).

Let *U* *⊂M* be an open subset. Let *α*^{(}^{p}^{)} *∈ Db*^{(}_{M}^{p}^{)}(U), *β*^{(}^{p}^{)} *∈ E*_{M}^{(}^{p}^{)}(U) be
sections. We suppose that*β*^{(}^{p}^{)}has a compact support in*U. Then the following*
integral is well-deﬁned:

(α^{(}^{p}^{)}*, β*^{(}^{p}^{)}) :=

*M**α*^{(}^{p}^{)}*, β*^{(}^{p}^{)}^{∗}*ω*^{1}*∧ · · · ∧ω*^{n}*.*

**Definition 1.4.** Let *α*^{(}^{p}^{)} *∈ Db*^{(}_{M}^{p}^{)}, *β*^{(}^{p−}^{1)} *∈ E*_{M}^{(}^{p−}^{1)} be sections. We
suppose *β*^{(}^{p−}^{1)} has a compact support. Then the sheaf-morphism*δ*:*Db*^{(}_{M}^{p}^{)}*→*
*Db*^{(}_{M}^{p−}^{1)}is deﬁned as

(δα^{(}^{p}^{)}*, β*^{(}^{p−}^{1)}) = (α^{(}^{p}^{)}*, dβ*^{(}^{p−}^{1)}).

Hence we have

*δ*= (*−*1)^{n}^{(}^{p−}^{1)+1}*∗d∗.*
**Definition 1.5.** LetX^{r}* _{s}*be the sheaf of

_{r}*T*_{x}*M⊗*_{s}

*T*_{x}^{∗}*M*-valued*C** ^{∞}*
functions, and

*Db*

^{r}*the sheaf of*

_{s}

_{r}*T*_{x}*M⊗*_{s}

*T*_{x}^{∗}*M*-valued distributions. Then,
the sheaf-morphisms *∇*:X^{r}_{s}*→*X^{r}* _{s}*+1

*,*

*Db*

^{r}

_{s}*→ Db*

^{r}

_{s}_{+1}are deﬁned as follows:

for*a(x)∈*X^{0}0*,* we have *∇a(x) =* *∂a*

*∂x*^{j}*dx*^{j}*,*
for *∂*

*∂x*^{j}*∈*X^{1}0*,* we have *∇*
*∂*

*∂x*^{j}

=*Γ*_{j k}^{i}*∂*

*∂x*^{i}*⊗dx*^{k}*,*
for*dx*^{j}*∈*X^{0}1*,* we have *∇*

*dx*^{j}

=*−Γ*_{i k}^{j}*dx*^{i}*⊗dx*^{k}*,*
for*e∈*X^{r}_{s}*, f* *∈*X^{r}_{s}^{}*,* we have *∇*(e*⊗f*) = (*∇e)⊗f*+*e⊗ ∇f.*

Here,

*Γ*_{i k}* ^{j}* =

*g*

^{jl}*Γ*

*=*

_{ilk}*g*

^{jl}*·*1 2

*∂g*_{il}

*∂x** ^{k}* +

*∂g*

_{lk}*∂x*^{i}*−∂g*_{ki}

*∂x** ^{l}*
are the Riemann-Christoﬀel symbols.

**Proposition 1.2.** *We set*
*e*=*e*^{j}_{i}^{1}^{···j}^{r}

1*···i**s**dx*^{i}^{1}*⊗ · · · ⊗dx*^{i}^{s}*⊗* *∂*

*∂x*^{j}^{1} *⊗ · · · ⊗* *∂*

*∂x*^{j}^{r}*∈*X^{r}_{s}*.*
*Then we have*

*∇e*=

*∂*_{k}*e*^{j}_{i}^{1}^{···j}^{r}

1*···i**s* +*e*^{j}_{i}^{1}^{···j}^{p−1}^{qj}^{p+1}^{···j}^{r}

1*···i**s* *Γ*_{q k}^{j}^{p}*−e*^{j}_{i}^{1}^{···j}^{r}

1*···i**p−1**qi**p+1**···i**s**Γ*_{i}^{q}

*p**k*

*×dx*^{i}^{1}*⊗ · · · ⊗dx*^{i}^{s}*⊗* *∂*

*∂x*^{j}^{1} *⊗ · · · ⊗* *∂*

*∂x*^{j}^{r}*⊗dx*^{k}*.*
Hence we call the following*the covariant diﬀerentiation:*

*∇**k**e*=

*∂*_{k}*e*^{j}_{i}^{1}^{···j}^{r}

1*···i**s* +*e*^{j}_{i}^{1}^{···j}^{p−1}^{qj}^{p+1}^{···j}^{r}

1*···i**s* *Γ*_{q k}^{j}^{p}*−e*^{j}_{i}^{1}^{···j}^{r}

1*···i**p−1**qi**p+1**···i**s**Γ*_{i}^{q}

*p* *k*

*×dx*^{i}^{1}*⊗ · · · ⊗dx*^{i}^{s}*⊗* *∂*

*∂x*^{j}^{1} *⊗ · · · ⊗* *∂*

*∂x*^{j}^{r}*.*

**§2.****Elastic Mechaniques on Riemannian**
**Manifolds**

Let *M* be an *n-dimensional Riemannian manifold with metric* *g. We*
consider an elastic body *G* in *M*. A motion of *G* is identiﬁed with an open
subset*G* of*M*with one parameter family of diﬀeomorphisms*h** _{t}*(

*·*) (t

*∈*R):

*h** _{s}*:

*G∩ {t*= 0

*}−→*

^{∼}*G∩ {t*=

*s}.*

Then the elastic wave equation for *G* is formulated as a time development
equation for small motions of*G; that is,h** _{t}*is close to the identity map and it
is expressed as

*h*^{i}* _{t}*(x) =

*x*

*+*

^{i}*u*

*(x, t) in a local coordinate system (x*

^{i}^{1}

*, . . . , x*

*), where*

^{n}*u*=

*u** ^{i}*(x, t)∂

*is the small displacement vector ﬁeld.*

_{i}Then, the diﬀerential *dh* of the map *h*and its dual map *dh** ^{∗}* are given as
follows:

*dh*:*T*_{x}*M* *ξ*^{i}*−→η** ^{i}*=

*ξ*

^{j}*∂h*

^{i}*∂x*^{j}

=*ξ** ^{i}*+

*∂u*

^{i}*∂x*^{j}*ξ*^{j}*∈T*_{h}_{(}_{x}_{)}*M,*

*dh** ^{∗}*:

*T*

_{h}

^{∗}_{(}

_{x}_{)}

*M*

*η*

_{i}*−→ξ*

*=*

_{j}*η*

_{i}*∂h*

^{i}*∂x*^{j}

=*η** _{j}*+

*∂u*

^{i}*∂x*^{j}*η*_{i}*∈T*_{x}^{∗}*M.*

Let us calculate the diﬀerence between the line element of*M*
*ds(x)*^{2}=*g** _{ij}*(x)

*dx*

^{i}*⊗dx*

^{j}and its pull-back ^{∗}*ds(x)*^{2} by*h:*

*ε** _{ij}*(x)

*dx*

^{i}*⊗dx*

*:=1 2*

^{j}*∗**ds(x)*^{2}*−ds(x)*^{2}

=1 2

*g** _{kl}*(h(x))

*∂h*

^{k}*∂x*^{i}

*∂h*^{l}

*∂x*^{j}*dx*^{i}*⊗dx*^{j}*−g** _{ij}*(x)dx

^{i}*⊗dx*

^{j}*.*
Here *ε** _{ij}* is called the strain tensor. By ignoring the non-linear terms of

*u*

*,*

^{m}*∂*_{k}*u** ^{m}*,

*∂*

_{l}*u*

*, we have*

^{m}*ε*

*=1*

_{kl}2(g_{km}*∂*_{l}*u** ^{m}*+

*g*

_{ml}*∂*

_{k}*u*

*+*

^{m}*u*

^{m}*∂*

_{m}*g*

*)*

_{kl}*.*On Riemannian spaces, equations

*∇**m**g** _{kl}*=

*∂*

_{m}*g*

_{kl}*−Γ*

_{ml}

^{n}*g*

_{kn}*−Γ*

_{mk}

^{n}*g*

*= 0*

_{nl}hold (cf. [2] Section 15), so we have
*ε** _{kl}*= 1

2*{g*_{km}*∂*_{l}*u** ^{m}*+

*g*

_{ml}*∂*

_{k}*u*

*+*

^{m}*u*

*(Γ*

^{m}

_{ml}

^{n}*g*

*+*

_{kn}*Γ*

_{mk}

^{n}*g*

*)*

_{nl}*}*(2.1)

= 1

2*{g** _{km}*(∂

_{l}*u*

*+*

^{m}*u*

^{n}*Γ*

_{nl}*) +*

^{m}*g*

*(∂*

_{ml}

_{k}*u*

*+*

^{m}*u*

^{n}*Γ*

_{nk}*)*

^{m}*}*

= 1

2(g_{km}*∇*_{l}*u** ^{m}*+

*g*

_{ml}*∇*

_{k}*u*

*)*

^{m}*.*

In physics, we assume that the stress tensor *σ** ^{ji}* has a linear relationship
with the strain tensor at each point. Hence there exists the elastic coeﬃcient
tensor

*E*

*such that*

^{jilk}(2.2) *σ*^{ji}

*√g* =*E*^{jilk}*ε*_{kl}*.*

As a physical assumption for*E** ^{ijkl}*, we have the equations:

*E** ^{ijkl}*=

*E*

*=*

^{jikl}*E*

^{ijlk}*.*

In particular, it is well-known that the elastic coeﬃcient tensor of an isotropic elastic body has the following form:

(2.3) *E** ^{ijkl}*=

*λg*

^{ij}*g*

*+*

^{kl}*µg*

^{ik}*g*

*+*

^{jl}*µg*

^{il}*g*

^{jk}*,*

where *λ* and *µ* are the two L´ame constants. Therefore, from (2.1), (2.2) and
(2.3), we get

*σ*^{ji}

*√g* =*E*^{ijkl}*ε** _{kl}*
(2.4)

= (λg^{ij}*g** ^{kl}*+

*µg*

^{ik}*g*

*+*

^{jl}*µg*

^{il}*g*

*) 1*

^{jk}2*g*_{mk}*∇**l**u** ^{m}*+1

2*g*_{ml}*∇**k**u*^{m}

=*λg*^{ij}*∇*_{l}*u** ^{l}*+

*µg*

^{jl}*∇*

_{l}*u*

*+*

^{i}*µg*

^{il}*∇*

_{l}*u*

^{j}*.*

Hence the equation of power-balance between the stress of the elastic body and the external force is written as follows:

*df** ^{i}*=

*σ*

^{ji}*√gdS*_{j}*,*

where *df** ^{i}* is the external force vector for the surface element

*dS*

*.*

_{j}In order to introduce the elastic wave equation, we consider a small neigh-
borhood *V* of a point *x* in *M*, whose boundary is given by a smooth closed