1. Introduction
Let
be an
n-dimensional Riemannian manifold with the Levi–Civita connection ∇. If
H is the vector bundle over
of all covariant tensors of rank 3 satisfying only the identities of the covariant derivative
of the Ricci tensor
, then
H decomposes into the pointwise orthogonal sum
of its three subbundles,
, and
(see [
1,
2] (pp. 432–433). Moreover
, and
are pointwise irreducible under the action of the orthogonal group
. Therefore, the Riemannian manifold is called an
an Einstein-like manifold of type (respectively,
or
) if
(respectively,
or
). In the first case
is called the
Killing–Ricci tensor (see [
3]), while in the second case
is called the
Codazzi–Ricci tensor (see [
4]) and in the third case
is called the
Sinyukov–Ricci tensor (see [
5,
6,
7]).
A discussion of the geometry of the above and other types of Riemannian Einstein-like manifolds can be found in the paper [
1] and the monograph [
2] (pp. 432–455). In turn, the application of such manifolds in general relativity can be found in [
7,
8]. For example, it is well known that the scalar curvature
is constant for an arbitrary Einstein-like manifold
belonging to either class
or
. Moreover, an arbitrary manifold belonging to
∩
must have a parallel Ricci tensor. An example of this type of Einstein-like manifold is a Riemannian
locally symmetric space (see [
9], p. 369).
In turn, we use the
Bourguignon Laplacian (see [
10]) and the
Sampson Laplacian (see [
11,
12,
13]) to study the global geometry of the above three classes of Einstein-like manifolds. Both of these Laplacians admit
Weitzenböck decompositions (see [
2], p. 53). We recall here that a Laplace operator
D permits a
Weitzenböck decomposition if
, where
ℜ is the Weitzenböck curvature operator, which depends linearly on the curvature
R and the Ricci tensors of
and
is the
Bochner Laplacian (see [
2], pp. 52–53). Weitzenböck decompositions are important because (see [
2], p. 53) there is a method, per Bochner [
14], of proving vanishing theorems for null space of a Laplace operator that allows a Weizenböck decomposition.
For example, several fundamental results formulated in various theorems are based on the Bochner technique, which usually shows that the assumption of positive or negative sectional curvatures of compact Riemannian manifolds yields the vanishing of certain geometric and topological invariants (such as the Betti numbers), geometrically interesting tensor fields (such as the Killing–Yano tensors), and mappings (such as harmonic mappings); see for example [
9] (pp. 333–364), [
15,
16,
17].
However, we have already entered the era of geometric analysis and its applications in studying geometric and topological properties of complete Riemannian manifolds (see for example [
18]). Therefore, in this article we discuss the global geometry of Einstein-like complete manifolds using a generalized version of the Bochner technique (see for example [
19]). Furthermore, in the three sections below we demonstrate the application of various methods of the generalized Bochner technique (see for example [
20]) to the study of the above-mentioned three classes of complete Einstein-like manifolds. In particular, the results obtained in our paper generalize well-known results on Einstein-like compact manifolds to the case of Einstein-like complete non-compact manifolds.
2. -Spaces and the Sampson Laplacian
A Riemannian manifold
is said to be Einstein-like of type
if its Ricci tensor
is cyclic parallel, that is, if
for all
(see [
21]). In particular, from [
2] (p. 451) it is known that if
is a compact (without boundary) Einstein-like manifolds of type
with nonpositive sectional curvature, then
. If, in addition, there exists a point in
M where the sectional curvature of every two-plane is strictly negative, then
is Einsteinian, i.e., its Ricci tensor satisfies
for some constant
(see [
2], p. 451).
On the other hand, Deng (see [
22]) studied the rigidity of complete
-manifolds and showed that
is an
n-dimensional complete Einstein-like manifold of type
with a
Yamabe constant and nonpositive scalar curvature, and is an Einstein manifold if there exists a small number
C depending on the dimension
n and
such that
for the Weyl curvature tensor,
W. In turn, Chu modernized this result in his article [
23]. We remark here that the above results were obtained using the methods of the classical Bochner technique.
This section studies complete Einstein-like manifolds of type
with nonpositive sectional curvature. As the starting point in the study of Riemannian manifolds of nonpositive curvature, we first recall the following well known Cartan–Hadamard theorem:
Let be an n-dimensional simply connected complete Riemannian manifold of nonpositive curvature; then, is diffeomorphic to the n-dimensional Euclidean space . Therefore, a simply connected complete Riemannian manifold of nonpositive curvature is called a
Hadamard manifold or a
Cartan–Hadamard manifold, after the Cartan–Hadamard theorem (see for example [
9], p. 241; [
18], pp. 391–381).
Remark 1. From the Cartan–Hadamard theorem, one can conclude, in particular, that no compact simply connected manifold admits a metric of nonpositive curvature (see [9], p. 162). Therefore, compact Hadamard manifolds do not exist. Here, we recall that the function is subharmonic if , where is the Beltrami Laplacian on functions. Then, we can formulate the following proposition.
Lemma 1. On a Hadamard manifold any non-negative subharmonic function such that for is equal to zero.
Proof. The following theorem holds (see [
24]): on a complete simply connected Riemannian manifold
of nonpositive sectional curvature, every nonnegative subharmonic function
for
is a constant
C. In this case, we have
. We observe that Hadamard manifolds have infinite volume (see [
25]); hence, from the last inequality, we obtain
. This finishes the proof of our Lemma 1. □
In turn, we introduce the Sampson Laplacian into consideration and consider several of its properties. In order to do this, we define the differential operator
as follows:
of degree 1 such that
, where Sym:
is the linear algebraic operator of symmetrization. This means that
is a symmetrized covariant derivative defined by the following equation (see [
2], pp. 355–356):
for any
, and
. Then, there exists its formal adjoint operator
, which is called the
divergence operator (see [
2], p. 356). Notice that
is nothing other than the
restriction of
to
(see [
2], p. 35).
Using the operators
and
, we can define the second order differential operator
, by the formula
. At the same time, if
, then the tensor field
is called a
-harmonic symmetric tensor, as it is an analog of the harmonic forms of the Hodge–de Rham theory (see [
9], p. 335).
The Weitzenböck decomposition formula for the Sampson Laplacian
has the form
The second component of the right-hand side of (
1) is called the Weitzenböck curvature operator of the Sampson Laplacian
.
Next, from (
1) by direct calculation, we obtain the
Bochner–Weitzenböck formula,
for
. If
, then for any point
there exists an orthonormal eigenframe
of
such that
for the Kronecker delta,
. In this case (see [
2], p. 436, [
26], p. 388), we have
where
is the
sectional curvature in the direction of the two-plane
of
at an arbitrary point
. In this case, the Formula (
2) can be rewritten in the form
We can now prove the following statement.
Theorem 1. Let be a Hadamard manifold and φ a -harmonic symmetric 2-tensor on such that for at least one ; then, .
Proof. Let
be a non-zero
-harmonic symmetric 2-tensor on a Riemannian manifold with non-positive sectional curvature; then, from Formula (
4), we obtain
On the other hand, we have
. Then, from the last formula and (
5), we obtain
where
due to the
Kato inequality .
Moreover, if we suppose that
is a manifold with nonpositive sectional curvature, then from (
6) we obtain the inequality
. It is known from [
27] that if the inequality
holds for a nonnegative function
defined on a complete Riemannian manifold, then either
for all
or
f = const. In particular,
q may even be less than one here (see [
27]). In this case, the inequality
becomes
. At the same time, we know that a Hadamard manifold
has an infinite volume; hence, from the last inequality we obtain
. This completes our proof. □
On the other hand, let
be the bundle of traceless symmetric
p-tensors on
. Then, the fact that sec
implies the negative semi-definiteness of the quadratic
for any
, while
is proven in [
28]. For this case, the following proposition holds:
Theorem 2. Let be a Hadamard manifold and φ a -harmonic traceless symmetric p-tensor on such that for at least one ; then, .
Let be a Riemannian Einstein-like manifold of type . Then, its Ricci tensor, , satisfies the equations and has a constant trace, i.e., the scalar curvature is a constant function. This means that . Therefore, is a -harmonic symmetric 2-tensor. Then, we can formulate the following lemma.
Lemma 2. A Killing–Ricci tensor is a -harmonic symmetric 2-tensor.
In this case, the following proposition is an immediate consequence of Lemma 2 and Theorem 1.
Corollary 1. Let an n-dimensional Riemannian Einstein-like manifold of type be a Hadamard manifold. If for at least one , then is isometric to .
Proof. We know that a Killing–Ricci tensor of
is a
-harmonic symmetric 2-tensor. Moreover, if
is a Hadamard manifold and
for at least one
, then
per Theorem 2. Next, we need to prove one obvious statement. If the sectional curvature is non-positive and the Ricci curvature is zero, then the Riemannian manifold is flat; that is, let
be a unit vector. We can complete it on an orthonormal basis,
, for
at an arbitrary point
; then, (see [
9], p. 86):
In this case, from the conditions and we obtain , i.e., the sectional curvature vanishes identically. In this case, is a flat Riemannian manifold. If is simply connected, it follows that is isometric to the Euclidean space . □
We now consider a three-dimensional Riemannian Einstein-like manifold of type . In this case, the following corollary holds.
Corollary 2. Let be a three-dimensional simply connected Riemannian Einstein-like manifold of type . If its Ricci tensor satisfies the conditions and for at least one , then is isometric to .
Proof. In dimensions up to three, Einstein-like manifolds of type
have been classified in [
29]; in particular, they are homogeneous (see [
30]). Therefore, a three-dimensional Einstein-like manifold of type
is complete, because any homogeneous Riemannian manifold is complete (see [
2], p. 181). Moreover, it is well known (see [
9], p. 86 and [
31]) that
for unit orthogonal vectors
at any point
such that
is orthogonal to the plane
. In this case, the condition
can be rewritten in the form
. Then, from Theorem 1 and Corollary 2, we can conclude that if the Ricci tensor
satisfies the two conditions
and
for at least one
, then
is a Ricci-flat manifold, and hence is a flat manifold due to equality (
7). In this case,
is isometric to the Euclidean space
. □
3. -Spaces and the Bourguignon Laplacian
A Riemannian manifold
is said to have a
harmonic curvaturetensor if
(see [
9], p. 362). This happens if and only if the Ricci tensor
is a Codazzi–Ricci tensor, i.e.,
for any
. This means that
is an Einstein-like manifold belonging to class
. There exist numerous examples of compact Riemannian manifolds with this property (see [
2], pp. 443–447; [
4]). On the other hand, the following classical
Berger–Ebin theorem is well known. If
is a compact (without boundary) Einstein-like manifold of type
with non-negative sectional curvature, then
. If, in addition, there exists a point in
M where the sectional curvature of every two-plane is strictly positive, then
is Einsteinian (see [
2], p. 445). Based on the results obtained above, we can supplement this theorem as follows: a three-dimensional compact Einstein-like manifold of type
with
has constant positive sectional curvature.
In this section, we generalize this result to the case of a complete Riemannian manifold. In order to do this, we use the
Bourguignon Laplacian (see [
10]).
Here, we consider a symmetric tensor
as a one-form with values in the cotangent bundle
on
M. This bundle is equipped with the Levi–Civita covariant derivative ∇; thus, there is an induced exterior differential
on
-valued differential one-forms such as
for any tangent vector fields
on
M and an arbitrary
(see [
2], pp. 133–134, 355; [
9], pp. 349–350; and [
32]). In this case,
is a
Codazzi tensor if and only if
(see [
9], p. 350). The formal adjoint of
is denoted by
(see [
2], p. 134). Moreover, Bourguignon proved in [
33] (see p. 271) that
for an arbitrary Codazzi tensor
. At the same time, he defined a
harmonic symmetric 2-tensor as a tensor
such that
(see [
9], p. 350 and [
33] p. 270). Next, Bourguignon defined the Laplacian
by the formula
(see [
33], p. 273). Then, the symmetric harmonic 2-tensors belong to the kernel of the Bourguignon Laplacian
. The converse is true in the compact case as well; namely, if
is a compact manifold (without boundary), then
denotes the usual Hilbert space of functions or tensors with the global product (with respect to the global norm)
Then, by direct computation, we obtain the following integral formula (see [
10]):
Next, an easy computation yields the
Weitzenböck decomposition formula (see [
2], p. 355 and [
33], p. 273)
The second component of the right-hand side of (
10) is called the Weitzenböck curvature operator of the Bourguignon Laplacian
(see [
2], p. 356).
Based on the last two formulas, we conclude that the Bourguignon Laplacian is a non-negative operator and its kernel is the finite dimensional vector space of harmonic symmetric 2-tensors (or, in other words, Codazzi tensors with constant trace). Therefore, the harmonic symmetric 2-tensor will be called the -harmonic tensor.
Using (
10), the Bochner-Weitzenböck formula (see [
10]) can be obtained:
where
for an arbitrary
and an orthonormal eigenframe
of
such that
at any point
. Let
be a
-harmonic; then, (
11) can be rewritten in the following form:
The following theorem supplements the Berger–Ebin theorem for the case of -harmonic tensors on a complete noncompact Riemannian manifold.
Theorem 3. Let be a connected complete and noncompact Riemannian manifold with nonnegative sectional curvature. Then, there is no non-zero -harmonic tensor such that for any .
Proof. Let
be a connected complete and noncompact Riemannian manifold with nonnegative sectional curvature, and let
be a non-zero
-harmonic symmetric 2-tensor; then,
. Therefore, from (
12), we can obtain the inequality
where
due to the
Kato inequality. Then, we can conclude that
on a connected complete and noncompact Riemannian manifold with nonnegative sectional curvature. For
, then, either
or
(see [
27]). In a case where
has infinite volume, all of the constant functions hold while zero is in
; that is, if the function
for some positive number
q is a constant function,
C, then the inequality
becomes
. If in addition
has an infinite volume, then we can obtain
from the last inequality. It must be recalled that a complete Riemannian manifold of non-negative sectional curvature has an infinite volume (see [
27,
34]). This remark completes the proof. □
We are now able to formulate the following lemma.
Lemma 3. The Codazzi–Ricci tensor is a -harmonic symmetric 2-tensor.
Proof. Let be a Riemannian Einstein-like manifold of type . Then, its Ricci tensor satisfies the equations and has a constant trace, i.e., the scalar curvature is a constant function. This means that ; therefore, is a -harmonic symmetric 2-tensor. □
In this case, the following proposition is an immediate consequence of the above Lemma 3 and Theorem 3.
Corollary 3. Let be a connected complete and noncompact Einstein-like manifold of type with non-negative sectional curvature. If for some , then is flat.
Proof. If
is a connected complete and noncompact manifold with non-negative sectional curvature, and
for some
, then
is Ricci-flat per Lemma 3 and Theorem 3. Next, we need to prove one obvious statement: if the sectional curvature is nonnegative and the Ricci curvature is zero, then the Riemannian manifold is flat. That is, let
be a unit vector which we complete on an orthonormal basis,
, for
at an arbitrary point
; then, (see [
9], p. 86)
In this case, from the conditions and we obtain , which completes the proof. □
In conclusion, we formulate one more obvious corollary.
Corollary 4. Let be a three-dimensional connected complete and noncompact Einstein-like manifold of type . If and for some , then is flat.
Remark 2. In Corollaries 3 and 4, we proved that is a flat Riemannian manifold. If is simply connected, then is isometric to the Euclidian space, .
4. On Compact Einstein-like Manifolds of the Type
A Riemannian manifold
is said to be
Einstein-like of type if its Ricci tensor
satisfies the condition
where
and
for any
. Riemannian manifolds satisfying condition (
13) are called
Sinyukov manifolds in [
5]. Besse defined these Equations (
13), but did not carry out any research for manifolds of class
. The local properties of such manifolds were studied in [
5]. In turn, the purpose of [
35] was the local classification of all three-dimensional Riemannian manifolds belonging to class
. The application of such manifolds in general relativity can be found in [
7,
8].
We now prove a theorem on compact Einstein-like manifolds of type .
Theorem 4. If is a compact (without boundary) Einstein-like manifold of type with non-positive sectional curvature, then . If, in addition, there exists a point in M where the sectional curvature of every two-plane is strictly negative, then is Einsteinian.
Proof. From (
13), we can obtain
for
for any
. Assume that the manifold
M is compact; then, from (
2) we can obtain the integral formula
where, by virtue of (
14), we have
and
for an orthonormal eigenframe
of
such that
at any point
. Therefore, if
is a compact (without boundary) Einstein-like manifolds of type
with non-positive sectional curvature, then from (
16) we obtain
. In this case, from (
13) we obtain
. In addition, if there exists a point in
M where the sectional curvature of every two-plane is strictly negative, then from (
15) and (
16) we can conclude that
is Einsteinian. □
All three-dimensional Riemannian manifolds belonging to class
are known from [
35]. In turn, we formulate a theorem for a four-dimensional compact Sinyukov manifold.
Theorem 5. A four-dimensional compact Sinyukov manifold with positive sectional curvature is diffeomorphic to the sphere or the real projective space.
Proof. Our proof is based on three facts. First, if
is a Sinyukov manifold and dim
M≤ 4, then
is locally conformally flat (see [
5]). We recall here that a locally conformally flat Riemannian manifold
is determined by the condition that any point
has a neighborhood
and a
-function
f on
such that the Riemannian manifold
is flat (see [
2], p. 60). Second, we proved in [
20] that in the case of a locally conformally flat Riemannian manifold of dimension
, the conditions
and
are equivalent for its
curvature operator,
(see [
9], p. 83), and its sectional curvature, sec, respectively. Third, it has been proven (see [
36]) that the
Ricci flow deforms
g to a metric of constant positive curvature, provided that
is compact and has the positive curvature operator
. In this case,
is diffeomorphic to the sphere
or the real projective space
. □
In conclusion, we formulate a theorem supplementing the previous assertion.
Theorem 6. Let be a four-dimensional complete Sinyukov manifold with ; then, belongs to one of the following classes: either flat, or locally isometric to the product of a sphere and a line, which are globally conformally equivalent to either or a spherical space form.
Proof. We know that a four-dimensional Sinyukov manifold
is locally conformally flat (see [
5]). Moreover, the main theorem of [
37] states that complete locally conformally flat manifolds of dimension
with Ricci tensor
belong to one of the following classes: either flat, or locally isometric to the product of a sphere and a line, and are globally conformally equivalent to either
or a spherical space form (see [
38], p. 69). □