Armenian Journal of Mathematics
Volume 7, Number 1, 2015, 6–31
Operator r on a submanifold of Riemannian
manifold and its applications
Guo Shun Zi
Sichuan University, Minnan Normal University
Abstract. The paper generalizes the self-adjoint differential
operator on hypersurfaces of a constant curvature manifold
to submanifolds, introduced by Cheng-Yau. Using the series of
such operators, a new way to prove Minkowski-Hsiung integral
formula is given and some integral formulas for compact submanifolds is derived. An application to a hypersurface of a Riemannian manifold with harmonic Riemannian curvature is presented.
Key Words: Newton tensor, operator, submanifold, hypersurface, Codazzi
tensor
Mathematics Subject Classification 2000: 53C40, 14E20; 46E25, 20C20
Introduction
Denote by V and W an n-dimensional and a p-dimensional vector spaces, respectively, V ∗ the dual space of V , {ei }(i = 1, . . . , n) and
P{eαα}(α = 1, . . . , p)
bases of V and of W , respectively. Let the tensor D =
Dij ωi ⊗ ωj ⊗ eα ∈
α,i,j
N
N
α
α
, where {ωi } is
V ∗ V ∗ W be symmetric which means that Dij
= Dji
the dual basis of {ei }. In this paper we first define the r-th Newton tensor
T(r) (D) (r = 0, 1, . . . , n), determined by the tensor D of type (1, 2) which will
be called the generalized Newton tensor. When V is the tangent space to
a submanifold at some point, and D is the second fundamental form of the
submanifold (associated with the metric), the r-th elementary symmetric
functions are called the modified mean curvatures. Following this, we define
α
in the paper the r-th modified mean curvatures of Dij
and call them Qr . We
also study some algebraic properties of the r-th Newton tensor associated
0
This work was partially supported by Natural Science Foundation of the Fujian
Province, China (No.2013J01030), the Postdoctoral Science Foundation funded project
(No.0020107602039) of Sichuan University and the projects (No.L206F2, No.SK09013,
No.2001 L21430) of Minnan Normal University
6
Operator r on a submanifold of Riemannian manifold and its applications
with r-th modified mean curvatures and the properties of them for a submanifold of a space with constant sectional curvature. We note that these
definitions and properties are natural generalizations of the classical Newton tensor and the r-th elementary symmetric polynomial’s definitions and
properties (see [17]). Then, following the operator introduced by Cheng-Yau
in [6] and using the Newton tensor we induce a series of differential operators r which are adjoint relative to the L2 -inner product. In the study
of those properties, we find a new way to prove Minkowski-Hsiung integral
formula and derive some integral formulas for compact submanifolds, which
are analogous to the usual Minkowski-Hsiung integral formula. Considering
the case r acts on Qr , we obtain two general conclusions. Finally, we focus on the 2 operator for a hypersurface of a Riemannian manifold with
harmonic Riemanian curvature to study and obtain a result of [20].
I’d like to thank Prof. Guo Zhen for brilliant guidance and stimulations.
1
The generalized Newton tensor and the
higher order mean curvatures
We begin with an algebra, first recalling some fundamental formulas. Let V
be a (real) n-dimensional vector space, and D : V −→ V be a diagonalizable
linear transformation. We fix a basis {vi , i = 1, . . . , n} of V, and denote the
matrix of D relative to this basis by (Dij ), and the eigenvalues of D relative
to this basis by k1 , . . ., kn .
The r-th elementary symmetric function is
Qr =
X
ki1 · · · kir =
1≤i1 ≤···≤ir ≤n
1 X
ki · · · kir .
r! i ,··· ,i 1
1
r
The r-th Newton tensor is
T(r) (D) = Qr I − Qr−1 D + · · · + (−1)r Dr ,
where Dr denotes the r-times linear transformation on the vector space V
by D. Relative to {vi }, the matrix of Tr (D) is
T(r)ij = Qr δij − Qr−1 Dij + · · · + (−1)r Dii1 · · · Dir j .
R. C. Reilly gave the following properties (see [17]):
1). T(r+1) (D) = Qr+1 I − DT(r) , r = 0, 1, . . . , n, where I is the identity
transformation.
2). T(r) (D) = DT(r) .
3). (r + 1)Qr+1 = Trace(DT(r) ).
7
8
Guo Shun Zi
4). Let D = D(t) be a smooth one-parameter family of diagonalizable
transformations of V. Then for r = 0, 1, . . . , n we have
∂D
∂Qr+1
= Trace(
T(r) ).
∂t
∂t
We recall the definition of the generalized Kronecker symbols (see [4]):

+1,
if (j1 , . . . , jr ) are distinct, and (j1 , . . . , jr )




is an even permutation of (i1 , . . . , ir );

i1 ,...,ir
−1,
if (j1 , . . . , jr ) are distinct, and (j1 , . . . , jr )
εj1 ,...,jr =


is an odd permutation of (i1 , . . . , ir );



0,
other case .
Remark 1.1 Moreover, the generalized Kronecker symbol can be expressed
in terms of the matrix
δi j · · · δi j 1 r 11
δi j · · · δi j 2 r 21
i1 ,...,ir
(1.1)
εj1 ,...,jr = ..
.. ,
.
.
.
.
. δir j1 . . . δir jr where δij is the standard Kronecker delta, which means:
+1,
if i = j,
δij =
0,
if i 6= j.
Lemma 1.1
1 X i1 ,...,ir
εj1 ,...,jr Di1 j1 · · · Dir jr ,
r!
1 X i1 ,...,ir ,i
εj1 ,...,jr ,j Di1 j1 · · · Dir jr .
T(r)ij =
r!
Proof. For Di1 j1 = ki1 δi1 j1 , . . . , Dir jr = kir δir jr , we have
1 X i1 ,...,ir
1 X i1 ,...,ir
εj1 ,...,jr Di1 j1 · · · Dir jr =
εi1 ...,ir ki1 · · · kir
r!
r!
1X
=
ki1 · · · kir
r!
= Qr .
Qr =
(1.2)
(1.3)
¿From the right part of (1.3), we know that the generalized Kronecker symbol can be expressed in terms of (1.1), then if we express the matrix of
,...,ir ,i
εii11 ,...,i
by unfolding the matrix along its last line, we obtain
r ,j
1 X i1 ,...,ir ,i
1 X i1 ,...,ir ,i
εj1 ,...,jr ,j Di1 j1 · · · Dir jr =
εi1 ,...,ir ,j ki1 · · · kir
r!
r!
= Qr δij − T(r−1)il Dlj .
Using the property 1 given by R. C. Reilly, we obtain that (1.3) is true. Operator r on a submanifold of Riemannian manifold and its applications
Remark 1.2 These can be viewed as the second expression of the r-th elementary symmetric function and the r-th Newton tensor (the papers [11, 16]
make use of this kind of expression).
Let V and W denote an n-dimensional and a p-dimensional vector spaces,
respectively, V ∗ denotes the dual space of V , {ei } (i = 1, . . . , n) and
{eα } (α = P
1, . . . , p) denote bases of N
V and
Nof W , respectively.
∗
∗
α
W be symmetric which means
V
Let D =
Dij ωi ⊗ ωj ⊗ eα ∈ V
α,i,j
α
α
, where {ωi } is the dual basis to {ei }. In this paper we begin with
= Dji
Dij
defining the r-th Newton tensor T(r) (D)(r = 0, 1, . . . , n). Closely following
the second exposition for the Newton tensor, and imitating the definition of
the mean curvature in [17, 11], we define the generalized Newton tensor as
follows:
Definition 1.1 1) If r is an odd integer,
N
Nr = 2k + 1 (k
N = 0, 1, . . .), then
T(r) (D) is a mapping T(r) (D) : V ∗ V ∗ W −→ V ∗ V ∗ such that for
Z = Zijα ωi ⊗ ωj ⊗ eα
T(r) (D)Z =
1 i1 ,...,ir ,i α1 α1
αk
α
α
k
εj1 ,...,jr ,l (Di1 j1 Di2 j2 ) · · · (Diαr−2
jr−2 Dir−1 jr−1 )(Dir jr Zlj )ωi ⊗ ωj .
r!
αk
αk
α1
α1
r ,i
α
α
(D) = r!1 εij11,...,i
Denoting T(r)il
,...,jr ,l (Di1 j1 , Di2 j2 ) · · · (Dir−2 jr−2 , Dir−1 jr−1 )Dir jr , we
have
α
α
T(r) (D)Z = T(r)il
Zljα ωi ⊗ ωj , (T(r) (D)Z)ij = T(r)il
Zljα .
2) If r is an even integer,
1, . . .),N
then T(r) (D) is detemined
N r∗ =
N2k (k = 0,
N
∗
∗
∗
as a map T(r) (D) : V
V
W −→ V
V
W such that
T(r) (D)Z =
1 i1 ,...,ir ,i α1 α1
αk
α
k
εj1 ,...,jr ,l (Di1 j1 Di2 j2 ) · · · (Diαr−1
jr−1 Dir jr )Zlj ωi ⊗ ωj ⊗ eα .
r!
Denoting T(r)il (D) =
αk
1 i1 ,...,ir ,i
k
ε
(Diα11j1 Diα21j2 ) · · · (Diαr−1
jr−1 Dir jr ),
r! j1 ,...,jr ,l
we obtain
T(r) (D)Z = T(r)il Zljα ωi ⊗ ωj ⊗ eα , (T(r) (D)Z)αij = T(r)il Zljα .
The map T(r) (D) is called the generalized Newton transformation (or
tensor) of D.
Remark 1.3 For convenience to compute, in this Section, we shall agree
that repeated indices are summed, and T(r) (D) is viewed as T(r) if r = 0,
T(0)ij = δij , if r = n, T(n)ij = 0. Also, we suppose T(r) = T(r) (D).
We are really interested only in the situation where V is the tangent space
to a submanifold, and D is the second fundamental form of the submanifold
(associated with the metric), the r-th elementary symmetric functions calling
the r-th modified mean curvatures. Then, following this we define the r-th
α
modified mean curvatures of Dij
and call them Qr .
9
10
Guo Shun Zi
Definition 1.2 1) If r is an odd integer, r = 2k + 1, then define
Qr :=
1 i1 ,...,ir α1 α1
αk
α
k
εj1 ,...,jr (Di1 j1 Di2 j2 ) · · · (Diαr−2
jr−2 Dir−1 jr−1 )Dir jr eα ,
r!
1 i1 ,...,ir α1 α1
αk
α
k
εj1 ,...,jr (Di1 j1 Di2 j2 ) · · · (Diαr−2
jr−2 Dir−1 jr−1 )Dir jr .
r!
2) If r is an even integer, r = 2k, then define
Qαr :=
Qr :=
1 i1 ,...,ir α1 α1
αk
k
εj1 ,...,jr (Di1 j1 Di2 j2 ) · · · (Diαr−1
jr−1 Dir jr ).
r!
Remark 1.4 If σr is a formal
of D, then it is not
r-th mean
curvature
n
n
n!
difficult to know that Qr = r σr , where r = (n−r)!r! . Suppose Q0 = 1, and
P
if r is 1, then Q1 = nσ1 = i,α Diiα eα .
We are going to prove some algebraic properties of the r-th Newton tensor
associated with the r-th modified mean curvatures. Those properties are
natural generalizations of the algebraic properties of classical Newton tensor
and the r-th elementary symmetric polynomial.
Lemma 1.2
(r + 1)Qr+1 = Trace(T(r) D).
(1.4)
Proof. If r is an odd integer,
α
Trace(T(r) D) = T(r)il
Dliα
1 i1 ,...,ir ,i α1 α1
αk
α
α
k
=
ε
(D D ) · · · (Diαr−2
jr−2 Dir−1 jr−1 )Dir jr Dli
r! j1 ,...,jr ,l i1 j1 i2 j2
(r + 1)!
1
i ,...,i ,i
αk
k
ε 1 r r+1 (Dα1 Dα1 ) · · · (Diαr−2
=
jr−2 Dir−1 jr−1 )
r! (r + 1)! j1 ,...,jr ,jr+1 i1 j1 i2 j2
α
αk+1
·(Dirk+1
jr Dir+1 jr+1 )
= (r + 1)Qr+1 .
If r is an even integer,
Trace(T(r) D) = T(r)il Dliα eα
1 i1 ,...,ir ,i α1 α1
αk
α
k
εj1 ,...,jr ,l (Di1 j1 Di2 j2 ) · · · (Diαr−1
=
jr−1 Dir jr )Dli eα
r!
(r + 1)!
1
i ,...,i ,i
αk
k
=
εj11 ,...,jrr ,jr+1
(Diα11j1 Diα21j2 ) · · · (Diαr−1
jr−1 Dir jr )
r+1
r! (r + 1)!
·Diαr+1 jr+1 eα
= (r + 1)Qr+1 .
Operator r on a submanifold of Riemannian manifold and its applications
11
Lemma 1.3 If r is an odd integer,
α
α
T(r)ij
= T(r)ji
(1.5)
T(r)ij = T(r)ji .
(1.6)
If r is an even integer,
Proof. Using the symmetry of D, if r is an odd integer, set r = 2k + 1,
α
T(r)ij
=
=
=
=
=
1 i1 ,...,ir ,i α1
αk
α
k
ε
(D , Dα1 ) · · · (Diαr−2
jr−2 , Dir−1 jr−1 )Dir jr
r! j1 ,...,jr ,j i1 j1 i2 j2
1 j1 ,...,jr ,j α1 α1
αk
α
k
ε
(D D ) · · · (Djαr−2
ir−2 Djr−1 ir−1 )Djr ir
r! i1 ,...,ir ,i j1 i1 j2 i2
1 j1 ,...,jr ,j α1 α1
αk
α
k
ε
(D D ) · · · (Diαr−2
jr−2 Dir−1 jr−1 )Dir jr
r! i1 ,...,ir ,i i1 j1 i2 j2
1 i1 ,...,ir ,j α1 α1
αk
α
k
ε
(D D ) · · · (Diαr−2
jr−2 Dir−1 jr−1 )Dir jr
r! j1 ,...,jr ,i i1 j1 i2 j2
α
T(r)ji
.
If r is an even integer, set r = 2k,
T(r)ij =
=
=
=
=
1 i1 ,...,ir ,i α1
αk
k
εj1 ,...,jr ,j (Di1 j1 , Diα21j2 ) · · · (Diαr−1
jr−1 , Dir jr )
r!
1 j1 ,...,jr ,j α1 α1
αk
k
εi1 ,...,ir ,i (Dj1 i1 Dj2 i2 ) · · · (Djαr−1
ir−1 Djr ir )
r!
1 j1 ,...,jr ,j α1 α1
αk
k
εi1 ,...,ir ,i (Di1 j1 Di2 j2 ) · · · (Diαr−1
jr−1 Dir jr )
r!
1 i1 ,...,ir ,j α1 α1
αk
k
εj1 ,...,jr ,i (Di1 j1 Di2 j2 ) · · · (Diαr−1
jr−1 Dir jr )
r!
T(r)ji .
Lemma 1.4 If r is an even integer,
α
T(r) (D) = Qr I − T(r−1)
Dα .
(1.7)
α
T(1)
(D) = Qα1 I − T(0) Dα .
(1.8)
If r = 1,
Proof. If r is an even integer,
1 i1 ,i2 ,...,ir ,i α1 α1
αk
k
ε
(D D ) · · · (Diαr−1
jr−1 Dir jr )
r! j1 ,j2 ,...,jr ,j i1 j1 i2 j2
1 i1 i2 ,i3 ,...,ir ,i
i i1 ,i2 ,...,ir−1 ,ir
r ,i
=
(δ ε
− δji2 εij11,i,j32,...,i
,...,jr−1 ,jr + . . . + δj εj1 ,j2 ,...,jr−1 ,jr )
r! j j1 ,j2 ,...,jr−1 ,jr
αk
k
·(Diα11j1 Diα21j2 ) · · · (Diαr−1
jr−1 Dir jr )
T(r)ij =
12
Guo Shun Zi
=
=
=
=
=
1 i2 ,i3 ,...,ir ,i
αk
α1
k
Diα21j2 ) · · · (Diαr−1
εj1 ,j2 ,...,jr−1 ,jr (Djj
jr−1 Dir jr )
1
r!
1
αk
αk
α1
α1
r ,i
− εij11,i,j32,...,i
,...,jr−1 ,jr (Di1 j1 Djj2 ) · · · (Dir−1 jr−1 Dir jr )
r!
+··· ··· ···
1 i ,i ,...,i ,ir
αk
i
k
+ εj11 ,j22 ,...,jr−1
· (Diα11j1 Diα21j2 ) · · · (Diαr−1
jr−1 Dir jr )δj
r−1 ,jr
r!
1
αk
αk
α2
α2
α1
α1
r ,i
− εij22,i,j33,...,i
...,jr ,j1 (Di3 j3 Di4 j4 ) · · · (Dir−1 jr−1 Dir jr )(Di2 j2 Djj1 )
r!
1
αk
αk
α1
α1
α2
α2
r ,i
− εij11,i,j33,...,i
...,jr ,j2 (Di3 j3 Di4 j4 ) · · · (Dir−1 jr−1 Dir jr )(Di1 j1 Djj2 )
r!
−··· ··· ···
+Qr δji
1 i ,i ,...,ir−1 ,i α1 α1
αk
αk
α
α
− εj11 ,j22 ...,jr−1
,l (Di1 j1 Di2 j2 ) · · · (Dir−3 jr−3 Dir−2 jr−2 )(Dir−1 jr−1 Dlj )
r!
1 i ,i ,...,ir−1 ,i α1 α1
αk
αk
α
α
− εj11 ,j22 ...,jr−1
,l (Di1 j1 Di2 j2 ) · · · (Dir−3 jr−3 Dir−2 jr−2 )(Dir−1 jr−1 Dlj )
r!
−··· ··· ···
+Qr δji
1 α
1 α
− T(r−1)il
Dljα − · · · − T(r−1)il
Dljα + Qr δji
r!
r!
α
Qr δji − T(r−1)il
Dljα .
If r = 1,
α
T(1)
(D) = εij11ij Diα1 j1
= δji11 δji Diα1 j1 − δji1 δji1 Diα1 j1
α
= Qα1 δji − δji1 Djj
1
α
= Qα1 δji − T(0)ij1 Djj
.
1
Lemma 1.5 Let D = D(t) be a smooth one-parameter family of D, then
for r = 1, . . . , n + 1 we have
If r is even,
∂Dα
∂Qr
α
= Trace(T(r−1)
).
(1.9)
∂t
∂t
If r = 1,
∂Qα1
∂Dα
(1.10)
= Trace(T(0)
).
∂t
∂t
Proof. If r is even, from the equation
(r)Qr = Trace(T(r−1) D)
Operator r on a submanifold of Riemannian manifold and its applications
13
and
Qr =
1 i1 ,...,ir α1 α1
αk
k
ε
(D D ) · · · (Diαr−1
jr−1 Dir jr ),
r! j1 ,...,jr i1 j1 i2 j2
we have
∂Qr
∂t
α1
=
1 i1 ,...,ir ∂Di1 j1
ε
( ∂t Diα21j2
r! j1 ,...,jr
α1
∂D
αk
k
Diα11j1 ∂ti2 j2 ) · · · (Diαr−1
jr−1 Dir jr )
+
+··· ··· ···
αk−1
αk−1
α1
α1
r
+ r!1 εij11,...,i
,...,jr (Di1 j1 Di2 j2 ) · · · (Dir−3 jr−3 Dir−2 jr−2 )
α
∂Di k
∂D
αk
ir jr
k
Diαrkjr + Diαr−1
jr−1 ∂t )
∂t
αk−1
αk−1
α1
α1
r
= r!1 εij11,...,i
,...,jr (Di1 j1 Di2 j2 ) · · · (Dir−3 jr−3 Dir−2 jr−2 )
(
r−1 jr−1
α
∂Di k
∂D
αk
ir jr
r−1
k
( r−1
Diαrkjr + Diαr−1
jr−1 ∂t )
∂t
+··· ··· ···
αk−1
αk−1
α1
α1
r
+ r!1 εij11,...,i
,...,jr (Di1 j1 Di2 j2 ) · · · (Dir−3 jr−3 Dir−2 jr−2 )
j
α
∂Di k
∂D
αk
ir jr
k
Diαrkjr + Diαr−1
jr−1 ∂t )
∂t
αk−1
αk−1
(r−1)!
1
=
εi1 ,...,ir (Diα11j1 Diα21j2 ) · · · (Dir−3
jr−3 Dir−2 jr−2 )
r! (r−1)! j1 ,...,jr
(
r−1 jr−1
α
∂Di k
r−1 jr−1
Diαrkjr
αk
∂Dir jr
k
Diαr−1
jr−1 ∂t )
+
(
∂t
+··· ··· ···
αk−1
αk−1
1
+ (r−1)!
εi1 ,...,ir (Diα11j1 Diα21j2 ) · · · (Dir−3
jr−3 Dir−2 jr−2 )
r! (r−1!) j1 ,...,jr
α
=
∂Di k
α
∂Dirkjr
)
∂t
∂t
α
∂Dirkjr
α
αk−1
i1 ,...,ir
αk
α1
α1
k−1
1
2
(D
ε
D
)
·
·
·
(D
D
)D
i1 j1 i2 j2
ir−3 jr−3 ir−2 jr−2
ir−1 jr−1 ∂t
r (r−1)! j1 ,...,jr
(
r−1 jr−1
+···
k
Diαrkjr + Diαr−1
jr−1
···
···
α
=
∂Dirkjr
αk−1
αk−1
αk
α1
α1
1
r
+ 2r (r−1)!
εij11,...,i
(D
D
)
·
·
·
(D
D
)D
,...,jr
i1 j1 i2 j2
ir−3 jr−3 ir−2 jr−2
ir−1 jr−1 ∂t
α
∂Dirkjr
2 αk
T
r (r−1)ir jr ∂t
+···
···
···
αk
=
=
∂Dir jr
αk
+ 2r T(r−1)i
∂t
r jr
α
∂Dirkjr
αk
T(r−1)ir jr ∂t
∂Dα
α
Trace(T(r−1)
).
∂t
If r = 1 ,
∂(εij11 Diα11j1 )
∂Qα1
=
∂t
∂tα1
∂D
i1 j1
= εij11
∂t
∂Dα
= Trace(T(0)
).
∂t
14
2
Guo Shun Zi
Operator r on a submanifold of a space
with constant sectional curvatures and it’s
applications
In this Section, we follow closely the exposition of the moving frame in
[3, 21], and we agree that Qr is a vector, formal in a submanifold like as
in the above Section, however being the modified mean curvature function
in a hypersurface. Let x : M n → N n+p be an isometric immersion of ndimensional Riemannian M n as a submanifold in (n + p)-dimensional space
N . We choose a local field of orthonormal frames e1 , . . . , en+p of N n+p such
that, restricted to M , the vectors e1 , . . . , en are tangent to M . We shall
make use of the following convention on the ranges of indices
1 ≤ A, B, C, . . . ≤ n + p,
1 ≤ i, j, k, . . . ≤ n,
n + 1 ≤ α, β, γ, . . . ≤ n + p,
and we shall agree that repeated indices are summed over the respective
ranges. With respect to the frame field of N chosen above, let ω1 , . . . , ωn+p
be the field of the dual frame.
Then the structure equations of N are given by
X
dωA =
ωB ∧ ωBA , ωBA + ωAB = 0,
(2.1)
1X
RABCD ωC ∧ ωD , (2.2)
2
where ωAB is the Levi-civita connection of N with respect to eA and RABCD
is the Riemannian curvature tensor of N . We know that RABCD satisfies the
following identities
dωAB =
X
ωAC ∧ ωCB + ΦAB , ΦAB = −
RABCD = −RABDC = −RBADC , RABCD = RCDAB ,
(2.3)
RABCD + RACDB + RADBC = 0.
(2.4)
We restrict these forms to M by the same letters. Then
ωα = 0.
(2.5)
The structure equations of M are
X
dωi =
ωj ∧ ωji , ωji + ωij = 0,
(2.6)
j
dωij =
X
k
ωik ∧ ωkj + Φij , Φij = −
1X
Rijkl ωk ∧ ωl .
2
(2.7)
Operator r on a submanifold of Riemannian manifold and its applications
Since 0 = dωα =
P
ωj ∧ ωjα , by Cartan’s lemma we may write
X
ωiα =
hαij ωj , hαij = hαji .
15
(2.8)
j
¿From these formulas we obtain
Rijkl = Rijkl +
X
(hαik hαjl − hαil hαjk ),
(2.9)
α
dωαβ =
X
ωαγ ∧ ωγβ + Φαβ , Φαβ = −
γ
Rαβkl = Rαβkl +
1X
Rαβkl ωk ∧ ωl .
2
X
(hαik hβil − hαil hβik ).
(2.10)
(2.11)
i
Here (ωij ) defines a connection
M , and (ωαβ ) a connection in the normal
P of
α
bundle of M . We call B =
hij ωi ⊗ ωj ⊗ eα the second fundamental form
of the immersed manifold M . We take exterior differentiation of (2.8) and
use hαij,k to denote the covariant derivatives by
X
hαij,k ωk = dhαij +
X
hαik ωkj +
X
hαkj ωki −
X
hβij ωαβ .
(2.12)
Then
hαij,k − hαik,j = Rαijk .
(2.13)
Now we introduce the operator r .
For a C ∞ function f defined on M , we define its gradient and Hessian by
the following formulas
X
X
X
df :=
f,i ωi ,
f,ij ωj := df,i +
f,j ωj (f,ij = f,ji ).
(2.14)
For a section ξ = ξ α eα of the normal bundle T ⊥ (M ) we define the covariant
derivative of ξ α by
X
X
ξ,iα ωi = dξ α +
ξ β ωβα
(2.15)
and the convariant derivative of ξ,iα by
X
α
ξ,ij
ωj = dξ,iα +
X
ξ,jα ωji −
X
ξ,iβ ωαβ .
(2.16)
When p > 1 and r is odd, we can define the differential operator r .
Definition 2.1 For a section ξ = ξ α eα of the normal bundle T ⊥ (M ) we
denote the differential operator
r∗ : C ∞ (T ⊥ (M )) −→ C ∞ (M )
16
Guo Shun Zi
by
r∗ ξ =
X
α
α
.
ξ,ij
T(r)ij
(2.17)
For a C ∞ function f of M we define the differential operator
r : C ∞ (M ) −→ C ∞ (T ⊥ (M ))
by
r f =
X
α
T(r)ij
f,ij eα .
(2.18)
If p > 1 and r is odd, we define differential operator r , and in the case
p = 1 we also define differential operator r as well as the above definitions.
Definition 2.2 For a C ∞ function f of M we can define the differential
operator
r : C ∞ (M ) −→ C ∞ (M )
X
r f =
T(r)ij f,ij .
(2.19)
Remark 2.1 If r = 0, then r f =
P
f,ii = ∆f .
i
Now we suppose that N is of constant curvature c, then
Rαjkl = 0, Rijkl = c(δik δjl − δil δjk )
So we have the Gauss equation
Rijkl = c(δik δjl − δil δjk ) +
X
(hαik hαjl − hαil hαjk )
(2.20)
α
and
hαij,k − hαik,j = 0.
(2.21)
So the fundamental form B must be a Codazzi tensor. We have a lemma as
follows:
Lemma 2.1 Let x : M n → N n+p (c) be an immersion of a compact orientable n-dimensional Riemannian manifold M n as a submanifold in the
(n + p)-dimensional Riemannian N n+p with constant sectional curvature c,
and let B be the second fundamental form of M n .
i) If p > 1 and r is an even integer, then the r-th Newton tensor of B is
divergence-free, i.e.,
X
T(r)ij,j = 0.
j
Operator r on a submanifold of Riemannian manifold and its applications
If p > 1 and r is an odd integer, then
X
α
T(r)ij,j
= 0.
j
ii) If p = 1 and r is any integer, then the r-th Newton tensor of B is
divergence-free, i.e.,
X
T(r)ij,j = 0.
j
Proof. Since ii) is proved in [17], we are going to do only i).
If r is an even integer, set r = 2k,
T(r)ij =
1 i1 ,i2 ,...,ir ,i α1 α1
αk
k
ε
(h h ) · · · (hαir−1
jr−1 hir jr ).
r! j1 ,j2 ,...,jr ,j i1 j1 i2 j2
We have
1 i1 ,i2 ,...,ir ,i α1
αk
k
ε
(h
hα1 + hαi11j1 hαi21j2 ,j ) · · · (hαir−1
jr−1 hir jr )
r! j1 ,j2 ,...,jr ,j i1 j1 ,j i2 j2
+··· ··· ···
1
αk
αk
αk
αk
α1
α1
r ,i
+ εij11,i,j22,...,i
,...,jr ,j (hi1 j1 hi2 j2 ) · · · (hir−1 jr−1 ,j hir jr + hir−1 jr−1 ,j hir jr ,j )
r!
1 i1 ,i2 ,...,ir ,i αk αk
α1
α1
α1
1
(h h ) · · · (hαir−1
ε
=
jr−1 ,j hir jr + hir−1 jr−1 ,j hir jr ,j )
r! j1 ,j2 ,...,jr ,j i1 j1 i2 j2
+··· ··· ···
1
αk
αk
αk
αk
α1
α1
r ,i
+ εij11,i,j22,...,i
,...,jr ,j (hi1 j1 hi2 j2 ) · · · (hir−1 jr−1 ,j hir jr + hir−1 jr−1 ,j hir jr ,j )
r!
k i1 ,i2 ,...,ir ,i α1 α1
αk
αk
αk
k
=
ε
(h h ) · · · (hαir−1
jr−1 ,j hir jr + hir−1 jr−1 ,j hir jr ,j )
r! j1 ,j2 ,...,jr ,j i1 j1 i2 j2
1
αk
k
=
εi1 ,i2 ,...,ir ,i (hα1 hα1 ) · · · (hαir−1
jr−1 ,j hir jr ,j ).
(r − 1)! j1 ,j2 ,...,jr ,j i1 j1 i2 j2
T(r)ij,j =
and we know that
i1 ,...,ir ,i
ri
εij11,...,i
,...,jr j + εj1 ,...,j,jr = 0
,
hαirkjr ,j = hαirkj,jr
so we have
X
j
T(r)ij,j = 0.
17
18
Guo Shun Zi
If r is an odd integer, set r = 2k + 1,
X
α
T(r)ij,j
=
j
1 i1 ,i2 ,...,ir−1 ,ir ,i α1
αk
α
k
εj1 ,j2 ,...,jr−1 ,jr ,j (hi1 j1 ,j hαi21j2 + hαi11j1 hαi21j2 ,j ) · · · (hαir−2
jr−2 hir−1 jr−1 )hir jr
r!
·hαir jr + · · · · · · · · ·
1 i ,i ,...,i ,ir ,i α1
αk
αk
αk
k
+ εj11 ,j22 ,...,jr−1
(hi1 j1 , hαi21j2 ) · · · (hαir−2
jr−2 ,j hir−1 jr−1 + hir−2 jr−2 ,j hir−1 jr−1 ,j )
r−1 ,jr ,j
r!
·hαir jr
1 i ,i ,...,i ,ir ,i α1 α1
αk
α
k
+ εj11 ,j22 ,...,jr−1
(hi1 j1 hi2 j2 ) · · · (hαir−2
jr−2 hir−1 jr−1 )hir jr ,j
r−1 ,jr ,j
r!
2 i1 ,i2 ,...,ir ,i α1 α1
αk
α
k
=
ε
(h h
) · · · (hαir−2
jr−2 hir−1 jr−1 ,j )hir jr
r! j1 ,j2 ,...,jr ,j i1 j1 i2 j2 ,j
+··· ··· ···
2 i ,i ,...,i ,ir ,i α1 α1
αk
α
k
+ εj11 ,j22 ,...,jr−1
(hi1 j1 hi2 j2 ) · · · (hαir−2
jr−2 hir−1 jr−1 ,j )hir jr
r−1 ,jr ,j
r!
1 i ,i ,...,i ,ir ,i α1 α1
αk
α
k
+ εj11 ,j22 ,...,jr−1
(hi1 j1 hi2 j2 ) · · · (hαir−2
jr−2 hir−1 jr−1 ,j )hir jr ,j
r−1 ,jr ,j
r!
=
↓
=
↓
=
↓
=
2k i1 ,i2 ,...,ir−1 ,ir ,i α1 α1
αk
α
k
εj1 ,j2 ,...,jr−1 ,jr ,j (hi1 j1 hi2 j2 ) · · · (hαir−2
jr−2 hir−1 jr−1 ,j )hir jr
r!
1 i ,i ,...,i ,ir ,i α1 α1
αk
α
k
(hi1 j1 hi2 j2 ) · · · (hαir−2
+ εj11 ,j22 ,...,jr−1
jr−2 hir−1 jr−1 ,j )hir jr ,j
r−1 ,jr ,j
r!
(by exchanging the second integers)
2k i1 ,i2 ,...,ir−1 ,ir ,i α1 α1
αk
α
k
εj1 ,j2 ,...,j,jr ,jr−1 (hi1 j1 hi2 j2 ) · · · (hαir−2
jr−2 hir−1 j,jr−1 )hir jr
r!
1 i ,i ,...,i ,ir ,i α1 α1
αk
α
k
(hi1 j1 hi2 j2 ) · · · (hαir−2
+ εj11 ,j22 ,...,jr−1
jr−2 hir−1 jr−1 ,j )hir j,jr
r−1 ,j,jr
r!
(B is a Codazzi tensor)
2k i1 ,i2 ,...,ir−1 ,ir ,i α1 α1
αk
α
k
εj1 ,j2 ,...,j,jr ,jr−1 (hi1 j1 hi2 j2 ) · · · (hαir−2
jr−2 hir−1 jr−1 ,j )hir jr
r!
1 i ,i ,...,i ,ir ,i α1 α1
αk
α
k
(hi1 j1 hi2 j2 ) · · · (hαir−2
+ εj11 ,j22 ,...,jr−1
jr−2 hir−1 jr−1 ,j )hir jr ,j
r−1 ,j,jr
r!
(the generalized Kronecker sign is anti-symmetric)
2k i ,i ,...,i ,ir ,i α1 α1
αk
α
k
(hi1 j1 hi2 j2 ) · · · (hαir−2
− εj11 ,j22 ,...,jr−1
jr−2 hir−1 jr−1 ,j )hir jr
r−1 ,jr ,j
r!
1 i ,i ,...,i ,ir ,i α1 α1
αk
α
k
− εj11 ,j22 ,...,jr−1
(hi1 j1 hi2 j2 ) · · · (hαir−2
jr−2 hir−1 jr−1 ,j )hir jr ,j .
r−1 ,jr ,j
r!
So we have
X
α
T(r)ij,j
= 0.
j
This completes the proof of the lemma 2.4. Operator r on a submanifold of Riemannian manifold and its applications
19
For any point q ∈ M , the Riemannian metric h, iq induces inner products in
the normal bundles over M , also denoted by h, iq . Then, we can define for
any ξ, η ∈ C ∞ (T ⊥ (M )) a function hξ, ηiq = hξq , ηq iq , since ξq , ηq ∈ T ⊥ (M )q .
It also induces the operator ∗ by requiring
hξ, ηiq dM = η ∧ ∗ξ.
So we can define the L2 -inner product on C ∞ (T ⊥ (M )) as
Z
Z
hξ, ηidM =
η ∧ ∗ξ,
(, ) : (ξ, η) =
M
M
denoting the volume form by dM , see [5], [7]. We recall Theorem 1.1 of [7].
Lemma 2.2 Let M be a compact oriented submanifold. If φ satisfies the
conditions
X
(i) φαij = φαji , (ii)
φαij,j = 0,
then ∗φ and φ are adjoint, which means
(f, ∗φ ξ) = (ξ, φ f ),
(2.22)
in particular
Z
∗φ ξdM = 0.
(2.23)
M
We have the following key Theorem.
Theorem 2.1 Let M be a compact oriented submanifold of a space with
constant sectional curvature, the definitions of r and r∗ being as above.
Then
i) if p > 1 and r is an odd integer, 1 ≤ r ≤ n, r and r∗ are adjoint
relative to the L2 -inner product of M , i.e.,
(r∗ ξ, f ) = (ξ, r f ),
(2.24)
in particular,
Z
r∗ ξdM = 0.
(2.25)
M
ii) if p > 1 and r is an even integer or p = 1, r is self-adjoint relative to
the L2 -inner product of M , i.e.,
(g, r f ) = (f, r g),
(2.26)
in particular
Z
M
r f dM = 0.
(2.27)
20
Guo Shun Zi
Suppose ξ is a vector of the tangent bundle T > (M ). We have
Z
r ξdM = 0,
M
where we denote the volume form by dM .
Proof. If p > 1 and r is an even integer, or p = 1,
X
X X
g r f : = g
T(r)ij f,ij =
(g
T(r)ij f,i ),j
i,j
−
j
X
g,j T(r)ij f,i − g
i
X
i,j
=
X
(g
X
j
+
T(r)ij,j f,i
i,j
T(r)ij f,i ),j −
X
i
X
i,j
g,j T(r)ij,i f,i − g
i,j
(g,j T(r)ij f ),i +
X
X
g,ji T(r)ij f
ij
T(r)ij,j f,i .
i,j
If r is even, we have from Lemma 2.1
X
X
T(r)ij,j = 0,
T(r)ij,i = 0.
j
Meanwhile, set ξj = g
P
i
i
T(r)ij f,i , ηi = f
P
T(r)ij g,i . Making use of Green’s
j
theorem we get
(g, r f ) = (f, r g).
R
If g = 1 we have M r f dM = 0.
If p > 1 and r is an odd integer, making use of lemma 2.1 and Lemma 2.2
we obtain
(r∗ ξ, f ) = (ξ, r f ).
This completes the proof of the theorem. Remark 2.2 When p = 1, 1 is the same as the operator derived by
Cheng-Yau, [6].
Now we choose a vector a ∈ Rn+p+1 of N n+p with the constant curvatures.
For any point p ∈ M n , let X(p) ∈ M n be the position vector. We define the
height function by ϕ := ha, Xi, where h, i is the inner product in M n .
We are going to calculate r ϕ.
The moving equation of M n in N n+p is

ωi ei ,
 dX =
dei = ωij ej + ωiα eα − cXωi ,
(2.28)

deα =
−ωiα ei + ωαβ eβ .
Operator r on a submanifold of Riemannian manifold and its applications
21
Hence
dϕ = hdX, ai = hωi ei , ai.
If we set ϕ,i = hei , ai from the definition (2.14) of the covariant derivatives
of ϕ,i we obtain
ϕ,ij ωj = dϕ,i + ϕ,j ωji .
(2.29)
Taking moving equations (2.28) into the above equation, we obtain
ϕ,ij ωj = ωij hej , ai + hαij heα , aiωj
−chX, aiωi + hej , aiωji
= [hαij heα , ai − chX, aiδij ]ωj .
(2.30)
ϕ,ij = hαij heα , ai − chX, aiδij .
(2.31)
So
If r is even and p > 1, then
P
r ϕ =
T(r)ij ϕ,ij
i,j
P
P
=
T(r)ij hαij heα , ai − c T(r)ij hX, aiδij
α,i,j
(2.32)
i,j
= hTrace(T(r) B), ai − chX, aiTrace(T(r) ).
From (1.4), we know Trace(T(r) B) = (r + 1)Qr+1 ,
α
and from (1.7), T(r) (B) = Qr I − T(r−1)
B α we have
Trace(T(r) ) = nQr − rQr = (n − r)Qr .
(2.33)
r ϕ = (r + 1)hQr+1 , ai − chX, ai(n − r)Qr .
(2.34)
Then
The same proof works for the case where p = 1 and r is any integer, so
r ϕ = (r + 1)Qr+1 hen+1 , ai − chX, ai(n − r)Qr ,
and
(2.35)
n
Qr =
σr .
r
Using Theorem 2.1, we obtain the following theorem.
Theorem 2.2 Let x : M n → N n+p (c) be an immersion of the compact
orientable n-dimensional Riemannian manifold M n as a submanifold in the
(n + p)-dimensional Riemannian N n+p ⊂ Rn+p+1 with constant sectional
curvature, let a be any fixed element of N n+p , eα (α = 1, . . . , p) be the unit
normal vector field of M n , σr (r = 0, 1, . . . , (n − 1)) be the mean curvature
22
Guo Shun Zi
on M n , and X be the position vector of M n .
i) If p > 1 and r is an even integer, then
Z
(hσr+1 , ai − chX, aiσr )dM = 0.
(2.36)
M
ii) If p = 1 and r is any integer, then
Z
(σr+1 hen+1 , ai − chX, aiσr )dM = 0.
(2.37)
M
Remark 2.3 For hypersurfaces in the unit sphere, from the theorem we can
deduce Theorem A of R. C. Reilly, in [15], so our Theorem is a generalization
of the mentioned result.
Corollary 2.1 (Reilly [15]) Let x : M n → S n+1 be an immersion of the compact orientable n-dimensional Riemannian manifold M n as a hypersurface in
the (n + 1)-dimensional unit sphere S n+1 ⊂ Rn+2 . Let a be any fixed element
of S n+1 , en+1 be the unit normal vector field of M n , σr (r = 0, 1, . . . , (n − 1))
be the mean curvature functions on M n , and X be the position vector of M n .
Then
Z
(σr+1 hen+1 , ai − hX, aiσr )dM = 0.
(2.38)
M
If p = 1, and en+1 is the unit normal vector field on M n ⊂ N n+1 (c), using
the moving equations (2.28) we are going to compute r en+1 . By
den+1 = −
X
ωi(n+1) ei =
X
−hij ei ωj ,
we know
en+1,j = −
X
i
hij ei , en+1,i = −
X
j
hij ej ,
(2.39)
Operator r on a submanifold of Riemannian manifold and its applications
23
and we have
X
X
en+1,ij ωj = den+1,i +
en+1,j ωji
j
j
X
X
= −
[(dhij )ej + hij dej +
hkj ωji ek ]
j
k
X
X
X
= −(
hij,k ωk ej +
hik ωjk ej +
hkj ωik ej
k,j
−cX
X
hij ωj +
k,j
k,j
X
X
j
+
X
hij ωjk ek +
k,j
hkj ωji ek
k,j
hkj hij ωk en+1 )
k,j
X
X
X
= −(
hij,k ωk ej +
hkj ωk en+1 ) + cX
hij ωj
k,j
j
k,j
XX
X
= −
(
hij,k ek +
hjk hki en+1 − cXhij )ωj .
j
k
k
So we can obtain
en+1,ij = −
X
hij,k ek −
X
On the other hand,
P
hjk hki en+1 + cXhij .
(2.40)
k
k
T(r)ij hik hkj =
P
(Qr+1 δjk − T(r)jk )hkj
j,k
i,j,k
(2.41)
= Qr+1 Q1 − (r + 2)Qr+2 .
Making use of this identity we obtain
P
r en+1 =
T(r)ij en+1,ij
i,j P
P
P
= −
T(r)ij hij,k ek −
T(r)ij hjk hki en+1 + cX T(r)ij hij
i,j
i,j,k
i,j,k
P
= − (r + 1)Qr+1,k ek − (Qr+1 Q1 − (r + 2)Qr+2 )en+1
k
+c(r + 1)Qr+1 X.
(2.42)
Theorem 2.3 Let x : M n → N n+1 (c) be an immersion of a compact orientable n-dimensional Riemannian manifold M n as a hypersurface in (n+1)dimensional Riemannian manifold N n+1 ⊂ Rn+2 . Let en+1 be the unit normal vector field of M n , Qr (r = 0, 1, . . . , (n − 1)) be the r-th modified mean
curvature functions on M n , and X be the position vector of M n . Then
Z X
( (r+1)Qr+1,k ek +(Qr+1 Q1 −(r+2)Qr+2 )en+1 −c(r+1)Qr+1 X)dM = 0.
M
k
(2.43)
24
Guo Shun Zi
Remark 2.4 If Qr+1 = const, then
Z
((Qr+1 Q1 − (r + 2)Qr+2 )en+1 − c(r + 1)Qr+1 X)dM = 0.
(2.44)
M
If Qr+1 = 0, then
Z
Qr+2 en+1 dM = 0.
(2.45)
M
Furthermore, if a is a fixed vector of N n+1 (c), it is easy to know
hen+1 , ai,ij = hen+1,ij , ai.
¿From above computing we obtain
P
r hen+1 , ai = − (r + 1)Qr+1,k hek , ai
k
−(Qr+1 Q1 − (r + 2)Qr+2 )hen+1 , ai + c(r + 1)Qr+1 hX, ai.
Combining this equation with the equation (2.34), we get
P
r hen+1 , ai − r+1 hX, ai = − (r + 1)Qr+1,k hek , ai
k
−Qr+1 Q1 hen+1 , ai + cnQr+1 hX, ai.
(2.46)
(2.47)
(2.48)
Corollary 2.2 Let x : M n → N n+1 (c) be an immersion of a compact orientable n-dimensional Riemannian manifold M n as a hypersurface in (n+1)dimensional Riemannian manifolds N n+1 ⊂ Rn+2 . Let en+1 be the unit normal vector field of M n , Qr (r = 0, 1, . . . , (n − 1)) be the r-th modified mean
curvature functions on M n , a be any fixed element of N n+1 , and X be the
position vector of M n . Then
Z X
( (r + 1)Qr+1,k hek , ai + Qr+1 Q1 hen+1 , ai − cnQr+1 hX, ai)dM = 0.
M
k
(2.49)
If N n+p is the Euclidean space Rn+p , for any point p ∈ M n let X > be the
projection of the position vector X on the tangent space Tp M at the point
p, X ⊥ be the projection of the position vector X on the normal space Tp⊥ M
at the point p, so
X > = hX > , ei iei = u,i ei ,
u,i : = hX, ei i,
(2.50)
X ⊥ = hX, eα ieα = pα eα ,
pα : = hX, eα i.
We are going to compute r X > .
¿From (2.14) we denote the covariant derivative of u,i by
u,ij ωj = du,i + u,j ωj .
(2.51)
Operator r on a submanifold of Riemannian manifold and its applications
25
Taking (2.28) and c = 0 into the above equation, we obtain by a direct
calculation
u,ij = hαij pα + δij .
(2.52)
So we have
r X > =
P
T(r)ij u,ij
i,j
(2.53)
= TraceT(r) (B) + Trace(T(r) hα )pα
= (n − r)Qr + (r + 1)hQr+1 , Xi.
Using Theorem 2.1 we get the following theorem
Theorem 2.4 (Reilly [16]). Let x : M n → Rn+p be an immersion of a
compact orientable n-dimensional Riemannian manifold M n as a submanifold in the (n + p)-dimensional Euclidean space Rn+p , eα (α = 1, . . . , p) be
the unit normal vector field of M n , σr (r = 0, 1, . . . , (n − 1)) be the mean
curvature on M n , and X be the position vector of M n .
i) If p > 1 and r is an even integer, then
Z
(hσr+1 , Xi + σr )dM = 0.
(2.54)
M
ii) If p = 1 and r is any integer, then
Z
(hσr+1 en+1 , Xi + σr )dM = 0.
(2.55)
M
Remark 2.5 The theorem is Lemma A in Reilly, [16]. Here ii) is the one
of classical Minkowski-Hsiung integral formulas, [8, 10].
3
Related results to r
Let ei (i = 1, . . . , n) be a local orthonormal frame field on an n-dimensional
Riemannian manifold M n , ωi be its dual coframe field. The structure
equaP
tions of M are equations (2.6) and (2.7) in Section 2. Let ϕ = ϕij ωi ⊗ ωj
ij
be a symmetric tensor defined on M n , T(r) the r-th Newton transformation
of ϕ, Qr the r-th modified curvature,
X
r Qr :=
T(r)ij Qr,ij .
(3.1)
Then
r Qr =
X
T(r)ij Qr,ij
X
X
=
(Qr δij −
T(r−1)il ϕlj )Qr,ij .
l
26
Guo Shun Zi
We suppose that C(r)ij = T(r)il ϕlj . It is clear that C(r) is a symmetric tensor
here and rQr = Trace C(r−1) , hence we have
r Qr =
1X 1
( trC(r−1) δij − C(r−1)ij )(trC(r−1) ),ij .
r
r
(3.2)
The convariant derivative of C(r)ij is defined by the following formula
X
C(r)ij,k ωk = dC(r)ij +
k
X
C(r)kj ωki +
X
k
C(r)ik ωkj ,
(3.3)
k
and the convariant derivative of C(r)ij,k is defined by
X
C(r)ij,kl ωl = dC(r)ij,k +
X
C(r)mj,k ωmi +
m
l
X
C(r)im,k ωmj +
m
X
C(r)ij,m ωmk .
m
(3.4)
Taking exterior differentiation of the equation (3.3), we obtain
X
X
X
C(r)ij,kl ωl ∧ ωk =
C(r)mj Φmi +
C(r)im Φmj .
m
lk
Therefore
X
X
X
C(r)im Φmj ).
(C(r)mj Φmi +
(C(r)ij,kl − C(r)ij,lk )ωl ∧ ωk = 2
Taking (2.7) into (3.6), we have the Ricci identity
X
X
C(r)im Rmjkl .
C(r)mj Rmikl +
C(r)ij,kl − C(r)ij,lk =
m
(3.6)
m
m
lk
(3.5)
m
(3.7)
m
Let C(r−1) be a Codazzi tensor, which satisfies
C(r−1)ij,k − C(r−1)ik,j = 0.
(3.8)
¿From the results of Theorem 2.1, it is easy to obtain the following lemma:
Lemma 3.1 Let M n be a compact orientable n-dimensional Riemannian
manifold, ϕ be a symmetric tensor on M, C(r) = T(r) ϕ. If C(r−1) is a Codazzi
tensor, then for (0 ≤ r ≤ n),
Z
r Qr dM = 0.
M
P
C(r−1)ij,kk being defined as the Laplacian of C(r−1)ij . Using equation (3.7),
k
and following the methods of Calabi, Simons, Chern, Cheng-Yau, [1, 2, 18,
Operator r on a submanifold of Riemannian manifold and its applications
27
3, 6, 18], one can compute the Laplacian of the tensor,
P
C(r−1)ij,kk
∆C(r−1)ij =
k
P
P
=
(C(r−1)ij,kk − C(r−1)ik,jk ) + (C(r−1)ik,jk − C(r−1)ik,kj )
kP
kP
+ (C(r−1)ik,kj − C(r−1)kk,ij ) + C(r−1)kk,ij
k
Pk
P
P
=
C(r−1)mk Rmijk +
C(r−1)im Rmkjk + (C(r−1)ij,kk − C(r−1)ik,jk )
m,kP
m,k
k
P
+ (C(r−1)ik,kj − C(r−1)kk,ij ) + C(r−1)kk,ij .
k
k
(3.9)
Using (3.8), we obtain
X
X
∆C(r−1)ij =
C(r−1)mk Rmijk +
C(r−1)im Rmkjk + (trC(r−1) ),ij . (3.10)
m,k
m,k
Let us set |C(r−1) |2 =
P
i,j
2
, |∇C(r−1) |2 =
C(r−1)ij
P
i,j,k
2
. Making use of
C(r−1)ij,k
the equation (3.10), we obtain
P
1
∆|C(r−1) |2 = |∇C(r−1) |2 +
C(r−1)ij C(r−1)mk Rmijk
2
i,j,m,k
P
P
C(r−1)ij C(r−1)im Rmkjk + C(r−1)ij (trC(r−1) ),ij .
+
i,j,m,k
i,j
(3.11)
Near a point p ∈ M n we choose an orthonormal frame fields ei (i = 1, . . . , n)
such that C(r−1)ij = C(r−1)ii δij at p. Then (3.11) is simplified to
1
∇|C(r−1) |2
2
P
= |∇C(r−1) |2 + 12 Rijij (C(r−1)ii − C(r − 1)jj )2
i,j
P
+ Cii (trC(r−1) ),ii .
(3.12)
i
From equations (3.2) and (3.12), we have
1X 1
( trC(r−1) δij − C(r−1)ij )(trC(r−1) ),ij
r
r
1 1
1
1
=
( ∆|trC(r−1) |2 − |∇trC(r−1) |2 − ∆|C(r−1) |2
r 2r
r
2
1X
2
+|∇C(r−1) | +
Rijij (C(r−1)ii − C(r−1)jj )2 ).
2 i,j
r Qr =
(3.13)
¿From Lemma 3.1 we obtain immediately the following theorem :
Theorem 3.1 Let M n be a compact orientable n-dimensional Riemannian
manifold, ϕ be a symmetric tensor. For (1 ≤ r ≤ n) set C(r−1) = T(r−1) ϕ.
Near a point P ∈ M we choose orthomormal frame fields {ei }(i = 0, 1, . . . , n)
28
Guo Shun Zi
such that C(r−1)ij = C(r−1)ii δij . Suppose C(r−1) is a Codazzi tensor. Then
Z
1X
1
Rijij (C(r−1)ii − C(r−1)jj )2 )dM = 0.
(|∇C(r−1) |2 − |∇trC(r−1) |2 +
r
2
M
i,j
If M n ⊂ N n+1 (c) and r = 1, the operator derived by Cheng-Yau (see [6])
identifies our operator
X
1 f =
((T rϕ)δij − ϕij )f,ij .
i,j
Supposing M a compact hypersurface, the integral formula in the Theorem
3.1 proves to be
Z
1X
Rijij (ϕii − ϕjj )2 )dM = 0.
(3.14)
(|∇ϕ|2 − |∇trϕ|2 +
2 i,j
M
Now we set ϕij = hij . When M is of nonnegative sectional curvature, it is
obvious from the above integral formula that
|∇B|2 − n2 |∇H|2 ≥ 0,
where H =
1
n
P
(3.15)
hii . When H is constant, the condition above is naturally
i
true. A lot of works have been done for this case, see [1], [14], [18] and
[19]. If R − C = const ≥ 0 where R is the normalized scalar curvature, the
condition is also true. Cheng-Yau ([6]), Yau ([22]), Li ([12], and [13]) have
discussed the geometric meaning of the case.
Let (Rij ) be the matrix of the Ricci curvature tensor on M, r be the scalar
curvature,
X
X
Rkk .
(3.16)
Rkikj , r :=
Rij :=
k
Schouten tensor S =
P
k
Sij ωi ⊗ ωj , where
ij
Sij := Rij −
1
rδij .
2(n − 1)
(3.17)
It is well known that Schouten tensor is a Codazzi tensor on a local conformal
symmetric space. In this situation we set ϕij = Sij , and then the integral
formula (3.14) exists. The geometric meaning of the case is discussed in [9].
When r = 2,
2 Q2 =
X
i,j
(T rC(1) δij − C(1)ij )(
T rC(1)
),ij .
2
Operator r on a submanifold of Riemannian manifold and its applications
29
Suppose M n ⊂ N n+1 (c) and M n is of harmonic Riemannian curvatures,
however from the definitions of the convariant derivatives of Rij and Rijkl
X
X
X
Rij,k ωk := dRij +
Rik ωkj +
Rkj ωki ,
(3.18)
P
P
P
Rijkl,m ωm : = dRP
RmjklP
ωmi + Rimkl ωmj ,
ijkl +
(3.19)
+ Rijml ωmk + Rijkm ωml .
Taking exterior differention of equation (2.7) we obtain the following Bianchi
identity
Rijkl,m + Rijlm,k + Rijmk,l = 0.
(3.20)
Combining (2.3) (2.4) with (3.19) we obtain
P
Rij,k − Rik,j =
Rlijk,l
↓ (Riemanian curvature is harmonic)
= 0.
(3.21)
We set ϕij = hij . By
Rij = C(1)ij + (n − 1)cδij ,
we know that C(1) is a Codazzi tensor. So the integral formula of Theorem
3.1 proves to be
!
Z
X
X
X
1
1
2
2
r,k
+
Rijij (Rii − Rjj )2 dM = 0.
(3.22)
Rij,k
−
2
2
M
i,j
k
i,j,k
However
X
X
Rij,j =
j
Rikjk,j
k,j
X
=
Rjkik,j = 0.
k,j
So we have
r,i =
X
Rjj,i
j
=
X
Rij,j = 0.
j
Hence the integral formula (3.22) is
Z X
1X
2
(
Rij,k
+
Rijij (Rii − Rjj )2 )dM = 0.
2
M i,j,k
i,j
(3.23)
Xia in [20] discussed the geometric meaning of this situation.
Corollary 3.1 Let M n be a compact Riemannian manifold with harmonic
curvature tensor and nonnegative sectional curvature. If M n can be immersed into S n+1 as a hypersurface, then M n is isometric with either
S k (a) × S n−k (b)(a2 + b2 = 1) or S n .
30
Guo Shun Zi
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Guo Shun Zi
School of Mathematics of Sichuan University
Chengdu, 610065, People’s Republic of China,
School of Mathematics and Statistics of Minnan Normal University,
Zhangzhou 363000, People’s Republic of China
guoshunzi@yeah.net
Please, cite to this paper as published in
Armen. J. Math., V. 7, N. 1(2015), pp. 6–31
31