Differential Geometry of Identity Maps: A Survey

An identity map idM:M→M is a bijective map from a manifold M onto itself which carries each point of M return to the same point. To study the differential geometry of an identity map idM:M→M, we usually assume that the domain M and the range M admit metrics g and g′, respectively. The main purpose of this paper is to provide a comprehensive survey on the differential geometry of identity maps from various differential geometric points of view.


Introduction
An identity map id M : M → M 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 id M on a manifold M is defined to be the map with domain and range M which satisfies id M (p) = p for any p ∈ M. (1) Obviously, the identity map id M : M → M 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 id M : M → M, 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 id M : (M, g) → (M, g) of a compact Riemannian manifold (M, g) onto itself is a stable map if and only if the diagonal map in M × M is stable as a minimal submanifold. Further, the author and Nagano proved in [2]  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 g , respectively. The identity map id M : (M, g) → (M, g ) is called a conformal change of metric if g = ρ 2 g for some positive function ρ. It is well-known that, in the case dim M ≥ 4, if the identity map id M : (M, g) → (M, g ) is a conformal change of metric, then (M, g) and (M, g ) have the same Weyl tensor (cf. [4]); and in the case dim M = 3, (M, g) and (M, g ) 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 W 4 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.

Basics on Pseudo-Riemannian Manifolds
Consider a smooth n-manifold M covered by a system of coordinate neighborhoods (U, {x i }), where U is a neighborhood and the {x i } denote local coordinates on U, where the indices i, j, k, , . . . take on values in the range 1, 2, ..., n. Then a Riemannian (or pseudo-Riemannian) metric g on M can be expressed as g = g ij dx i dx j 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 ∂ i = ∂ ∂x i the basis vectors on the coordinate neighborhood {U, (x 1 , . . . , x n )}. Let X and Y be two vector fields on M. We then have where X h and Y h are the local components of the vector fields X and Y, respectively. With respect to the natural frame ∂ h , we have which are the local components of the metric tensor field g. The Christoffel symbols of the Riemannian manifold (M, g) are given by where (g k ) denotes the inverse matrix of (g ij ). Thus, we have g ij g jk = δ k i , where δ k i = 1 or 0 according to i = k or i = k.
Let ∇ denote the Levi-Civita connection of (M, g). Then for vector fields X, Y on M the covariant derivative ∇ X Y has local components The covariant derivative of a tensor T ij of type (0,2) is defined by Let X, Y and Z be three vector fields in M. Then defines a tensor field K of type (1, 3). If we put then K(X, Y) 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 Ric of (M, g) is the (0, 2)-tensor with local components given by The scalar curvature r is given by Let T r s denote the space of all tensors of type (r, s) on (M, g). Following [2], we denote by δ the codifferential of T r s defined by (δT) Then δ : T r s+1 → T r s . As in [2], a tensor T is called co-closed if it satisfies δT = 0.

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

Gradient, Divergence and Laplacian
Assume (M, g) is a pseudo-Riemannian manifold and f is a smooth function on M. Then the gradient of f , denote by ∇ f (or by grad f ), is the vector field dual to the differential d f . In other word, ∇ f is defined by In terms of a local coordinate system {x i } of M, the divergence of a vector field X = X j ∂ j , denoted by div X, is given by The Laplacian of f , denoted by ∆ f , is defined by ∆ f = −div(∇ f ), we have

Basics on Submanifolds
Assume that φ : (M, g) → (N, h) is an isometric immersion of a pseudo-Riemannian manifold into another. Denote by ∇ and ∇ N the Levi Civita connections on M and N, respectively. The formulas of Gauss and Weingarten are given respectively by for tangent vector fields X, Y and a normal vector field ξ, where B, A and D are the second fundamental form, the shape operator and the normal connection. For each ξ ∈ T ⊥ p M, the shape operator A ξ is a symmetric endomorphism of the tangent space The shape operator and the second fundamental form are related by for X, Y tangent to M and ξ normal to M. The mean curvature vector is defined by H = 1 2 trace h. For a vector X ∈ T p N, p ∈ M, we denote by X and X ⊥ the tangential and the normal components of X, respectively. The equations of Gauss, Codazzi and Ricci are given respectively by for vector fields X, Y, Z tangent to M, ξ normal to M, and∇B is given by and R D is the curvature tensor associated to the normal connection D.
The submanifold M is called totally geodesic if B = 0 holds identically; and parallel if∇B = 0 identically. The submanifold is said to have parallel mean curvature vector if DH = 0 identically (cf. [11] for a detailed survey on submanifolds with parallel mean curvature vector).

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 E m is said to be of finite type if its immersion φ is a finite sum of eigenmaps of the Laplacian ∆, i.e., where c 0 is a constant map and φ i , . . . , φ k are non-constant maps satisfying If the eigenvalues λ t 1 , . . . , λ t k of ∆ in (27) are distinct, then M (or the immersion φ) is said to be k-type. In particular, if one of λ t 1 , . . . , λ t k is zero, then M is said to be of null k-type.
Analogously, a smooth map ψ : M → E m of a Riemannian manifold M is called a finite type map if ψ is a finite sum of E m -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 E m is said to be of infinite type if it is not of finite type.
For each φ i in the spectral decomposition (27) of φ, we put Then each E i is a linear subspace of E m .

Linearly Independent and Orthogonal Immersions
Consider a k-type isometric immersion φ : M → E m whose spectral decomposition is given by (26). Then φ is said to be linearly independent if the subspaces E 1 , . . . , E k defined by (28) are linearly independent, i.e., the dimension of the subspace spanned by all vectors in The immersion φ is said to be orthogonal if the subspaces E 1 , . . . , E k are mutually orthogonal in E m (see [20,21]).
If we choose a Euclidean coordinate system {u i } on E m with c as its origin, then the spectral decomposition (26) of φ reduces to For each i ∈ {1, . . . , k}, we choose a basis {c ij : j = 1, . . . , m i } of E i , where m i denotes the dimension of E i . Put = m 1 + . . . + m k and let E denote the subspace of E m spanned by E 1 , . . . , E k . If the immersion φ is linearly independent, then {c ij : i = 1, . . . , k; j = 1, . . . , m i } are linearly independent vectors in E . Further, we may choose the Euclidean coordinate system {u i } on E m in such way that E is defined by u +1 = . . . = u m = 0.
We regard each c ij as a column -vector and put where λ i repeats m i -times for i = 1, . . . , k. If we put A = SDS −1 , then we get Ac ij = λ i c ij for any i ∈ {1, . . . , k} and j ∈ {1, . . . , m i }. Thus, we obtain for the immersion φ : M → E induced from φ : M → E m . We may regard the ( × )-matrix A as an (m × m)-matrix in a natural way (with zeros for each of the additional entries). For this (m × m)-matrix A, we also have By using the (m × m)-matrix A in (30), the author introduced in [20] the notion of adjoint hyperquadric as follows.
Let φ : M → E m be a non-minimal, linearly independent immersion whose spectral decomposition is given by (26). Let {u i } be a Euclidean coordinate system on E m with c as its origin. Let A be the (m × m)-matrix given in (30). Then, for a given point p ∈ M, Au, u := m ∑ i,j a ij u i u j = c p , (c p = Aφ, φ (p)) defines a hyperquadric Q p in E m , called the adjoint hyperquadric of φ at p. In particular, if φ(M) lies in an adjoint hyperquadric Q p of φ for some point p ∈ M, then all of the adjoint hyperquadrics {Q p : p ∈ M} give the same adjoint hyperquadric which is denoted by Q, called the adjoint hyperquadric of the linearly independent immersion φ.

Harmonic Maps
Let φ : (M, g) → (N,ḡ) be a map between Riemannian manifolds. We use φ −1 (TN) to denote the pull-back bundle of the tangent bundle TN by φ (or the induced bundle of φ). The pull-back bundle has base M and the fiber T φ(p) N over p ∈ M. The sections of φ −1 (TN) are called vector fields along φ. The differential dφ of φ can be regarded as a section of Let ∇ φ denote the pull-back connection on φ −1 (TN) defined by The second fundamental form σ of a map φ : (M, g) → (N,ḡ) is given by the covariant differential ∇(dφ). A map φ is called totally geodesic if its second fundamental form vanishes identically.
The energy density e(φ) of φ is a non-negative function given by where dφ denotes the differential of φ, ||dφ|| is the Hilbert-Schmidt norm of dφ, and "tr" denotes the trace. The Dirichlet energy of φ over any compact domain D of M is defined by A variation of a map φ is a smooth map F : M × (−ε, ε) → N such that F(p, 0) = φ(p), p ∈ M. A variation {φ t } is said to be supported in a domain D if φ t = φ on M \D for all t ∈ I, whereD denotes the interior of D. Note that ∂φ t ∂t | t=0 = V(p) defines a vector field on the pull back bundle φ −1 (TN) and it is called the vector field of the variation {φ t }.
It was well-known that for a smooth variation {φ t } of φ : (M, g) → (N,ḡ) supported in D, we have the first variation (cf. [22]) where V = ∂φ t ∂t | t=0 is the variation vector field, and τ(φ) is the tension field of φ given by It follows form Definition 1 that a harmonic map φ : (M, g) → (N,ḡ) is a solution of the corresponding Euler-Lagrange equation for the Dirichlet energy functional. Thus, a map φ is harmonic if and only if its tension field τ(φ) vanishes identically [22].

Harmonic Map and Tension Field
For a given map φ : (M, g) → (N,ḡ), let {x i } and {y α } be the local coordinates on M, and N, respectively. Then will be the local expression of the map φ. Let {∂ i } and {∂ α } be the natural frames of the local coordinates on M and N respectively. For simplicity, we put φ α i = ∂φ α ∂x i and φ α ij = ∂ 2 φ α ∂x i ∂x j . Then, in terms of local coordinates, we have where Γ k ij andΓ γ αβ denote the Christoffel symbols of (M, g) and (N,ḡ) with respect to the local coordinate systems {x i } and {y α }, respectively. Remark 1. Definition 1 makes sense for harmonic maps between pseudo-Riemannian manifolds as well.

Second Variation of Energy
Assume that φ : (M, g) → (N,ḡ) is a harmonic map from a compact manifold M into (N,ḡ). Let be a smooth two parameter variation of φ.
Let v(p) = ∂F ∂s | (0,0,p) ∈ T φ(p) N, w(p) = dF dt | (0,0,p) ∈ T φ(p) N be the variation vector fields. Then the Hessian, H φ , of the energy at φ is defined by Then we have Theorem 1. [23] If φ : (M, g) → (N,ḡ) is a harmonic map from a compact manifold. Then the Hessian of the energy at φ is given by for vector fields v, w along φ, where K N denotes the curvature tensor of (N,ḡ) and the Jacobi operator J φ is given by

Stability, Index and Nullity
The nullity of φ, denoted by Null(φ) is the dimension of the kernel of J φ . It follows from the spectral properties of elliptic operators on compact manifolds that both Ind(φ) and Null(φ) are finite.
where dφ(i(M)) is considered as variation vector fields along φ. The reduced nullity of φ, denoted by Null r (φ), is defined by (see [23]) The following two results on stable harmonic maps are well-known.

Theorem 2.
[24] For n ≥ 3, there exists no non-constant stable harmonic map from any compact Riemannian manifold to S n .

Theorem 3. [25]
For n ≥ 3, there exists no non-constant stable harmonic map from S n to any Riemannian manifold.

Relative Harmonic Metrics
Let φ : (M, g) → (N, h) be an immersion and g = φ * h be the Riemannian metric on M induced via φ. Let ϕ : (M, g ) → (N, h) be the associated isometric immersion, and id M : (M, g) → (M, g ) be the identity map. Then the tension field of the composition ϕ • id M of maps ϕ and id M is given by where "tr g " is the trace with respect to g and B is the second fundamental form of the associated isometric immersion ϕ : (M, g ) → (N, h). Thus, it follows from (40) that the composition ϕ • id M is a harmonic map if and only if the identity map id M : (M, g) → (M, g ) is harmonic and tr g B = 0. 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 g on a Riemannian manifold (M, g) is called harmonic with respect to g if the identity map id M : (M, g) → (M, g ) is a harmonic map.
From (35) we find the following.

Lemma 1. A metric g on M is harmonic with respect to a metric g on M if and only if
For the metric tensor g , if we put then we have Put Then L = (L k ji ) is a tensor of type (1, 2) satisfying L k ji = L k ij . 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 g is harmonic with respect to a given one g on a manifold M. Proposition 1. Let g and g be two Riemannian metrics on a manifold M. Then g is harmonic with respect to g if and only if we have where f = tr g g and ω i = ∇ k g ki .

Space of Harmonic Tensors
Let S denote the space of all symmetric covariant tensor fields of degree 2 on a Riemannian manifold (M, g). If we regard a tensor T ∈ S as an energy-momentum tensor, the equation δT = 0 is known in [26] as the conservation law of energy momentum.
Let M denote the space of all Riemannian metrics on (M, g). We put Then H g is a star set centered at g in the space S (see [2]). The following notion of (relative) harmonic tensors was introduced in [2]. Definition 3. Let (M, g) be a Riemannian manifold. A tensor T ∈ S on (M, g) 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 (M, g) be a Riemannian manifold of dimension ≥ 3. Then we have the following linear isomorphism of vector spaces: This isomorphism is given by S g → g − 1 2 (tr g g )g ∈ ker(δ).
where F (M) 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 (M, g) be Riemannian manifold. Then (a) The Ricci tensor of (M, g) is a harmonic tensor with respect to g. (b) The Ricci tensor of (M, g) is co-closed if and only if (M, g) has a constant scalar curvature.
In particular, Theorem 6 implies the following.  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.

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]): On the other hand, the notion of geodesic vector fields was introduced by Yano and Nagano in [29] as follows.
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 (M, g) is a geodesic vector field if and only if g = L v g is a harmonic tensor with respect to g, where L v denotes the Lie derivative with respect to v.

Remark 4.
Analogous to Definition 2, a pseudo-Riemannian metric g on a manifold M is said to be harmonic with respect to another pseudo-Riemannian metric g on M if the identity map id M : (M, g) → (M, g ) is harmonic. It was shown in [30] that the results given in Sections 4.2 and 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.

A Volume-Decreasing Phenomenon
If g is a co-closed harmonic metric with respect to a given metric g on a Riemannian manifold M, then we have (i) tr g g is constant and (ii) for any positive number c, cg 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 (M, g) is Riemannian manifold. If g is a co-closed harmonic metric w.r.t. g on M satisfying tr g g = tr g g, then (1) The volume form dv g ≤ dv g at each point of M; hence the identity map i M : (M, g) → (M, g ) is volume-decreasing. (2) dv g = dv g on M if and only if g = g on M.

Identity Maps on Warped Products
Let (B, g 1 ) and (N, g 2 ) be Riemannian manifolds and f a positive function on B. Then the product manifold B × N equipped with the warped product metric g = g 1 + f 2 g 2 is called a warped product manifold with f as its warping function. We denote this warped product manifold by B × f N.
Pyo, Kim and Park proved the following result in [33].
Theorem 10. Let (B, g 1 ) and (N, g 2 ) be Einstein manifolds such that the Ricci tensors R B and R N of B and N satisfy R B = c 1 g 1 and R N = c 2 g 2 , respectively. If the identity map Remark 5. Eells and Sampson proved in [22] that the identity map id S m of an m-sphere S m (m ≥ 3) 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.

Identity Maps on Kaehler and Hyper-Kaehler Manifolds
Let (M, J, g) be a Kaehler manifold. Then, with respect to a system of local coordinates (x h ), we have The following proposition was proved in [36] by Watanabe and Dohira.
Proposition 2. Let (M, J, g) be a Kaehler manifold. If a symmetric tensor T on (M, J, g) is represented by , then T is a harmonic tensor with respect to g.
Watanabe and Dohira also considered in [36] two Kaehler structures (J, g) and (J, g ) on a complex manifold M. Let Ω and Ω denote the Kaehler forms corresponding to g and g , respectively. By applying Proposition 2, they proved following.

Theorem 11.
If Ω belongs to the Kaehler class [Ω], then the identity map id M : (M, J, g) → (M, J, g ) is harmonic, i.e., g is harmonic with respect to g.
for some 1-form ϕ = (ϕ i ), then T is a harmonic tensor with respect to g.

Walker Manifolds
Definition 5. A Walker manifold (W, g, D) is a pseudo-Riemannian manifold (W n , g) equipped with a lightlike distribution D (i.e., g restricted to D is zero) which is parallel with respect to the Levi-Civita connection of g (cf. [38,39]).
In the 4-dimensional case, where the distribution D has the maximum dimension (i.e., when dim D = 2), the metric g is of neutral signature and there exists an adapted local coordinates (x, y, z, t) such that the matrix Q associated with the metric g takes the form where 0 and I denote 2 × 2 zero and identity matrices, respectively; and with a, b, c being functions of (x, y, z, t).

Equivalence Classes of Walker 4-Manifolds
In [40], Bejan and Druţȃ-Romaniuc introduced an equivalence relation on the set of all Walker metrics on a 4-dimensional manifold as follows: Two Walker metrics g andĝ on a Walker 4-manifold W 4 are said to be equivalent, denoted by g ∼ĝ, if around any point there exist some adapted local coordinates (x, y, z, t) with respect to which the entries of the associated matrices Q a,b,c of g andQâ ,b,ĉ ofĝ satisfy the following relation: for some local smooth functions α(y, z, t), β(x, z, t) and λ(x, y, z, t).

Example 1.
Let M denote the set of all Walker metrics on R 4 , whose associated matrices with respect to the coordinate frame are given by (47). Then we may identify M with Q a,b,c (a, b, c ∈ F (R 4 )). Consider the group with respect to the additive operation. Then G acts on M by the relation (49), and "∼" is the equivalence relation whose classes are the orbits. On the canonical Walker space (R 4 , g 0 , D) with a canonical neutral metric g 0 , the set of all Walker metrics harmonic with respect to g 0 is in one-to-one correspondence with the transformation group G from (50) (see [40]).

Relative Harmonic Metrics on Walker 4-Manifolds
Bejan and Drctȃ-Romaniuc determined all Walker metrics which are harmonic with respect to a given Walker metric on a Walker 4-manifold. Theorem 13. [40] Let (W 4 , g, D) be a Walker 4-manifold. Then, for any Walker metricĝ on W 4 , the identity map id W 4 : (W 4 , g, D) → (W 4 ,ĝ, D) is harmonic if and only if g ∼ĝ.

Index and Nullity of Identity Maps
In this section we present results on nullity and stability on identity maps.

Nullity of Identity Maps
Let us consider the identity map id M : (M, g) → (M, g). Lemma 1 shows that id M is always harmonic. In this case, the second variation of the harmonic id M : (M, g) → (M, g) is given by for vector fields v on M. Thus, the nullity of id M : (M, g) → (M, g) is the dimension of the space of the vector fields v satisfying for any vector field u with compact support in M. The following useful formula can be found in ( [28] page 57).
where div v denotes the divergence of v.
We have the following general result.   Corollary 4 follows from combining Theorem 14 and a result in [41] given by of Yano and Nagano.

Index and Relative Nullity of Identity Maps
In view of (53), Smith [23] obtained the following. Let ∆ H be the Hodge Laplacian acting on vector fields via duality. Then the Jacobi operator where the Ricci tensor Ric of M is regarded as a linear map on the tangent bundle TM. In particular, if (M, g) is Einstein, (54) becomes where I denotes the identity operator and c is a constant. By applying this formula, Smith [23] was able to relate the index and nullity of the identity map id M to the eigenvalues of the Laplacian ∆ acting on the space of smooth functions F (M). By putting λ(r) = #{eigenvalues λ : 0 < λ < r} and m(r) the multiplicity of r with m(0) = 0, Smith obtained the following result in [23].
In particular, Theorem 15 implies the following.

Index and Nullity of Spheres and Flat Torus
Smith determined in [23] the index, relative nullity and Killing nullity of the identity maps of S n : Null k (id S n ) = n(n + 1) n (n ≥ 1); and for the flat torus T n , he obtained index(id T n ) = 0 for all n ≥ 2, Null k (id T n ) = n for all n ≥ 2.

Index and Nullity of Product Maps
be two harmonic maps. Then the product map The product map of two harmonic maps is also harmonic. Fardoun and Ratto studied the index of nullity of product maps and obtained the following.
Additionally, in [42], Fardoun and Ratto determined the index and nullity of the identity maps for S n × S m and S n × T m .
For S n × T m , they obtained the following.
Theorem 16. [42] For the identity map id S n ×T m of the product manifolds S n × T m , we have:

Identity Maps on Compact Symmetric Spaces
If the identity map id M : (M, g) → (M, g) is stable, we simply say that the Riemannian manifold M is stable. Otherwise, M is said to be unstable. It is well-known that M is stable whenever M is Kaehlerian [23] (see also [1] page 131). Theorems 2 and 3 imply that the standard n-sphere S n is unstable for n ≥ 3.

Stability of Compact Symmetric Spaces
The following result was obtained by combining Theorem 14 and Nagano's table of spectra given in [43]. For exceptional symmetric spaces, Nagano proved in [1] that the Cayley projective plane F 4 /Spin(9) is also unstable.

Remark 6. Theorem 18 shows that a compact irreducible symmetric space is simply-connected if its identity map is unstable.
For stable compact symmetric space, Nagano [1] proved the following.

Instability of Compact Symmetric Spaces
Nagano proved the following properties for unstable symmetric spaces. Then SU(3)/T(k, ) is a non-symmetric 7-dimensional homogeneous space which admit a positively curved Riemannian metric (cf. [49]).

Theorem 24. Let SU(3)/T(k, ) have the SU(3)-invariant metric g which is canonically induced from an
Ad(SU(3))-invariant inner product on the Lie algebra su(3). If k and are relatively prime, then the identity map of (SU(3)/T(k, ), g) is stable.

Stability of Spheres with Deformed Metrics
From Section 6, we know that the standard unit m-sphere (S m , g 0 ) is unstable for m ≥ 3. In [51], Tanno studied the stability of a family of deformed metrics g(t) on S m with m = 2n + 1 as follows: For m = 2n + 1, we have the Hopf fibration π : (S m , g 0 ) → (CP n , h 0 ), where (CP n , h 0 ) is the complex projective n-space of constant holomorphic sectional curvature 4. Let ξ be a vector field on S m which is tangent to the fibers and of unit length. Then ξ is a Killing vector field with respect to the standard metric g 0 and let η be 1-form dual to ξ with respect to g 0 .
Consider the one-parameter family of metrics g(t) on S m is defined by where 0 < t < ∞. It is known that the metric given by (56) on S 2n+1 are homogeneous Riemannian metrics which are not symmetric nor Einstein. With respect to these metrics (g(t)), the volume elements remain unchanged (cf. [51,52]). Tanno proved the following.

Identity Maps and Riemannian Submersions
In Theorem 3, Xin gave an important result which says that each non-constant harmonic map from the unit n-sphere S n , n ≥ 3, into a Riemannian manifold is always unstable. Thus, Urakawa asked in [54] the following Question 1. Does there exist a deformation g t , 0 < t < ∞, of the standard metric g 1 on S n such that if g t is far from g 1 , then (S n , g t ) admits a stable harmonic map?
To provide an answer to this question, Urakawa [54] investigated a Riemannian submersion φ : (M, g) → (N, h) with totally geodesic fibers. Then he considered the canonical variation g t , 0 < t < ∞, of the metric g which also gives a Riemannian submersion φ : (M, g t ) → (N, h) with totally geodesic fibers. More precisely, Urakawa [54] proved the following theorem which gave an answer to this question.

Theorem 27.
Assume that the identity map id N of (N, h) is stable. Then there exists a small number ε such that, for every 0 < t < ε, the Riemannian submersion φ : (M, g t ) → (N, h) is stable.
Since the identity map id CP n of the complex projective space (CP n , h) is stable, the Hopf fibering π : (S 2n+1 , g t ) → (CP n , h) is stable for 0 < t < ε, for the canonical variation g t , 0 < t < ε, with g 1 = canonical metric. This provides an example which contrasts with the instability theorem of Xin (Theorem 3). Some further results on identity maps were also obtained by Urakawa in [54].

Identity Maps on Generalized Sasakian Space Forms
A Riemannian (2n + 1)-manifold (M 2n+1 , g) is called an almost contact metric manifold [55] if there exist a (1, 1) tensor field ϕ, a vector field ξ (called the structure vector field), and a 1-form η onM such that and for any vector fields X, Y onM. An almost contact metric manifold (M 2n+1 , ϕ, ξ, η, g) is called a Sasakian manifold if it satisfies for X, Y tangent toM 2n+1 , where ∇ is the Levi-Civita connection ofM 2n+1 . A Sasakian manifold with constant ϕ-sectional curvature c is called Sasakian space form. In this case, the curvature tensor of M 2n+1 is given by for vector fields X, Y, Z tangent toM 2n+1 . Sasakian space forms can be modeled based on c > −3, c = −3 or c < −3.
Alegre, Blair and Carriazo introduced and studied in [56] the notion of a generalized Sasakian space form. An odd-dimensional manifoldM 2n+1 equipped with an almost contact metric structure (φ, ξ, η, g) is called generalized Sasakian space form if there exist three functions f 1 , f 2 , f 3 onM 2n+1 such that We denote such a manifold byM 2n+1 [57] the stability of generalized Sasakian space forms and obtained the following results.
Theorems 28 and 29 imply the following corollaries, respectively.

Corollary 7.
Let M 2n+1 be a compact Sasakian space form of constant ϕ-sectional curvature c ≤ 1. If the first eigenvalue of the Laplacian satisfies λ 1 < (n + 1)c + 3n − 1, then the identity map id M is unstable.

Biharmonic Maps and Biharmonic Submanifolds
A biharmonic map is a map φ : (M, g) → (N, h) between Riemannian manifolds that is a critical point of the bienergy functional where D is any compact domain D of M and τ(φ) is the tension field of the map φ. The Euler-Lagrange equation of the bienergy functional was computed by Jiang in [58]. The first variational formula for the bienergy functional is given by where η is the variational vector field along φ and the bitension field τ 2 (φ) is given by and K N being the curvature tensor of (N, h). Hence, φ is biharmonic if and only if its bitension field τ 2 (φ) vanishes identically. A biharmonic submanifold of a Riemannian manifold is one whose defining isometric immersion is a biharmonic map. On the other hand, in his program to study finite type submanifolds, the author defined in the 1980s the notion of biharmonic submanifolds M of Euclidean spaces as those whose position vector field x satisfies ∆ 2 x = 0 (cf. [16,59,60]).

Biharmonic Maps and Relative Biharmonic Metrics
The following definition was made in [62] (see also [63] page 449).

Definition 6.
A Riemannian metricḡ on M is called a (relative) biharmonic metric with respect to another metric g if the identity map id M : (M, g) → (M,ḡ) is biharmonic. A biharmonic metricḡ of (M, g) is called proper biharmonic metric if it is non-harmonic.

Example 2.
For λ = |x| −1 , |x| −2 , 2 1±|x| 2 , the conformally flat metrics λ 2 g 0 is biharmonic with respect to Euclidean metric g 0 on an open set of U ⊂ E 4 , i.e., each of the following identity maps is a biharmonic map if and only if where Ric g is the Ricci operator of (M, g). In particular, for m = 4, the identity map (64) is biharmonic if and only if is a biharmonic map if and only if In particular, for m = 6, the identity map (67) is biharmonic if and only if From Definition 6 and Proposition 5, we have Corollary 8. A conformally related metricḡ = λ 2 g on a Riemannian manifold (M m , g) with m ≥ 3 is biharmonic with respect to g if and only if the function λ is solution of (65). In particular, a conformally related metricḡ = λ 2 g on a 4-manifold (M 4 , g) is biharmonic with respect to g if and only if the function λ solves (66).
A conformally related metricḡ = e 2γ g is called biharmonic in [62] if the identity map id M : (M,ḡ) → (M, g) is biharmonic. Thus, by statement (b) of Proposition 5, the equation for the "biharmonic metric" defined in [62] is Equation (68). The 4-dimensional conformally flat space (E 4 \ {0}, |x| −2 g 0 ) appeared in a study of Yang-Mills equation in [65]. The following corollary shows that this conformally flat metric is one of the proper biharmonic metrics in a Euclidean 4-space.

Obstructions to Biharmonic Metrics on Einstein Manifolds
The next three results of Baird, Fardoun and Ouakkas provide obstructions to the existence of proper biharmonic metric on compact Einstein manifolds.

A Biharmonicity Reduction Theorem For Submersions
The next result was obtained by Balmuş.
This theorem implies that the study of biharmonicity of a harmonic Riemannian submersion π : (M, g) → (N, h) reduces to the study of biharmonicity of the identity map id N : (N, h) → (N,h).

Gauss and Identity Maps
Let Q m−n be the Grassmann manifold consisting of oriented linear n-subspaces of the Euclidean m-space (E m ,ḡ 0 ) equipped with the standard Euclidean metricḡ 0 . Then Q m−n admits a standard Riemannian metric G 0 which makes Q m−n a compact symmetric space. In particular, Q m−2 with a natural complex structure is holomorphically isometric to the complex quadric of complex dimension m − 2.

Gauss Images
Consider an immersion φ : M n → (E m ,ḡ 0 ) from an n-manifold M n into a Euclidean m-space E m . Let g 0 denote the induced metric on M n via φ : M n → (E m ,ḡ 0 ). The Gauss map ν : M n → (Q m−n , G 0 ) is the map obtained by parallel displacement of the tangent plane dφ(T p M n ) (p ∈ M n ) in E m . In this section, we always assume that the Gauss map ν is a regular map so that the Gauss map ν induces a metric G 0 on M n . In this way, M n admits two metrics g 0 and G 0 induced via φ and ν, respectively.

Definition 7.
A submanifold M n of E m is said to have totally geodesic Gauss image if geodesics of (M, G 0 ) are carried to geodesics of (Q m−n , G 0 ) by the Gauss map.
The following result was proved by the author and Yamaguchi.
for vector fields X, Y, Z tangent to M n , where the Gauss image ∇ G 0 is the Levi-Civita connection of (M n , G 0 ).

Geometry of Gauss Transformations
In this subsection, we discuss the identity map id M : (M, G 0 ) → (M, g 0 ) associated with an isometric immersion φ : (M, g 0 ) → E m . We call this identity map the Gauss transformation of φ. More precisely, we present results for a surface in E m to have harmonic, conformal, homothetic, or affine Gauss transformation. Here the identity map id M : (M, G 0 ) → (M, g 0 ) is called affine if it carries geodesics into geodesics.

Laplace and Identity Maps
Consider an isometric immersion φ : (M, g 0 ) → (E m ,ḡ 0 ) of a Riemannian n-manifold M into a Euclidean m-space E m . Then the Laplace operator ∆ of (M, g 0 ) gives rise to a differentiable map In this section, we assume that the Laplace map L φ is a regular map, so that the Laplace image L φ (M) admits an induced metric g L φ via L φ . In this way, M admits two metrics g 0 and g L φ . Hence, we may consider the differential geometry of the identity map id M : (M, g 0 ) → (M, g L φ ). Note that the identity map id M : (M, g 0 ) → (M, g L φ ) 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 id M : In this section we present some answers for this question from [7].

Submanifolds with Homothetic Laplace Transformations
Let For an isometric immersion of (M, g) into a Euclidean m-space E m with homothetic identity map id M : (M, g 0 ) → (M, g L φ ), we have the following results from [7]. Theorem 41. Let φ : (M, g 0 ) → (E m ,ḡ 0 ) be an isometric immersion. If id M : (M, g 0 ) → (M, g L φ ) is homothetic, then we have: (1) If φ is of finite type, the Laplace map L φ : M → E m is of finite type.
(2) If M is compact, then the Laplace map L φ 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 L φ is of non-null k-type; and if φ is of null k-type, then L φ is of non-null (k − 1)-type.

Geometry of Laplace-Gauss Identity Maps
Suppose φ : (M n , g 0 ) → (E m ,ḡ 0 ) is an isometric immersion from a Riemannian n-manifold M n into E m . For simplicity, we assume that the Gauss map and the Laplace map associated with φ are regular. Thus we have the metrics G 0 and g L φ on M n 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 φ for short.
The following results on LG-identity maps are also obtained in [7]. (1) The gradient of the mean curvature function is an eigenvector of the shape operator A of M n in S n+1 .
(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. For further results in this respect, see the book [7].

Identity Maps of Tangent Bundles
For general information on geometry of tangent bundles, we refer to [69].

Tangent Bundles
Let (M, g) be a pseudo-Riemannian manifold with Levi-Civita connection ∇. The tangent bundle TM of M consists of pairs (x, X), where x is a point in M and X a tangent vector to M at x. The map π : TM → M : (x, X) → x is the natural projection.
The tangent space T (x,X) TM at a point (x, X) in TM is a direct sum of the vertical subspace V (x,X) = ker(dπ| (x,X) ) and the horizontal subspace H (x,X) with respect to the Levi-Civita connection ∇ of M: T (x,X) TM = H (x,X) ⊕ V (x,X) .
For any vector Y ∈ T x M, there exists a unique vector Y h ∈ H (x,X) at (x, X) ∈ TM which is called the horizontal lift of Y to the point (x, X) such that dπ(Y h ) = Y; and a unique vector Y v ∈ V (x,X) which is called the vertical lift of Y to the point (x, X) such that Y v (d f ) = Y( f ) for all f ∈ F (M). Hence, every vectorȲ ∈ T (x,X) TM can be decomposed as for uniquely determined vectors Y 1 , Y 2 ∈ T x M. The horizontal lift (respectively, vertical lift) of a vector field X on M to TM is the vector field X h (respectively, the vector field X v ) on TM whose value at (x, X) is the horizontal lift (respectively, vertical lift) of X x to (x, X). The tangent bundle TM of (M, g) equips with a metric g S in a natural way; called the Sasaki metric, which depends only on the metric g of the base manifold M. The Sasaki metric g S on TM is defined by the relations: for vector fields X, Y on TM. Intuitively, the Sasaki metric g S is constructed in such a way that the vertical and horizontal sub-bundles are orthogonal and the bundle map π : (TM, g S ) → (M, g) is a pseudo-Riemannian submersion.

Identity Maps of Tangent Bundles with Lift-Complete Metrics
Let (M, g) be a Riemannian n-manifold. Oniciuc defined in [70] a pseudo-Riemannian metric g c on TM, called the lift-complete of g, defined by the following relations: Via a non-linear connection on the tangent bundle TM, Oniciuc defined on TM in [70] the tensor field G of type (0, 2), called of lift-complete type, by Oniciuc proved the following. Theorem 55. [70] We have: (TM, G) → (M, g) is totally geodesic. (b) The identity map id TM : (TM, G) → (TM, g c ) is biharmonic, i.e., g c is biharmonic with respect to G.
For detailed results in this respect, see [70].

Identity Maps of Tangent Bundles with g-Natural Metrics
For a Riemannian manifold M, the Sasaki metric g S is only one possible choice inside a very large family of Riemannian metrics on the tangent bundle TM 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 R + to R has been completely described by Abbassi and Serih in [72] as follows.
To state the next two theorems, we put φ i (t) = α i (t) + tβ i (t), for all t ∈ R + . The following two results were proved by Abbassi and Calvaruso in [73].
Theorem 57. Let (M, g) be a Riemannian n-manifold and let G be an arbitrary g-natural metric on TM. Then the identity map id TM : (TM, g S ) → (TM, G) is harmonic if and only if the equation holds and either (1) the horizontal and vertical distributions are orthogonal with respect to G, or (2) (M, g) is an Einstein manifold with Ric(u) = cu, and 2φ 2 + (n − 1)β 2 = cα 2 holds.
Theorem 58. Let (M, g) be a Riemannian manifold and G an arbitrary g-natural metric on TM. Then id TM : (TM, G) → (TM, g S ) is harmonic if and only if the following two conditions are satisfied:

Harmonic Metrics on Non-Reductive Homogeneous Manifolds
Assume g andĝ are G-invariant pseudo-Riemannian metrics on a non-reductive homogeneous 4-manifold M 4 = G/K. Letĝ S (respectively,ĝ h andĝ c ) denote the Sasaki metric (respectively, the horizontal lift and the complete lift) ofĝ on the tangent bundle TM 4 . In addition, let g S (respectively, g h and g c ) denote the Sasaki metric (respectively, the horizontal lift and the complete lift) of g.
In [74] by Zaeim and Atashpeykar proved the following.

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.

Identity Maps of Tangent Bundles of Walker 4-Spaces
Bejan and Drctȃ-Romaniuc investigated the total space of the tangent bundle TW 4 of a Walker 4-manifold (W 4 , g, D) and proved the following. Theorem 60. [40] Let (W 4 , g, D) be a Walker 4-manifold. Then the following three statements are equivalent: (a) The identity map id W 4 : (W 4 , g, D) → (W 4 ,ĝ, D) is harmonic, i.e., a Walker metricĝ on W 4 is harmonic with respect to g. (b) The Sasaki metricĝ S of a Walker metricĝ is harmonic with respect to the Sasaki metric g S of the Walker metric g.
(c) The horizontal liftĝ h of a Walker metricĝ is harmonic with respect to the horizontal lift g h of the Walker metric g.

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 (r, ϕ, z) for r ≥ 0, ϕ, z ∈ R (undetermined for r = 0). Further, H(r) and P(r) are arbitrary smooth functions on M and g is non-degenerate where P(r) = 0.
The following three theorems were proved by Zaeim, Jafari and Yaghoubi in [80].
Then the identity map id M 4 : (M 4 , g) → (M 4 ,ĝ) is harmonic (equivalently, the metric tensorĝ is harmonic with respect to g) if and only if H (r)(Ĥ(r) − H(r)) −P(r)P (r) + P(r)P (r) = 0. Theorem 62. Let (M 4 , g) be homogeneous Gödel-type spacetime. In this case, a metric tensorĝ is harmonic with respect to g if and only if g =ĝ. Theorem 63. Let (M 4 , g) be a Gödel-type spacetime with the metric g given by (77) and letĝ be an arbitrary Gödel-type metric of Equation (78). Then the following four statements are equivalent: (a) The identity map id M 4 : (M 4 , g) → (M 4 ,ĝ) is harmonic, i.e.,ĝ is harmonic with respect to g. (b) The Sasaki metricĝ S is harmonic with respect to the Sasaki metric g S . (c) The horizontal liftĝ h is harmonic with respect to the horizontal lift g h . (d) The complete liftĝ c is harmonic with respect to the complete lift g c .