1. Introduction
An identity map
from a differentiable manifold
M into itself, also known as an identity transformation, is a map that always returns to the same point that was used in as its argument. In other words, the identity map
on a manifold
M is defined to be the map with domain and range
M which satisfies
Obviously, the identity map is the simplest map which is both continuous and bijective.
The main purpose of studying the differential geometry of identity maps is to investigate the following problem:
“To what extent does the identity map of a manifold determine the geometry of the manifold?”
In general, to study the differential geometry of an identity map
, we shall assume that the domain
M and the range
M admit two geometric structures of the same kind. In [
1], Nagano proved that the identity map
of a compact Riemannian manifold
onto itself is a stable map if and only if the diagonal map in
is stable as a minimal submanifold. Further, the author and Nagano proved in [
2] that a Riemannian manifold
admits a geodesic vector field
v if and only if the identity map
is a harmonic map, where
denotes the Lie derivative with respect to
v. For two linear connections
and
on a manifold
M, the identity map
is called affine if the identity map
carries every parallel vector field along each curve
of
into a parallel vector field along
in
. It is known from ([
3] page 225) that the identity map
is affine if and only if it carries each geodesic of
into a geodesic of
.
To study the conformal geometry of the identity map on a manifold
M, we assume that domain
M and range
M equipped with metrics
g and
, respectively. The identity map
is called a conformal change of metric if
for some positive function
. It is well-known that, in the case
, if the identity map
is a conformal change of metric, then
and
have the same Weyl tensor (cf. [
4]); and in the case
,
and
have the same Cotton tensor (cf. e.g., [
5,
6]).
The main purpose of this paper is to provide a survey on known results about identity maps from various Riemannian geometric points of view.
This article is organized as follows. The basic materials on pseudo-Riemannian manifolds is given in the second section.
Section 3 provides a brief introduction on harmonic maps including the first and the second variations of Dirichlet energy functionals as well as the stability, index and nullities of harmonic maps. In
Section 4, we discuss the relationship between identity maps and (relative) harmonic metrics, (relative) harmonic tensor and geodesic vector fields. We also present a volume-decreasing phenomenon for co-closed harmonic metrics. We mention a necessary and sufficient conditions for the warped product metric of two Einstein manifolds to be harmonic as well as identity maps on Kaehler and hyper-Kaehler manifolds. In
Section 5, we present the result for a Walker metric (with neutral signature) on a Walker 4-manifold
to be harmonic with respect to another Walker metric. In
Section 6, we discuss index, relative nullity and Killing nullity of identity maps for several important Riemannian manifolds. In addition, results on stabilities and nullities of compact symmetric spaces are presented in this section. In
Section 7, we discuss identity maps on compact symmetric spaces. In
Section 8, we survey results on the stability of some other important spaces; including homogeneous spaces, spheres, flat tori, and their products, as well as the stability of generalized Sasakian space forms. In
Section 9, we present some known results on biharmonic identity maps; including biharmonic metrics, obstructions to the existence of biharmonic metrics on Einstein manifolds, and biharmonicity of Riemannian submersions. In
Section 10, we provide the relationship between Gauss maps and identity maps, mainly for surfaces. In
Section 11, we present results on Laplace maps and the related identity maps proved in the book [
7]. In the last section, we present results on the identity maps of tangent bundles of manifolds equipped with Sasakian, lift-complete, or
g-natural metrics. In addition, we discuss the identity maps on the tangent bundles of Walker 4-manifolds and of Gödel-type spacetimes.
2. Pseudo-Riemannian Manifolds and Submanifolds
We follow the notations given in [
6,
8,
9,
10].
2.1. Basics on Pseudo-Riemannian Manifolds
Consider a smooth
n-manifold
M covered by a system of coordinate neighborhoods
, where
U is a neighborhood and the
denote local coordinates on
U, where the indices
take on values in the range
. Then a Riemannian (or pseudo-Riemannian) metric
g on
M can be expressed as
where we have applied the Einstein convention, i.e., repeated indices, with one upper index and one lower index, denoted summation over its range.
In the following, we denote by
the basis vectors on the coordinate neighborhood
. Let
X and
Y be two vector fields on M. We then have
where
and
are the local components of the vector fields
X and
Y, respectively. With respect to the natural frame
, we have
which are the local components of the metric tensor field
g. The Christoffel symbols of the Riemannian manifold
are given by
where
denotes the inverse matrix of
. Thus, we have
, where
or 0 according to
or
.
Let ∇ denote the Levi–Civita connection of
. Then for vector fields
on
M the covariant derivative
has local components
The covariant derivative of a tensor
of type (0,2) is defined by
Let
and
Z be three vector fields in
M. Then
defines a tensor field
K of type
. If we put
then
is a tensor field of type (1, 1) which is linear both in
X and
Y. In terms of local components, (
6) can be written as
where
are the local components of the curvature tensor
K.
The Ricci tensor
of
is the
-tensor with local components given by
The scalar curvature
r is given by
Let
denote the space of all tensors of type
on
. Following [
2], we denote by
the codifferential of
defined by
Then
. As in [
2], a tensor
T is called co-closed if it satisfies
.
2.2. Bianchi Identities
The curvature tensor
K satisfies the first Bianchi identity
or in local components
The curvature tensor
K also satisfies the second Bianchi identity
or in local components
2.3. Gradient, Divergence and Laplacian
Assume
is a pseudo-Riemannian manifold and
f is a smooth function on
M. Then the gradient of
f, denote by
(or by grad
), is the vector field dual to the differential
. In other word,
is defined by
In terms of a local coordinate system
of
M, the divergence of a vector field
, denoted by div
, is given by
The Laplacian of
f, denoted by
, is defined by
, we have
2.4. Basics on Submanifolds
Assume that
is an isometric immersion of a pseudo-Riemannian manifold into another. Denote by ∇ and
the Levi Civita connections on
M and
N, respectively. The formulas of Gauss and Weingarten are given respectively by
for tangent vector fields
and a normal vector field
, where
B,
A and
D are the second fundamental form, the shape operator and the normal connection.
For each , the shape operator is a symmetric endomorphism of the tangent space at .
The shape operator and the second fundamental form are related by
for
tangent to
M and
normal to
M. The mean curvature vector is defined by
.
For a vector
,
, we denote by
and
the tangential and the normal components of
, respectively. The equations of Gauss, Codazzi and Ricci are given respectively by
for vector fields
tangent to
M,
normal to
M, and
is given by
and
is the curvature tensor associated to the normal connection
D.
The submanifold
M is called totally geodesic if
holds identically; and parallel if
identically. The submanifold is said to have parallel mean curvature vector if
identically (cf. [
11] for a detailed survey on submanifolds with parallel mean curvature vector).
2.5. Submanifolds of Finite Type
The theory of finite type submanifolds was initiated in [
12,
13,
14]. The first results on finite type submanifolds were collected in the book [
15]. Since that time, the subject has developed rapidly. A detailed survey on results up to 1996 was given in author’s report [
16]. The most recent comprehensive survey on this subject was given in author’s book [
9] (see also [
17,
18,
19]).
A submanifold
M of
is said to be of finite type if its immersion
is a finite sum of eigenmaps of the Laplacian
, i.e.,
where
is a constant map and
are non-constant maps satisfying
If the eigenvalues
of
in (
27) are distinct, then
M (or the immersion
) is said to be
k-type. In particular, if one of
is zero, then
M is said to be of null
k-type.
Analogously, a smooth map of a Riemannian manifold M is called a finite type map if is a finite sum of -valued eigenmaps of . In the same way, we also have the notion of k-type maps and null k-type maps. An isometric immersion (or a map) of a Riemannian manifold into is said to be of infinite type if it is not of finite type.
For each
in the spectral decomposition (
27) of
, we put
Then each is a linear subspace of .
2.6. Linearly Independent and Orthogonal Immersions
Consider a
k-type isometric immersion
whose spectral decomposition is given by (
26). Then
is said to be linearly independent if the subspaces
defined by (
28) are linearly independent, i.e., the dimension of the subspace spanned by all vectors in
is equal to
. The immersion
is said to be orthogonal if the subspaces
are mutually orthogonal in
(see [
20,
21]).
If we choose a Euclidean coordinate system
on
with
c as its origin, then the spectral decomposition (
26) of
reduces to
For each , we choose a basis of , where denotes the dimension of . Put and let denote the subspace of spanned by . If the immersion is linearly independent, then are ℓ linearly independent vectors in . Further, we may choose the Euclidean coordinate system on in such way that is defined by
We regard each
as a column
ℓ-vector and put
then
S is a nonsingular
-matrix. Let
D denote the diagonal
-matrix
where
repeats
-times for
. If we put
, then we get
for any
and
. Thus, we obtain
for the immersion
induced from
. We may regard the
-matrix
A as an
-matrix in a natural way (with zeros for each of the additional entries). For this
-matrix
A, we also have
By using the
-matrix
A in (
30), the author introduced in [
20] the notion of adjoint hyperquadric as follows.
Let
be a non-minimal, linearly independent immersion whose spectral decomposition is given by (
26). Let
be a Euclidean coordinate system on
with
c as its origin. Let
A be the
-matrix given in (
30). Then, for a given point
,
defines a hyperquadric
in
, called the adjoint hyperquadric of
at
p. In particular, if
lies in an adjoint hyperquadric
of
for some point
, then all of the adjoint hyperquadrics
give the same adjoint hyperquadric which is denoted by
Q, called the adjoint hyperquadric of the linearly independent immersion
.
3. Harmonic Maps
Let
be a map between Riemannian manifolds. We use
to denote the pull-back bundle of the tangent bundle
by
(or the induced bundle of
). The pull-back bundle has base
M and the fiber
over
. The sections of
are called vector fields along
. The differential
of
can be regarded as a section of
Let
denote the pull-back connection on
defined by
where
. Denote by
the space of all sections of
.
The second fundamental form of a map is given by the covariant differential . A map is called totally geodesic if its second fundamental form vanishes identically.
The energy density
of
is a non-negative function given by
where
denotes the differential of
,
is the Hilbert–Schmidt norm of
, and “tr” denotes the trace. The Dirichlet energy of
over any compact domain
D of
M is defined by
A variation of a map is a smooth map such that . A variation is said to be supported in a domain D if on for all , where denotes the interior of D. Note that defines a vector field on the pull back bundle and it is called the vector field of the variation .
It was well-known that for a smooth variation
of
supported in
D, we have the first variation (cf. [
22])
where
is the variation vector field, and
is the tension field of
given by
3.1. Harmonic Map and Tension Field
Definition 1. [
22]
A map between two Riemannian manifolds is called a harmonic map if it is a critical point of the energy functional defined by (32), where D is any compact domain of M. It follows form Definition 1 that a harmonic map
is a solution of the corresponding Euler-Lagrange equation for the Dirichlet energy functional. Thus, a map
is harmonic if and only if its tension field
vanishes identically [
22].
For a given map
, let
and
be the local coordinates on
M, and
N, respectively. Then
will be the local expression of the map
. Let
and
be the natural frames of the local coordinates on
M and
N respectively. For simplicity, we put
and
. Then, in terms of local coordinates, we have
where
and
denote the Christoffel symbols of
and
with respect to the local coordinate systems
and
, respectively.
Remark 1. Definition 1 makes sense for harmonic maps between pseudo-Riemannian manifolds as well.
3.2. Second Variation of Energy
Assume that
is a harmonic map from a compact manifold
M into
. Let
be a smooth two parameter variation of
.
Let
,
be the variation vector fields. Then the Hessian,
, of the energy at
is defined by
Then we have
Theorem 1. [
23]
If is a harmonic map from a compact manifold. Then the Hessian of the energy at ϕ is given byfor vector fields along ϕ, where denotes the curvature tensor of and the Jacobi operator is given by 3.3. Stability, Index and Nullity
A vector field v along satisfying is called a Jacobi field. The index of , denoted by , is the dimension of the largest subspace of on which is negative definite.
A harmonic map is called stable if then index . Geometrically, a harmonic map is stable if its energy integral is a local minimum of for any variation with . Otherwise, it is called unstable.
The nullity of , denoted by is the dimension of the kernel of . It follows from the spectral properties of elliptic operators on compact manifolds that both and are finite.
For a Riemannian manifold
, let
denote the space of all infinitesimal isometries on
. Then the Killing nullity of a harmonic map
is defined by
where
is considered as variation vector fields along
. The reduced nullity of
, denoted by
, is defined by (see [
23])
The following two results on stable harmonic maps are well-known.
Theorem 2. [
24]
For , there exists no non-constant stable harmonic map from any compact Riemannian manifold to . Theorem 3. [
25]
For , there exists no non-constant stable harmonic map from to any Riemannian manifold. 4. Identity Maps, Harmonic Metrics and Harmonic Tensors
4.1. Relative Harmonic Metrics
Let
be an immersion and
be the Riemannian metric on
M induced via
. Let
be the associated isometric immersion, and
be the identity map. Then the tension field of the composition
of maps
and
is given by
where “
” is the trace with respect to
g and
B is the second fundamental form of the associated isometric immersion
. Thus, it follows from (
40) that the composition
is a harmonic map if and only if the identity map
is harmonic and
.
Partly motivated by this fact, the author and Nagano introduced in 1984 the following notion of (relative) harmonic metrics on a Riemannian manifold.
Definition 2. [
2]
A metric on a Riemannian manifold is called harmonic with respect to g if the identity map is a harmonic map. Let g and be metrics on an n-manifold M. Denote by , the Levi–Civita connection, Christoffel symbols, of ; and by , the corresponding quantities of , respectively.
From (
35) we find the following.
Lemma 1. A metric on M is harmonic with respect to a metric g on M if and only if For the metric tensor
, if we put
then we have
Then
is a tensor of type
satisfying
. Moreover, it follows from the definition of Christoffel symbols that
Now, we may state the following useful result from [
2] which provides an easy way to determine whether a metric
is harmonic with respect to a given one
g on a manifold
M.
Proposition 1. Let g and be two Riemannian metrics on a manifold M. Then is harmonic with respect to g if and only if we havewhere and . 4.2. Space of Harmonic Tensors
Let
denote the space of all symmetric covariant tensor fields of degree 2 on a Riemannian manifold
. If we regard a tensor
as an energy–momentum tensor, the equation
is known in [
26] as the conservation law of energy momentum.
Let
denote the space of all Riemannian metrics on
. We put
Then
is a star set centered at
g in the space
(see [
2]). The following notion of (relative) harmonic tensors was introduced in [
2].
Definition 3. Let be a Riemannian manifold. A tensor on is called a harmonic tensor with respect to g if it satisfies (46). The next two theorems describe the space of all harmonic tensors with respect to g.
Theorem 4. [
2]
Let be a Riemannian manifold of dimension . Then we have the following linear isomorphism of vector spaces:This isomorphism is given by .
Theorem 5. [
2]
Let be a Riemannian 2-manifold. Thenwhere is the space of all smooth functions on M. Remark 2. Theorems 4 and 5 imply that the space of all harmonic tensors with respect to g is infinite-dimensional.
The following theorem follows from Proposition 1 and the second Bianchi identity (see [
2] page 399).
Theorem 6. Let be Riemannian manifold. Then
- (a)
The Ricci tensor of is a harmonic tensor with respect to g.
- (b)
The Ricci tensor of is co-closed if and only if has a constant scalar curvature.
In particular, Theorem 6 implies the following.
Corollary 1. Let be a Riemannian manifold whose Ricci tensor is definite. Then the identity map is harmonic.
For 2-dimensional manifolds, we also have the following.
Theorem 7. [
2]
If is an orientable Riemannian 2-manifold, then is linearly isomorphic with the space of holomorphic quadratic differentials on M with the natural complex structure. From Theorem 7 and Riemann–Roch’s theorem, we have
Corollary 2. If M is a 2-sphere or a real projective plane, then we have
Remark 3. One of the reviewers of this article pointed out that there is a discussion of the harmonicity of the identity map and its relation to DeTurck’s method for breaking gauge invariance by Graham and Lee [27] published in 1991. 4.3. Links between Geodesic Vector Fields and Harmonic Tensors
A Killing vector field on an orientable compact Riemannian manifold is characterized by the following two conditions (see [
28]):
- (1)
and
- (2)
divergence .
On the other hand, the notion of geodesic vector fields was introduced by Yano and Nagano in [
29] as follows.
Definition 4. A vector field on a Riemannian manifold is called a geodesic vector field if it satisfies .
The next theorem provides a simple link between geodesic vector fields and harmonic metrics.
Theorem 8. [
2]
A vector field v on a Riemannian manifold is a geodesic vector field if and only if is a harmonic tensor with respect to g, where denotes the Lie derivative with respect to v. Remark 4. Analogous to Definition 2, a pseudo-Riemannian metric on a manifold M is said to be harmonic with respect to another pseudo-Riemannian metric g on M if the identity map is harmonic.
It was shown in [30] that the results given in Section 4.2 and Section 4.3 hold true in pseudo-Riemannian setting. Some further results on harmonic metrics in pseudo-Riemannian setting can also be found in [31,32] by Bejan and Duggal. 4.4. A Volume-Decreasing Phenomenon
If is a co-closed harmonic metric with respect to a given metric g on a Riemannian manifold M, then we have
- (i)
is constant and
- (ii)
for any positive number c, is also a co-closed harmonic metric with respect to g.
For co-closed harmonic metrics, we have the following volume-decreasing phenomenon.
Theorem 9. [
2]
Let is Riemannian manifold. If is a co-closed harmonic metric w.r.t. g on M satisfying , then- (1)
The volume form at each point of M; hence the identity map is volume-decreasing.
- (2)
on M if and only if on M.
4.5. Identity Maps on Warped Products
Let and be Riemannian manifolds and f a positive function on B. Then the product manifold equipped with the warped product metric is called a warped product manifold with f as its warping function. We denote this warped product manifold by .
Pyo, Kim and Park proved the following result in [
33].
Theorem 10. Let and be Einstein manifolds such that the Ricci tensors and of B and N satisfy and , respectively. If the identity mapis harmonic, then is Einstein if and only if . Remark 5. Eells and Sampson proved in [22] that the identity map of an m-sphere is deformable to maps of arbitrary small energy. For further results on identity maps from homotopic points of view, we refer to the two reports [34,35] by Eells and Lemaire. 4.6. Identity Maps on Kaehler and Hyper-Kaehler Manifolds
Let
be a Kaehler manifold. Then, with respect to a system of local coordinates
, we have
The following proposition was proved in [
36] by Watanabe and Dohira.
Proposition 2. Let be a Kaehler manifold. If a symmetric tensor T on is represented by for some 1-form , then T is a harmonic tensor with respect to g.
Watanabe and Dohira also considered in [
36] two Kaehler structures
and
on a complex manifold
M. Let
and
denote the Kaehler forms corresponding to
g and
, respectively. By applying Proposition 2, they proved following.
Theorem 11. If belongs to the Kaehler class , then the identity map is harmonic, i.e., is harmonic with respect to g.
Proposition 2 was extended to hyper-Kaehler manifolds by A. D. Vîlcu and G. E. Vîlcu as follows.
Theorem 12. [
37]
Let be a hyper-Kaehler manifold. If a symmetric tensor T is represented byfor some 1-form , then T is a harmonic tensor with respect to g. 10. Gauss and Identity Maps
Let be the Grassmann manifold consisting of oriented linear n-subspaces of the Euclidean m-space equipped with the standard Euclidean metric . Then admits a standard Riemannian metric which makes a compact symmetric space. In particular, with a natural complex structure is holomorphically isometric to the complex quadric of complex dimension .
10.1. Gauss Images
Consider an immersion from an n-manifold into a Euclidean m-space . Let denote the induced metric on via . The Gauss map is the map obtained by parallel displacement of the tangent plane in . In this section, we always assume that the Gauss map is a regular map so that the Gauss map induces a metric on . In this way, admits two metrics and induced via and , respectively.
Definition 7. A submanifold of is said to have totally geodesic Gauss image if geodesics of are carried to geodesics of by the Gauss map.
The following result was proved by the author and Yamaguchi.
Theorem 35. [
67,
68]
A Euclidean submanifold in has totally geodesic Gauss image if and only if the second fundamental form B of in satisfiesfor vector fields tangent to , where the Gauss image is the Levi–Civita connection of . 10.2. Geometry of Gauss Transformations
In this subsection, we discuss the identity map associated with an isometric immersion . We call this identity map the Gauss transformation of . More precisely, we present results for a surface in to have harmonic, conformal, homothetic, or affine Gauss transformation. Here the identity map is called affine if it carries geodesics into geodesics.
Theorem 36. [
2]
An isometric immersion of a surface is harmonic if and only if is harmonic. Theorem 36 implies the following.
Corollary 11. [
2]
Let be an immersion of a surface M into . Then the Gauss map is harmonic and the identity map is homothetic if and only if- (1)
has constant Gauss curvature and
- (2)
there exists a hypersphere of such that is harmonic.
Remark 9. Theorem 36 is false in general if . In fact, for , there exists a hypersurface in such that is harmonic, but is not harmonic.
Theorem 37. [
2]
Let be an immersion of a surface M into . Then the identity map is conformal and the Gauss map is harmonic if and only if either- (a)
is harmonic, or
- (b)
there exists a hypersphere of such that is harmonic.
Theorem 38. [
2]
Let be an immersion of a surface M into . If is affine, then either is homothetic or both and are flat surfaces. Theorem 39. [
2]
Let be an immersion of a surface M into . Then is affine and the Gauss map is harmonic if and only if has constant Gauss curvature and either (1)
is immersed in a hypersphere of as a minimal surface via ϕ, or (2)
is immersed as an open portion of the product surface of two planar circles via ϕ. 11. Laplace and Identity Maps
Consider an isometric immersion
of a Riemannian
n-manifold
M into a Euclidean
m-space
. Then the Laplace operator
of
gives rise to a differentiable map
called the Laplace map, where
is regarded as the immersion as well as the position vector field of
M in
. The image
of
M under map
is called the Laplace image of
M.
In this section, we assume that the Laplace map
is a regular map, so that the Laplace image
admits an induced metric
via
. In this way,
M admits two metrics
and
. Hence, we may consider the differential geometry of the identity map
. Note that the identity map
was called as the Laplace transformation in the book [
7] of the author and Verstraelen.
The following question was asked and initially studied in [
7].
Question 2. To what extent does the identity map determine the immersion ?
In this section we present some answers for this question from [
7].
11.1. Submanifolds with Homothetic Laplace Transformations
Let be a homogeneous Riemannian manifold and be the group of isometries on M. Then an isometric immersion is said to be G-equivariant if there exists a homomorphism such that for each and .
Theorem 40. Let be an equivariant isometric immersion of a compact irreducible Riemannian homogeneous manifold into . Then the identity map is homothetic.
For an isometric immersion of
into a Euclidean
m-space
with homothetic identity map
, we have the following results from [
7].
Theorem 41. Let be an isometric immersion. If is homothetic, then we have:
- (1)
If ϕ is of finite type, the Laplace map is of finite type.
- (2)
If M is compact, then the Laplace map is of k-type if and only if the immersion ϕ is of k-type.
- (3)
If ϕ is of finite type, then the Laplace map is of non-null finite type; in particular, if x is of non-null k-type, then is of non-null k-type; and if ϕ is of null k-type, then is of non–null -type.
Theorem 42. Let be an isometric immersion. If is homothetic, then the Laplace image lies in a hypersphere of as a minimal submanifold if and only if either M is a minimal submanifold of a hypersphere of or ϕ is of null 2-type.
Theorem 43. If is an isometric immersion with parallel mean curvature vector, then is homothetic if and only if M is immersed as a minimal submanifold of a hypersphere of via ϕ.
Theorem 44. Let be a hypersurface of with or 3. Then is homothetic if and only if is an open part of a hypersphere in via ϕ.
Theorem 45. Let be an isometric immersion with constant mean curvature. Then is homothetic if and only if is a minimal surface in a hypersphere of via ϕ.
11.2. Geometry of Conformal or Harmonic Laplace Transformations
Theorem 46. Let be a hypersurface with . If the identity map is conformal and has positive semi-definite Ricci tensor, then is conformally flat.
Theorem 47. Let be an isometric immersion of a surface into . If is conformal, then the Laplace image is a minimal surface of via if and only if ϕ is biharmonic.
Theorem 48. Let be an isometric immersion with parallel mean curvature vector. Then the identity map is harmonic.
11.3. Geometry of Laplace-Gauss Identity Maps
Suppose
is an isometric immersion from a Riemannian
n-manifold
into
. For simplicity, we assume that the Gauss map and the Laplace map associated with
are regular. Thus we have the metrics
and
on
via Gauss and Laplace maps, respectively. Therefore, we may consider the identity map
between the Laplace and Gauss images. We call this map the Laplace–Gauss identity map, or the LG-identity map of
for short.
The following results on LG-identity maps are also obtained in [
7].
Theorem 49. Let be an isometric immersion. Then the LG-identity map is conformal if and only if locally either ϕ has constant mean curvature function or M is immersed as product of a planar curve and an affine -space via ϕ.
Remark 10. A surface in is called a surface of Delaunay if it has constant mean curvature function. Theorem 49 implies that every surface of Delaunay in has homothetic LG-identity map.
Theorem 50. Let be a hypersurface. Then the LG-identity map is homothetic if and only if locally either M has constant mean curvature function in or M is the product of an affine -space with a planar curve whose curvature function is given byfor some positive constants . Theorem 51. Let be a hypersurface of a hypersphere of . Then M has constant mean curvature function and is homothetic if and only if either
- (1)
is an open part of the product of two spheres with some suitable radii a and b, or
- (2)
is an open part of a hypersphere of .
Theorem 52. Let be a hypersurface of a hypersphere of . If M has non-constant mean curvature function and is conformal, then
- (1)
The gradient of the mean curvature function is an eigenvector of the shape operator A of in .
- (2)
The shape operator A has at most three distinct eigenvalues.
- (3)
The eigenvalue corresponding to the eigenvector given by the gradient of the mean curvature function is of multiplicity one.
Theorem 53. Let be an isometric immersion. Then has constant mean curvature in and the LG-identity map is conformal if and only if is either a totally umbilical surface or a minimal surface in .
Theorem 54. Let be an isometric immersion. Then has constant Gauss curvature and is homothetic if and only if locally is a totally umbilical surface or a Clifford torus in .
For further results in this respect, see the book [
7].
12. Identity Maps of Tangent Bundles
For general information on geometry of tangent bundles, we refer to [
69].
12.1. Tangent Bundles
Let be a pseudo-Riemannian manifold with Levi–Civita connection ∇. The tangent bundle of M consists of pairs , where x is a point in M and X a tangent vector to M at x. The map is the natural projection.
The tangent space
at a point
in
is a direct sum of the vertical subspace
and the horizontal subspace
with respect to the Levi–Civita connection ∇ of
M:
For any vector
, there exists a unique vector
at
which is called the horizontal lift of
Y to the point
such that
; and a unique vector
which is called the vertical lift of
Y to the point
such that
for all
. Hence, every vector
can be decomposed as
for uniquely determined vectors
. The horizontal lift (respectively, vertical lift) of a vector field
X on
M to
is the vector field
(respectively, the vector field
) on
whose value at
is the horizontal lift (respectively, vertical lift) of
to
.
The tangent bundle
of
equips with a metric
in a natural way; called the Sasaki metric, which depends only on the metric
g of the base manifold
M. The Sasaki metric
on
is defined by the relations:
for vector fields
on
. Intuitively, the Sasaki metric
is constructed in such a way that the vertical and horizontal sub-bundles are orthogonal and the bundle map
is a pseudo-Riemannian submersion.
12.2. Identity Maps of Tangent Bundles with Lift-Complete Metrics
Let
be a Riemannian
n-manifold. Oniciuc defined in [
70] a pseudo-Riemannian metric
on
, called the lift-complete of
g, defined by the following relations:
Via a non-linear connection on the tangent bundle
, Oniciuc defined on
in [
70] the tensor field
G of type
, called of lift-complete type, by
Oniciuc proved the following.
Theorem 55. - (a)
The identity map is totally geodesic if and only if the projection is totally geodesic.
- (b)
The identity map is biharmonic, i.e., is biharmonic with respect to G.
For detailed results in this respect, see [
70].
12.3. Identity Maps of Tangent Bundles with g-Natural Metrics
For a Riemannian manifold M, the Sasaki metric is only one possible choice inside a very large family of Riemannian metrics on the tangent bundle of M, known as Riemannian g-natural metrics. Those metrics are constructed in a “natural” way from the Riemannian metric g over M.
The introduction of
g-natural metrics by Kowalski and Sekizawa in [
71] converts the classification of second order natural transformations of Riemannian metrics on manifolds to that of metrics on tangent bundles.
The set of
g-natural metrics which depend on six smooth functions from
to
has been completely described by Abbassi and Serih in [
72] as follows.
Theorem 56. Let be a Riemannian manifold and G be a g-natural metric on . Then there are six smooth functions , , , such that for every , we havewhere . For , the same holds with , . To state the next two theorems, we put
for all
. The following two results were proved by Abbassi and Calvaruso in [
73].
Theorem 57. Let be a Riemannian n-manifold and let G be an arbitrary g-natural metric on . Then the identity map is harmonic if and only if the equationholds and either (1)
the horizontal and vertical distributions are orthogonal with respect to G, or (2)
is an Einstein manifold with , and holds. Theorem 58. Let be a Riemannian manifold and an arbitrary g-natural metric on . Then is harmonic if and only if the following two conditions are satisfied:and 12.4. Harmonic Metrics on Non-Reductive Homogeneous Manifolds
Assume g and are G-invariant pseudo-Riemannian metrics on a non-reductive homogeneous 4-manifold . Let (respectively, and ) denote the Sasaki metric (respectively, the horizontal lift and the complete lift) of on the tangent bundle . In addition, let (respectively, and ) denote the Sasaki metric (respectively, the horizontal lift and the complete lift) of g.
In [
74] by Zaeim and Atashpeykar proved the following.
Theorem 59. Let be a non-reductive homogeneous 4-manifold. Then the following four statements are equivalent:
- (a)
The identity map is harmonic, i.e., the pseudo-Riemannian metric on is harmonic with respect to the pseudo-Riemannian metric g.
- (b)
The identity map is harmonic.
- (c)
The identity map is harmonic.
- (d)
The identity map is harmonic.
Remark 11. A complete classification of non-reductive pseudo-Riemannian homogeneous spaces up to dimension four was obtained earlier in [75] by Fels and Renner. 12.5. Identity Maps of Tangent Bundles of Walker 4-Spaces
Bejan and Drctă-Romaniuc investigated the total space of the tangent bundle of a Walker 4-manifold and proved the following.
Theorem 60. [
40]
Let be a Walker 4-manifold. Then the following three statements are equivalent:- (a)
The identity map is harmonic, i.e., a Walker metric on is harmonic with respect to g.
- (b)
The Sasaki metric of a Walker metric is harmonic with respect to the Sasaki metric of the Walker metric g.
- (c)
The horizontal lift of a Walker metric is harmonic with respect to the horizontal lift of the Walker metric g.
12.6. Identity Maps of Gödel-Type Spacetimes
The Gödel metric, introduced by Kurt Gödel in [
76], is an exact solution of the Einstein field equations such that the stress-energy tensor, which contains two terms, the first term representing the matter density of a homogeneous distribution of swirling dust particles (dust solution), and the second associated with a nonzero cosmological constant. Gödel spacetimes have been deeply investigated in physics (cf. e.g., [
77,
78,
79]).
Gödel-type 4D spacetimes can be described by the Lorentzian metrics
where
t is the time variable and
for
(undetermined for
). Further,
and
are arbitrary smooth functions on
M and
g is non-degenerate where
.
The following three theorems were proved by Zaeim, Jafari and Yaghoubi in [
80].
Theorem 61. Let g and be Gödel-type metrics on a 4-manifold with the metric g given by (77) and given by Then the identity map is harmonic (equivalently, the metric tensor is harmonic with respect to if and only if Theorem 62. Let be homogeneous Gödel-type spacetime. In this case, a metric tensor is harmonic with respect to g if and only if .
Theorem 63. Let be a Gödel-type spacetime with the metric g given by (77) and let be an arbitrary Gödel-type metric of Equation (78). Then the following four statements are equivalent: - (a)
The identity map is harmonic, i.e., is harmonic with respect to g.
- (b)
The Sasaki metric is harmonic with respect to the Sasaki metric .
- (c)
The horizontal lift is harmonic with respect to the horizontal lift .
- (d)
The complete lift is harmonic with respect to the complete lift .