AN INTRODUCTION TO
DIFFERENTIAL GEOMETRY WITH
APPLICATIONS TO ELASTICITY
Philippe G. Ciarlet
City University of Hong Kong

Contents
Preface 5
1 Three-dimensional differential geometry 9
Introduction 9
1.1 Curvilinear coordinates . . . . . . . . . . . . . . . . . . . . . . . 11
1.2 Metrictensor 13
1.3 Volumes, areas, and lengths in curvilinear coordinates . . . . . . 16
1.4 Covariantderivativesofavectorfield 19
1.5 Necessary conditions satisfied by the metric tensor; the Riemann
curvaturetensor 24
1.6 ExistenceofanimmersiondefinedonanopensetinR
3
with a
prescribedmetrictensor 25
1.7 Uniqueness up to isometries of immersions with the same metric
tensor 36
1.8 Continuity of an immersion as a function of its metric tensor . . 44
2 Differential geometry of surfaces 59
Introduction 59
2.1 Curvilinear coordinates on a surface . . . . . . . . . . . . . . . . 61
2.2 First fundamental form . . . . . . . . . . . . . . . . . . . . . . . 65
2.3 Areas and lengths on a surface . . . . . . . . . . . . . . . . . . . 67
2.4 Second fundamental form; curvature on a surface . . . . . . . . . 69
2.5 Principal curvatures; Gaussian curvature . . . . . . . . . . . . . . 73
2.6 Covariant derivatives of a vector field defined on a surface; the

4.4 Existence and uniqueness theorems for the linear Koiter shell
equations; covariant derivatives of a tensor field defined on a
surface 185
4.5 A brief review of linear shell theories . . . . . . . . . . . . . . . . 193
References 201
Index 209
PREFACE
This book is based on lectures delivered over the years by the author at the
Universit´e Pierre et Marie Curie, Paris, at the University of Stuttgart, and at
City University of Hong Kong. Its two-fold aim is to give thorough introduc-
tions to the basic theorems of differential geometry and to elasticity theory in
curvilinear coordinates.
The treatment is essentially self-contained and proofs are complete. The
prerequisites essentially consist in a working knowledge of basic notions of anal-
ysis and functional analysis, such as differential calculus, integration theory
and Sobolev spaces, and some familiarity with ordinary and partial differential
equations.
In particular, no aprioriknowledge of differential geometry or of elasticity
theory is assumed.
In the first chapter, we review the basic notions, such as the metric tensor
and covariant derivatives, arising when a three-dimensional open set is equipped
with curvilinear coordinates. We then prove that the vanishing of the Riemann
curvature tensor is sufficient for the existence of isometric immersions from a
simply-connected open subset of R
n
equipped with a Riemannian metric into
a Euclidean space of the same dimension. We also prove the corresponding
uniqueness theorem, also called rigidity theorem.
In the second chapter, we study basic notions about surfaces, such as their
two fundamental forms, the Gaussian curvature and covariant derivatives. We

covariant derivatives of tensor fields, are also introduced in Chapters 3 and 4,
where they appear most naturally in the derivation of the basic boundary value
problems of three-dimensional elasticity and shell theory.
Occasionally, portions of the material covered here are adapted from ex-
cerpts from my book “Mathematical Elasticity, Volume III: Theory of Shells”,
published in 2000 by North-Holland, Amsterdam; in this respect, I am indebted
to Arjen Sevenster for his kind permission to rely on such excerpts. Other-
wise, the bulk of this work was substantially supported by two grants from the
Research Grants Council of Hong Kong Special Administrative Region, China
[Project No. 9040869, CityU 100803 and Project No. 9040966, CityU 100604].
Last but not least, I am greatly indebted to Roger Fosdick for his kind
suggestion some years ago to write such a book, for his permanent support
since then, and for his many valuable suggestions after he carefully read the
entire manuscript.
Hong Kong, July 2005 Philippe G. Ciarlet
Department of Mathematics
and
Liu Bie Ju Centre for Mathematical Sciences
City University of Hong Kong


Chapter 1
THREE-DIMENSIONAL DIFFERENTIAL
GEOMETRY
INTRODUCTION
Let Ω be an open subset of R
3
,letE
3
denote a three-dimensional Euclidean

j
, where g
i
= ∂
i
Θ and g
j
· g
i
= δ
j
i
.
The vector fields g
i
:Ω→ R
3
and g
j
:Ω→ R
3
respectively form the
covariant,andcontravariant, bases in the set Θ(Ω).
Itisshowninparticularhowvolumes, areas,andlengths,inthesetΘ(Ω)
are computed in terms of its curvilinear coordinates, by means of the functions
g
ij
and g
ij
(Theorem 1.3-1).

ijq
and Γ
p
ij
be defined by
Γ
ijq
=
1
2
(∂
j
g
iq
+ ∂
i
g
jq
− ∂
q
g
ij
)andΓ
p
ij
= g
pq
Γ
ijq
, where (g

ij
are the Christoffel symbols of the first,andsecond,
kind and the functions
R
qijk
= ∂
j
Γ
ikq
− ∂
k
Γ
ijq
+Γ
p
ij
Γ
kqp
− Γ
p
ik
Γ
jqp
are the covariant components of the Riemann curvature tensor of the set Θ(Ω).
We then focus our attention on the reciprocal questions:
GivenanopensubsetΩofR
3
and a smooth enough symmetric and positive-
definite matrix field (g
ij

ij
= ∂
i

Θ · ∂
j

Θ in Ω,
there then exist a vector c ∈ E
3
and an orthogonal matrix Q of order three such
that
Θ(x)=c + Q

Θ(x) for all x ∈ Ω.
Together, the above existence and uniqueness theorems constitute an impor-
tant special case of the fundamental theorem of Riemannian geometry and as
such, constitute the core of Chapter 1.
We conclude this chapter by showing (Theorem 1.8-5) that the equivalence
class of Θ, defined in this fashion modulo isometries of E
3
, depends continu-
ously on the matrix field (g
ij
) with respect to appropriate Fr´echet topologies.
Sect. 1.1] Curvilinear coordinates 11
1.1 CURVILINEAR COORDINATES
To begin with, we list some notations and conventions that will be consistently
used throughout.
All spaces, matrices, etc., considered here are real.

according to the context.
Let E
3
denote a three-dimensional Euclidean space,leta ·b and a∧b denote
the Euclidean inner product and exterior product of a, b ∈ E
3
,andlet|a| =
√
a ·a denote the Euclidean norm of a ∈ E
3
.ThespaceE
3
is endowed with
an orthonormal basis consisting of three vectors

e
i
=

e
i
.Letx
i
denote the
Cartesian coordinates of a point x ∈ E
3
and let

∂
i

3
/∂x
i
∂x
j
∂x
k
.
Let there be given an open subset

ΩofE
3
and assume that there exist an
open subset Ω of R
3
and an injective mapping Θ :Ω→ E
3
such that Θ(Ω) =

Ω.
Then each point x ∈

Ω can be unambiguously written as
x = Θ(x),x∈ Ω,
and the three coordinates x
i
of x are called the curvilinear coordinates of x
(Figure 1.1-1). Naturally, there are infinitely many ways of defining curvilinear
coordinates in a given open set


e
2
e
3
e
1
R
3
Θ
ˆe
2
ˆe
3
ˆe
1
ˆx

Ω
g
2
(x)
g
3
(x)
g
1
(x)
E
3
Figure 1.1-1: Curvilinear coordinates and covariant bases in an open set

ˆ
Ω
E
3
ψ
Figure 1.1-2: Two familiar examples of curvilinear coordinates. Let the mapping Θ be
defined by
Θ :(ϕ, ρ, z) ∈ Ω → (ρ cos ϕ, ρ sin ϕ, z) ∈ E
3
.
Then (ϕ, ρ, z) are the cylindrical coordinates of bx = Θ(ϕ, ρ, z). Note that (ϕ +2kπ, ρ, z)or
(ϕ + π +2kπ,−ρ, z),k∈ Z, are also cylindrical coordinates of the same point bx and that ϕ is
not defined if bx is the origin of E
3
.
Let the mapping Θ be defined by
Θ :(ϕ, ψ, r) ∈ Ω → (r cos ψ cos ϕ, r cos ψ sin ϕ, r sin ψ) ∈ E
3
.
Then (ϕ, ψ, r) are the spherical coordinates of bx = Θ(ϕ, ψ, r). Note that (ϕ +2kπ, ψ +2π, r)
or (ϕ +2kπ, ψ + π +2π, −r) are also spherical coordinates of the same point bx and that ϕ
and ψ are not defined if bx is the origin of E
3
.
In both cases, the covariant basis at bx and the coordinate lines are represented with
self-explanatory notations.
Sect. 1.2] Metric tensor 13
If Θ ∈C
1
(Ω; E

⎝
∂
1
Θ
1
∂
2
Θ
1
∂
3
Θ
1
∂
1
Θ
2
∂
2
Θ
2
∂
3
Θ
2
∂
1
Θ
3
∂

⎛
⎝
∂
i
Θ
1
∂
i
Θ
2
∂
i
Θ
3
⎞
⎠
(x),
i.e., g
i
(x) is the i-th column vector of the matrix ∇Θ(x). Then the expansion
of Θ about x may be also written as
Θ(x + δx)=Θ(x)+δx
i
g
i
(x)+o(δx).
If in particular δx is of the form δx = δte
i
,whereδt ∈ R and e
i

.
14 Three-dimensional differential geometry [Ch. 1
(there exist t
0
and t
1
with t
0
< 0 <t
1
such that the i-th coordinate line is
given by t ∈ ]t
0
,t
1
[ → f
i
(t):=Θ(x + te
i
) in a neighborhood of x; hence
f

i
(0) = ∂
i
Θ(x)=g
i
(x)); see Figures 1.1-1 and 1.1-2.
Returning to a general increment δx = δx
i

algebra. For instance, δx
T
stands for the transpose of the column vector δx
and ∇Θ(x)
T
designates the transpose of the matrix ∇Θ(x), the element at the
i-th row and j-th column of a matrix A is noted (A)
ij
,etc.
In other words, the principal part with respect to δx of the length between
the points Θ(x + δx)andΘ(x)is{δx
i
g
i
(x) · g
j
(x)δx
j
}
1/2
. This observation
suggests to define a matrix (g
ij
(x)) of order three, by letting
g
ij
(x):=g
i
(x) · g
j

(x)=X
ik
(x)g
k
(x)intherelationsg
i
(x) · g
j
(x)=δ
i
j
.Thisgives
X
ik
(x)g
kj
(x)=δ
i
j
;consequently,X
ik
(x)=g
ik
(x), where
(g
ij
(x)) := (g
ij
(x))
−1

(x)g
j
(x)g
k
(x)=g
ik
(x)δ
j
k
= g
ij
(x),
and thus the vectors g
i
(x)arelinearly independent since the matrix (g
ij
(x)) is
positive definite. We would likewise establish that g
i
(x)=g
ij
(x)g
j
(x).
The three vectors g
i
(x)formthecontravariant basis at the point x = Θ(x)
and the elements g
ij
(x) of the symmetric positive definite matrix (g

(x)g
j
(x).
Sect. 1.2] Metric tensor 15
A mapping Θ :Ω→ E
3
is an immersion if it is an immersion at each point
in Ω, i.e., if Θ is differentiable in Ω and the three vectors g
i
(x)=∂
i
Θ(x)are
linearly independent at each x ∈ Ω.
If Θ :Ω→ E
3
is an immersion, the vector fields g
i
:Ω→ R
3
and g
i
:Ω→ R
3
respectively form the covariant,andcontravariant bases.
To conclude this section, we briefly explain in what sense the components of
the “metric tensor” may be “covariant” or “contravariant”.
Let Ω and

ΩbetwodomainsinR
3


Θ(x) ∈ E
3
are linearly independent. Let g
i
(x)and

g
i
(x)bethe
vectors of the corresponding contravariant bases at the same point x.Asimple
computation then shows that
g
i
(x)=
∂χ
j
∂x
i
(x)

g
j
(x)andg
i
(x)=
∂ χ
i
∂x
j

ij
(x) be the covariant components, and let g
ij
(x)andg
ij
(x)
be the contravariant components, of the metric tensor at the same point Θ(x)=

Θ(x) ∈ E
3
. Then a simple computation shows that
g
ij
(x)=
∂χ
k
∂x
i
(x)
∂χ

∂x
j
(x)g
k
(x)andg
ij
(x)=
∂ χ
i

Chapter 11, Sections 1 to 3]; it is also shown in ibid. that the same “tensor”
also has “mixed” components g
i
j
(x), which turn out to be simply the Kronecker
symbols δ
i
j
. 
In fact, analogous justifications apply as well to the components of all the
other “tensors” that will be introduced later on. Thus, for instance, the co-
variant components v
i
(x)andv
i
(x), and the contravariant components v
i
(x)
and v
i
(x) (both with self-explanatory notations), of a vector at the same point
Θ(x)=

Θ(x) satisfy (cf. Section 1.4)
v
i
(x)g
i
(x)=v
i

i
∂x
j
(x)v
j
(x).
In other words, the components v
i
(x) “vary like” the vectors g
i
(x)ofthe
covariant basis under a change of curvilinear coordinates, while the components
v
i
(x) of a vector “vary like” the vectors g
i
(x)ofthecontravariant basis. This
is why they are respectively called “covariant” and “contravariant”. A vector
is an example of a “first-order” tensor.
Likewise, it is easily checked that each exponent in the “contravariant” com-
ponents A
ijk
(x) of the three-dimensional elasticity tensor in curvilinear coor-
dinates introduced in Section 3.4 again “varies like” that of the corresponding
vector of the contravariant basis under a change of curvilinear coordinates.
Remark. See again Antman [1995, Chapter 11, Sections 1 to 3] to deci-
pher the “fourth-order tensor” hidden behind such contravariant components
A
ijk
(x). 

Given a domain D ⊂ R
3
with boundary Γ, we let dx denote the volume
element in D,dΓdenotethearea element along Γ, and n = n
i

e
i
denote the
unit (|n| =1)outer normal vector along Γ (dΓ is well defined and n is defined
dΓ-almost everywhere since Γ is assumed to be Lipschitz-continuous).
Note also that the assumptions made on the mapping Θ in the next theorem
guarantee that, if D is a domain in R
3
such that D ⊂ Ω, then {

D}
−
⊂

Ω,
Sect. 1.3] Volumes, areas, and lengths in curvilinear coordinates 17
{Θ(D)}
−
= Θ(D), and the boundaries ∂

D of

D and ∂D of D are related by
∂

D is given in terms of the area element dΓ(x) at x ∈ ∂D by
d

Γ(x)=|Cof ∇Θ(x)n(x)|dΓ(x)=

g(x)

n
i
(x)g
ij
(x)n
j
(x)dΓ(x),
where n(x):=n
i
(x)e
i
denotes the unit outer normal vector at x ∈ ∂D.
(c) The length element d

(x) at x = Θ(x) ∈

Ω is given by
d

(x)=

δx
T

Indications about the proof of the relation between the area elements d

Γ(x)
and dΓ(x) given in (b) are found in Ciarlet [1988, Theorem 1.7-1] (in this for-
mula, n(x)=n
i
(x)e
i
is identified with the column vector in R
3
with n
i
(x)as
its components). Using the relations Cof (A
T
)=(Cof A)
T
and Cof(AB)=
(Cof A)(Cof B), we next have:
|Cof ∇Θ(x)n(x)|
2
= n(x)
T
Cof

∇Θ(x)
T
∇Θ(x)

n(x)

D := Θ(D), and let

f ∈ L
1
(

D)
be given. Then

b
D

f(x)dx =

D
(

f ◦ Θ)(x)

g(x)dx.
18 Three-dimensional differential geometry [Ch. 1
t
x
Θ(x)=ˆx
x+δx
Θ(x+δx)
I
R
f
C

b
(bx)atbx = Θ(x) ∈
b
Ωareexpressedintermsofdx, dΓ(x), and δx at x ∈ Ωby
means of the covariant and contravariant components of the metric tensor; cf. Theorem 1.3-1.
Given a domain D such that
D ⊂ Ω and a dΓ-measurable subset Σ of ∂D, the corresponding
relations are used for computing the volume of
b
D = Θ(D) ⊂
b
Ω, the area of
b
Σ=Θ(Σ) ⊂ ∂
b
D,
and the length of a curve
b
C = Θ(C) ⊂
b
Ω, where C = f(I)andI is a compact interval of R.
In particular, the volume of

D is given by
vol

D :=

b
D

g(x)

n
i
(x)g
ij
(x)n
j
(x)dΓ(x).
In particular, the area of

Σisgivenby
area

Σ:=

b
Σ
d

Γ(x)=

Σ

g(x)

n
i
(x)g
ij

g
ij
(f(t))
df
dt
i
(t)
df
dt
j
(t)dt.
Sect. 1.4] Covariant derivatives of a vector field 19
This relation shows in particular that the lengths of curves inside the open
set Θ(Ω) are precisely those induced by the Euclidean metric of the space E
3
.
For this reason, the set Θ(Ω) is said to be isometrically imbedded in E
3
.
1.4 COVARIANT DERIVATIVES OF A VECTOR
FIELD
Suppose that a vector field is defined in an open subset

ΩofE
3
by means of its
Cartesian components v
i
:


v
3
(ˆx)
ˆe
1
ˆe
2
ˆe
3
E
3
ˆ
Ω
v
i
(ˆx) ˆe
i
ˆx
Figure 1.4-1: A vector field in Cartesian coordinates. At each point bx ∈
b
Ω, the vector
bv
i
(bx)
b
e
i
is defined by its Cartesian components bv
i
(bx) over an orthonormal basis of E

i
(x)g
i
(x):=v
i
(x)

e
i
for all x = Θ(x),x∈ Ω,
where the three vectors g
i
(x)formthecontravariant basis at x = Θ(x) (Section
1.2). Using the relations g
i
(x) ·g
j
(x)=δ
i
j
and

e
i
·

e
j
= δ
i


e
i
= v
j
(x)g
j
(x) ·

e
i
.
The three components v
i
(x) are called the covariant components of the
vector v
i
(x)g
i
(x) at x, and the three functions v
i
:Ω→ R defined in this
fashion are called the covariant components of the vector field v
i
g
i
:
Ω → E
3
.

1
e
2
e
3
Ω
R
3
Θ
ˆe
1
ˆe
2
ˆe
3
g
1
(x)
g
2
(x)
v
i
(x)g
i
(x)
g
3
(x)
v

i
(x)g
i
(x)=bv
i
(bx)e
i
, bx = Θ(x).
An example of a vector field in curvilinear coordinates is provided by the displacement
field of an elastic body with {
b
Ω}
−
= Θ(Ω) as its reference configuration; cf. Section 3.2.
Theorem 1.4-1. Let Ω be an open subset of R
3
and let Θ :Ω→ E
3
be an
injective immersion that is also a C
2
-diffeomorphism of Ω onto

Ω:=Θ(Ω).
Given a vector field v
i

e
i
:

Then v
i
∈C
1
(Ω) and for all x ∈ Ω,

∂
j
v
i
(x)=

v
k
[g
k
]
i
[g

]
j

(x), x = Θ(x),
where
v
ij
:= ∂
j
v


e
1
,

e
2
,

e
3
}.
Proof. The following convention holds throughout this proof: The simul-
taneous appearance of x and x in an equality means that they are related by
x = Θ(x) and that the equality in question holds for all x ∈ Ω.
(i) Another expression of [g
i
(x)]
k
:= g
i
(x) ·

e
k
.
Let Θ(x)=Θ
k
(x)


Θ(x):=(

∂
k

Θ
i
(x)) (the row index is i)satisfy

∇

Θ(x)∇Θ(x)=I,
or equivalently,

∂
k

Θ
i
(x)∂
j
Θ
k
(x)=


∂
1

Θ

⎞
⎠
= δ
i
j
.
The components of the above column vector being precisely those of the
vector g
j
(x), the components of the above row vector must be those of the
vector g
i
(x)sinceg
i
(x) is uniquely defined for each exponent i by the three
relations g
i
(x) · g
j
(x)=δ
i
j
,j =1, 2, 3. Hence the k-th component of g
i
(x)over
the basis {

e
1
,

g
k
∈C
0
(Ω).
We next compute the derivatives ∂

g
q
(x) (the fields g
q
= g
qr
g
r
are of class
C
1
on Ω since Θ is assumed to be of class C
2
). These derivatives will be needed
in (iii) for expressing the derivatives

∂
j
u
i
(x)asfunctionsofx (recall that u
i
(x)=

k
(x)=Γ
q
m
(x)δ
m
k
=Γ
q
m
(x)g
m
(x) · g
k
(x)=−∂

g
q
(x) · g
k
(x).
Hence, noting that ∂

(g
q
(x) · g
k
(x)) = 0 and [g
q
(x)]

q
k
(x).
Since Θ ∈C
2
(Ω; E
3
)and

Θ ∈C
1
(

Ω; R
3
) by assumption, the last relations
show that Γ
q
k
∈C
0
(Ω).
(iii) The partial derivatives

∂
i
v
i
(x) of the Cartesian components of the vector
field v

v
k
(x):=∂

v
k
(x) −Γ
q
k
(x)v
q
(x),
and [g
k
(x)]
i
and Γ
q
k
(x) are defined as in (i) and (ii).
We compute the partial derivatives

∂
j
v
i
(x)asfunctionsofx by means of the
relation v
i
(x)=v

∂
j
v
i
(x)=

∂
j
v
k
(

Θ(x))[g
k
(x)]
i
+ v
q
(x)

∂
j
[g
q
(

Θ(x))]
i
= ∂


q
k
(x)v
q
(x)) [g
k
(x)]
i
[g

(x)]
j
,
since ∂

g
q
(x)=−Γ
q
k
(x)g
k
(x) by (ii). 
The functions
v
ij
= ∂
j
v
i

be as in
Theorem 1.4-1, and let there be given a vector field v
i
g
i
:Ω→ R
3
with covariant
components v
i
∈C
1
(Ω).
(a) The first-order covariant derivatives v
ij
∈C
0
(Ω) of the vector field
v
i
g
i
:Ω→ R
3
, which are defined by
v
ij
:= ∂
j
v


∂
j
(v
k
g
k
)

· g
i
.
(b) The Christoffel symbols Γ
p
ij
:= g
p
·∂
i
g
j
=Γ
p
ji
∈C
0
(Ω) satisfy the relations
∂
i
g

p
,
may be equivalently defined by the relations
∂
j
(v
i
g
i
)=v
ij
g
i
.
These relations unambiguously define the functions v
ij
= {∂
j
(v
k
g
k
)}·g
i
since
the vectors g
i
are linearly independent at all points of Ω by assumption. To
this end, we simply note that, by definition, the Christoffel symbols satisfy
∂

i
)g
i
− v
i
Γ
i
jk
g
k
= v
ij
g
i
.
To establish the other relations ∂
j
g
q
=Γ
i
jq
g
i
,wenotethat
0=∂
j
(g
p
· g

j
g
q
· g
p
)g
p
=Γ
p
qj
g
p
.

Remark. The Christoffel symbols Γ
p
ij
can be also defined solely in terms of
the components of the metric tensor; see the proof of Theorem 1.5-1. 
IftheaffinespaceE
3
is identified with R
3
and Θ(x)=x for all x ∈ Ω, the
relation ∂
j
(v
i
g
i

CURVATURE TENSOR
It is remarkable that the components g
ij
= g
ji
:Ω→ R of the metric tensor of
an open set Θ(Ω) ⊂ E
3
(Section 1.2), defined by a smooth enough immersion
Θ :Ω→ E
3
, cannot be arbitrary functions.
As shown in the next theorem, they must satisfy relations that take the
form:
∂
j
Γ
ikq
− ∂
k
Γ
ijq
+Γ
p
ij
Γ
kqp
− Γ
p
ik

:= ∂
i
Θ · ∂
j
Θ
denote the covariant components of the metric tensor of the set Θ(Ω).Letthe
functions Γ
ijq
∈C
1
(Ω) and Γ
p
ij
∈C
1
(Ω) be defined by
Γ
ijq
:=
1
2
(∂
j
g
iq
+ ∂
i
g
jq
− ∂

Γ
kqp
− Γ
p
ik
Γ
jqp
=0inΩ.
Proof. Let g
i
= ∂
i
Θ. It is then immediately verified that the functions Γ
ijq
are also given by
Γ
ijq
= ∂
i
g
j
· g
q
.
For each x ∈ Ω, let the three vectors g
j
(x) be defined by the relations g
j
(x) ·
g

since ∂
i
g
j
=(∂
i
g
j
· g
p
)g
p
. Differentiating the same relations yields
∂
k
Γ
ijq
= ∂
ik
g
j
· g
q
+ ∂
i
g
j
· ∂
k
g

g
j
· g
q
= ∂
k
Γ
ijq
− Γ
p
ij
Γ
kqp
.
Since ∂
ik
g
j
= ∂
ij
g
k
,wealsohave
∂
ik
g
j
· g
q
= ∂

in the form of the
equivalent relations ∂
ik
g
j
· g
q
= ∂
ki
g
j
· g
q
.
The functions
Γ
ijq
=
1
2
(∂
j
g
iq
+ ∂
i
g
jq
− ∂
q

Finally, the functions
R
qijk
:= ∂
j
Γ
ikq
− ∂
k
Γ
ijq
+Γ
p
ij
Γ
kqp
− Γ
p
ik
Γ
jqp
are the covariant components of the Riemann curvature tensor of the
set Θ(Ω). The relations R
qijk
= 0 found in Theorem 1.4-1 thus express that
the Riemann curvature tensor of the set Θ(Ω) (equipped with the metric tensor
with covariant components g
ij
) vanishes.
1.6 EXISTENCE OF AN IMMERSION DEFINED ON