A comprehensive survey on parallel submanifolds in Riemannian and pseudo-Riemannian manifolds

A submanifold of a Riemannian manifold is called a parallel submanifold if its second fundamental form is parallel with respect to the van der Waerden-Bortolotti connection. From submanifold point of view, parallel submanifolds are the simplest Riemannian submanifolds next to totally geodesic ones. Parallel submanifolds form an important class of Riemannian submanifolds since extrinsic invariants of a parallel submanifold do not vary from point to point. In this paper we provide a comprehensive survey on this important class of submanifolds.


Introduction
In Riemannian geometry, parallel transport is a way of transporting geometrical data along smooth curves in a Riemannian manifold. Following an important idea of T. Levi-Civita [1] in 1917, one can transport vectors of a Riemannian manifold along curves so that they stay parallel with respect to the Levi-Civita connection (or Riemannian connection). Afterwards, a general theory of parallel transportation of tensor fields in Riemannian geometry was studied in the 1920s by T. Levi-Civita, J. A. Schouten, J. D. Struik, H. Weyl, E. Cartan, B. L. van der Waerden and E. Bortolotti among others (cf. e.g., [2]).
For an immersed submanifold M of a Riemannian manifold (N,g), there exist two important symmetric tensor fields; namely, the first fundamental form which is the induced metric tensor field g of M and the second fundamental form h which is a normal bundle valued (1, 2)-tensor field.
It is well known that the first fundamental form g is a parallel tensor field with respect to the Levi-Civita connection. The submanifold M is called a parallel submanifold if its second fundamental form h is a parallel tensor field with respect to the van der Waerden-Bortolotti connection. Thus the extrinsic invariants of a parallel submanifold M do not vary from point to point. Obviously, parallel submanifolds are natural extensions of totally geodesic submanifolds for which the second fundamental form vanishes identically.
Parallel surfaces in a Euclidean 3-space E 3 are classified in 1948 by V. F. Kagan in [3]. Kagan's result states that open parts of planes E 2 , of spheres S 2 and of round cylinders S 1 × E 1 are the only parallel surfaces in E 3 . For n > 2, parallel hypersurfaces in Euclidean spaces are classified by U. Simon and A. Weinstein in [4]. A general classification theorem of parallel submanifolds in any Euclidean space is archived in 1974 by D. Ferus [5]. Since then the study of parallel submanifolds became a very interesting and important research subject in differential geometry.
In this paper, we provide a comprehensive survey on this important subject in differential geometry from classical results to the most recent ones.

Preliminaries
An immersion from a manifold M into a pseudo-Riemannian manifold (N,g) is called a pseudo-Riemannian submanifold if the induced metric g on M is a pseudo-Riemannian metric. For a pseudo-Riemannian submanifold M of N , let ∇ and∇ be the Levi-Civita connection of g andg, respectively. Let us denote the Riemann curvature tensors of M and N by R andR, respectively, and let , denote the associated inner product for both g andg. A pseudo-Riemannian manifold is called a Lorentzian manifold if its index is one at each point.
A tangent vector v of a pseudo-Riemannian manifold is called space-like (respectively, time-like) if v = 0 or v, v > 0 (respectively, v, v < 0). A vector v is called light-like or null if v, v = 0 and v = 0. A pseudo-Riemannian submanifold M is called spatial (or space-like) if each tangent vector vector of M is space-like.
A submanifold M of a pseudo-Riemannian manifold is called non-degenerate if the induced metric on M is non-degenerate. In particular, a non-degenerate surface of a pseudo-Riemannian manifold is either spatial or Lorentzian. Throughout this article, we assume that every parallel surface M is non-degenerate, i.e., the induced metric on M is non-degenerate.

Basic definitions, formulas and equations. The formulas of Gauss and
Weingarten of a pseudo-Riemannian submanifold M of a pseudo-Riemannian manifold (N,g) are given respectively by (cf. [6,7,8]) for vector fields X, Y tangent to M and ξ normal to M , where h, A and D are the second fundamental form, the shape operator and the normal connection of M . The shape operator and the second fundamental form are related bỹ g(h(X, Y ), ξ) = g(A ξ X, Y ) for vector fields X, Y tangent to M and ξ normal to M . The equations of Gauss, Codazzi and Ricci of M in N are given respectively by for vectors X, Y, Z, W tangent to M and vector ξ, η normal to M , where R D is the normal curvature tensor defined by and∇h denotes the covariant derivative of h with respect to the van der Waerden-Bortolotti connection∇ = ∇ ⊕ D, defined by The mean curvature vector H of M in N is given by where {e 1 , . . . , e n } is an orthonormal frame of M such that e j , e k = ǫ j δ jk .
The relative null subspace N p of a pseudo-Riemannian submanifold M in N at p ∈ M is defined by The dimension ν p of N p is called the relative nullity at p.
The first normal space at a point p of a pseudo-Riemannian submanifold M iñ M is, by definition, the image space, Im h(p), of the second fundamental form of M at p, i.e., Im h(p) = {h(X, Y ) : X, Y ∈ T p M }.

2.2.
Indefinite real space forms. Let (N,g) be a pseudo-Riemannian manifold. At a point p ∈ N , a 2-dimensional linear subspace π of the tangent space T p N is called a plane section. For a given basis {v, w} of a plane section π, we define a real number by A plane section π is called nondegenerate if Q(u, v) = 0. For a nondegenerate plane section π ⊂ T p N at p, the number is independent of the choice of basis {u, v} for π and is called the sectional curvaturẽ K(π) of π.
A pseudo-Riemannian manifold is said to have constant curvature if its sectional curvature function is constant. It is well known that if a pseudo-Riemannian manifold N is of constant curvature c, then its curvature tensorR satisfies R(X, Y )Z = c{ Y, Z X − X, Z Y }.
Example 2.1. (see, e.g., [6]) Let E n t denote the pseudo-Euclidean n-space equipped with the canonical pseudo-Euclidean metric of index t given by where (u 1 , . . . , u n ) is a rectangular coordinate system of E n t . For a nonzero real number c, we put H k (c) = x ∈ E k+1 1 : x, x = c −1 < 0 and x 1 > 0 , (15) where , is the associated scalar product. S k s (x 0 , c) and H k s (x 0 , c) are pseudo-Riemannian manifolds of constant curvature c with index s, known as a pseudosphere and a pseudo-hyperbolic space, respectively. The point x 0 is called the center of S m s (x 0 , c) and H m s (x 0 , c). If x 0 is the origin o of the pseudo-Euclidean spaces, we denote S k s (o, c) and H k s (o, c) by S k s (c) and H k s (c), respectively. The pseudo-Riemannian manifolds E k s , S k s (c), H k s (c) are the standard models of the indefinite real space forms. In particular, E k 1 , S k 1 (c), H k 1 (c) are the standard models of Lorentzian space forms.
The Riemannian manifolds E k , S k (c) and H k (c) (with s = 0) are of constant curvature, called real space forms. The Euclidean k-space E k , the k-sphere S k (c) and the hyperbolic k-space H k (c) are complete simply-connected Riemannian manifolds of constant curvature 0, c > 0 and c < 0, respectively. A complete simply-connected pseudo-Riemannian k-manifold, k ≥ 3, of constant curvature c and with index s is isometric to E k s , or S k s (c) or H k s (c) according to c = 0, or c > 0 or c < 0, respectively. In the following, we denote a k-dimensional indefinite space form of constant curvature c and index s by R k s (c). Also we denote an indefinite space form R k 0 (c) (with index s = 0) simply by R k (c).
For a pseudo-Riemannian submanifold M of a pseudo-Riemannian manifold R k s (c) of constant curvature c with index s, the equations of Gauss, Codazzi and Ricci reduce to (see, e.g., [6]) for vectors X, Y, Z, W tangent to M and ξ, η normal to M .
2.3. Gauss image. The classical Gauss map of a surface in E 3 was introduced by C. F. Gauss in his fundamental paper on the theory of surfaces [9]. He used it to define the Gauss curvature. Since then Gauss maps became one of the important tools in differential geometry. The classical Gauss map can be extended to arbitrary Euclidean submanifolds as follows: Let G(n, m − n) denote the Grassmann manifold consisting of linear n-subspaces of E m . Then the Grassmann manifold G(n, m − n) admits a canonical Riemannian metric via Plücker embedding which makes G(n, m−n) into a symmetric space. For an n-dimensional submanifold M of E m , the Gauss map Γ of M in E m is defined to be the mapping Γ : M → G(n, m − n) which carries a point p ∈ M into the linear n-subspace of E m obtained via the parallel displacement of the tangent space [11,15]). In the following, we shall assume that the Gauss maps are regular maps.
The following result of B.-Y. Chen and S. Yamaguchi in [11] provides a simple characterization of Euclidean submanifolds having totally geodesic Gauss image.
for any vector fields X, Y, Z tangent to M , where ∇ is the Levi-Civita connection of M and ∇ G is the Levi-Civita connection of the Gauss image with the induced metric via Γ.

Some general properties of parallel submanifolds
In this section, we present some basic properties of parallel submanifolds.  The following properties of parallel submanifolds are also well-known. Lemma 3.2. Every parallel submanifold M of a Riemannian manifold (N,g) has constant relative nullity, i.e., the dimension of the relative null subspace is constant. Further, every parallel submanifold in E m is of finite type in the sense of Chen (cf. e.g., [12,13,14]). Also, if a rank one compact symmetric space N is regarded as a submanifold of a Euclidean space E m via its first standard embedding, then any parallel submanifold of N via its first standard embedding is of finite type in E m (see, e.g., [13,14]).

Parallel submanifolds of Euclidean spaces
In this section, we present basic properties, characterizations and classification of parallel submanifolds of Euclidean spaces. 4.1. Gauss map and parallel submanifolds. As before, let G(n, m − n) denote the Grassmann manifold of n-planes through the origin in E m endowed with its natural Riemannian symmetric space metric and let G(E m ) denote the group of Euclidean motions on E m .
The following result was obtained by J. Vilms in [10].
If M is complete, then we have: (i) If the relative nullity ν = 0, then M is a complete totally geodesic submanifold of G(n, m − n).
is a complete totally geodesic submanifold of G(n, m − n) and the fibres are the leaves of the relative nullity foliation. The metric of M is composed from those on base and fibre, and the fibration admits an integrable connection with totally geodesic horizontal leaves (i.e. it is a totally geodesic Riemannian submersion). (iii) The original Riemannian connection of M , or its projection onto B, respectively, coincides with the connection induced from G(n, m − n). (iv) M has nonnegative curvature, and is locally symmetric.
As an application of Theorem 2.1, Chen and S. Yamaguchi [11] classified surfaces with totally geodesic Gauss image as follows. Another application of Theorem 2.1 is the following result of Chen and Yamaguchi obtained in [15].

4.2.
Normal sections and parallel submanifolds. Let M be an n-dimensional submanifold in a Euclidean m-space E m . For a given point p ∈ M and a given unit vector t at p tangent to M , the vector t and the normal space T ⊥ p M of M determine an (m − n + 1)-dimensional subspace E(p, t) in E m . The intersection of M and E(p, t) gives a curve γ t (in a neighborhood of p) which is called the normal section of M at p in the direction t (cf. [8,17,18]). In general, the normal section γ t is a space curve in E(p, t).
For normal sections, Chen proved the following result in [8,17].
for any unit vector t tangent to M .
An immediate consequence of this theorem is the following. By a vertex of a planar curve γ(s) we mean a point x on the curve such that the curvature function κ(s) of γ satisfies dκ 2 ds = 0 at x. Another application of Theorem 4.5 is the following simple geometric characterization of parallel submanifolds obtained by Chen in [8,17].
Theorem 4.7. An n-dimensional (n > 2) submanifold M of a Euclidean space is a parallel submanifold if and only if, for each p ∈ M , each normal section of M at any point p is a planar curve with p as one of its vertices.
For further applications of normal sections, see e.g., [18,20,19,21,22,23,24,25,26,27,28] 4.3. Symmetric submanifolds and parallel submanifolds. The notion of extrinsic symmetric submanifolds was defined by D. Ferus in [29]. More precisely, an isometric immersion ψ : M → E m is called extrinsic symmetric if for each p ∈ M there exists an isometry φ of M into itself such that φ(p) = p and ψ • φ = σ p • ψ, where σ p denotes the reflection at the normal space T ⊥ p M at p, i.e., the motion of E m which fixes the space through ψ(p) normal to ψ * (T p M ) and reflects ψ(p)+ψ * (T p M ) at ψ(p). The immersed submanifold ψ : M → E m is said to be extrinsic locally symmetric if each point p ∈ M has a neighborhood U and an isometry φ of U into itself such that φ(p) = p and ψ • φ = σ p • ψ on U . In other words, a submanifold M of E m is extrinsic locally symmetric if each point p ∈ M has a neighborhood which is invariant under the reflection of E m with respect to the normal space at p. D. Ferus [29] proved the following result. Symmetric submanifolds were classified completely by D. Ferus in [5] as being a very special class of orbits of isotropy representations of semisimple symmetric spaces. For some symmetric spaces N , a distinguished class of isotropy orbits (the symmetric R-spaces) are symmetric spaces. They are symmetric submanifolds in the corresponding tangent space T o N of N . If N is non-compact, the projection of these symmetric submanifolds from T o N into N via the exponential map at o provides examples of symmetric submanifolds in N .
In [30], J. Berndt et. al. extended these symmetric submanifolds to larger oneparameter families of symmetric submanifolds, and proved that if N is irreducible and of rank greater than or equal to 2, then every symmetric submanifold of N arises in this way. This result yields the full classification of symmetric submanifolds in Riemannian symmetric spaces. For symmetric submanifolds in non-flat Riemannian manifolds of constant curvature, see [31,32,33].

4.4.
Extrinsic k-symmetric submanifolds as ∇ c -parallel submanifolds. A canonical connection on a Riemannian manifold (M, g) is defined as any metric connection ∇ c on M such that the difference tensorD between ∇ c and the Levi-Civita connection ∇ of (M, g) is ∇ c -parallel. An embedded submanifold M of E m is said to be extrinsic homogeneous with constant principal curvatures if, for any given p, q ∈ M and a given piecewise differentiable curve γ from p to q, there is an isometry C. Olmos and C. Sánchez extended Ferus' result in [34] to the following.
Theorem 4.9. Let M be a compact Riemannian submanifold fully in E m and let h be its second fundamental form. Then the following three statements are equivalent: (1) M admits a canonical connection ∇ c such that ∇ c h = 0, (2) M is an extrinsic homogeneous submanifold with constant principal curvatures, (3) M is an orbit of an s-representation, that is, of an isotropy representation of a semisimple Riemannian symmetric space.
Furthermore, C. Sánchez defined in [35] the notion of extrinsic k-symmetric submanifolds of E m and classified such submanifolds for odd k. Moreover, he proved in [36] that the extrinsic k-symmetric submanifolds are essentially characterized by the property of having parallel second fundamental form with respect to the canonical connection of k-symmetric space. In particular, the above result implies that every extrinsic k-symmetric submanifold of a Euclidean space is an orbit of an s-representation.

5.
Symmetric R-spaces and parallel submanifolds of real space forms Symmetric spaces are the most beautiful and important Riemannian manifolds. Such spaces arise in a wide variety of situations in both mathematics and physics. This class of spaces contains many prominent examples which are of great importance for various branches of mathematics, like compact Lie groups, Grassmannians and bounded symmetric domains. Symmetric spaces are also important objects of study in representation theory, harmonic analysis as well as in differential geometry.
We refer to [37,38,39,40,41] for general information on compact symmetric spaces. 5.1. Symmetric R-spaces and Borel subgroups. An isometry s of a Riemannian manifold is called an involutive if s 2 = id. A Riemannian manifold M is called a symmetric space if for each p ∈ M there is an involutive isometry s p such that p is an isolated fixed point of s p ; the involutive isometry s p = id is called the symmetry at p.
Let M be a symmetric space. Denote by G = G M the closure of the group of isometries on M generated by {s p : p ∈ M } in the compact-open topology. Then G is a Lie group which acts transitively on the symmetric space. Thus the typical isotropy subgroup K, say at a point o ∈ M , is compact and M = G/K. Let I 0 (M ) denote the connected group of isometries of a compact symmetric Riemannian manifold M .
A symmetric R-space is a special type of compact symmetric space for which several characterizations were known. Originally in 1965, T. Nagano defined in [42] a symmetric R-space as a compact symmetric space M which admits a Lie transformation group P which is non-compact and contains the identity component of the isometric group I 0 (M ) of M as a subgroup.
In the theory of algebraic groups, a Borel subgroup of an algebraic group G is a maximal Zariski closed and connected solvable algebraic subgroup (cf. [43,44]). Subgroups between a Borel subgroup B and the ambient group G are called parabolic subgroups. Working over algebraically closed fields, the Borel subgroups turn out to be the minimal parabolic subgroups in this sense. Thus B is a Borel subgroup when the homogeneous space G/B is a complete variety which is as large as possible.
In 1965, M. Takeuchi used the terminology symmetric R-space for the first time in [45]. He gave a cell decomposition of an R-space in [45], which is a kind of generalization of a symmetric R-space. Here, by an R-space we mean M = G/U where G is a connected real semisimple Lie group without center and U is a parabolic subgroup of G. A compact symmetric space M is said to have a cubic lattice if a maximal torus of M is isometric to the quotient of E r by a lattice of E r generated by an orthogonal basis of the same length.
In 1985, O. Loos [46] provided another intrinsic characterization of symmetric R-spaces which states that a compact symmetric space M is a symmetric R-spaces if and only if the unit lattice of the maximal torus of M is a cubic lattice. The proof of Loos is based on the correspondence between the symmetric R-spaces and compact Jordan triple systems.

5.2.
Classification of symmetric R-spaces. An affine subspace of E m or a symmetric R-space M ⊂ E m , which is minimally embedded in a hypersphere of E m as described in [47] by M. Takeuchi and S. Kobayashi, is a parallel submanifold of E m . The class of symmetric R-spaces includes (see [47] (f) the Cayley projective plane OP 2 , and (g) the three exceptional spaces E 6 /Spin(10) × T, E 7 /E 6 × T, and E 6 /F 4 .

Ferus' theorem.
A classification theorem of parallel submanifolds in Euclidean spaces was obtained in 1974 by D. Ferus [5]. He proved that essentially these submanifolds mentioned above exhaust all parallel submanifolds of E m in the following sense.  For parallel submanifolds of spaces of constant curvature, see also [48,49].

5.5.
Parallel submanifolds in hyperbolic spaces. Parallel submanifolds of a hyperbolic space were classified by M. Takeuchi [49] in 1981 as follows.
(1) If M is not contained in any complete totally geodesic hypersurface of H m (c), then M is congruent to the product is a parallel submanifold as described in Ferus' result.
(2) If M is contained in a complete totally geodesic hypersurface N of H m (c), then N is isometric to an (m − 1)-sphere, or to a Euclidean (m − 1)-space, or to a hyperbolic (m − 1)-space. Consequently, such parallel submanifolds reduce to the parallel submanifolds described before.
B.-Y. Chen [50] and Chen and W. E. Kuan [51,52] proved the following simple characterization for Segre embeddings for n = 2 and for n ≥ 3, respectively (see, also [53,55,56,54]). A complex projective n-space CP n (c) of constant holomorphic sectional curvature c can be holomorphically isometrically embedded into an n+ν ν −1 -dimensional complex projective space of constant holomorphic sectional curvature µc as which is called the ν-th Veronese embedding of CP n (c). The degree of the ν-th Veronese embedding is ν (cf. e.g., page 83 of [57]). The Veronese embeddings were characterized by A. Ros [58] in terms of holomorphic sectional curvature H in the following result.
then M = CP n ( c ν ) and the immersion is given by the ν-th Veronese embedding. 6.2. Classification of parallel Kaehler submanifolds of CP m and CH m . In 1972, K. Ogiue classified parallel complex space forms in complex space forms in [59]. More precisely, he proved the following. Theorem 6.3. Let M n (c) be a complex space form holomorphically isometrically immersed in another complex space form M m (c). If the second fundamental form of the immersion is parallel, then either the immersion is totally geodesic orc > 0 and the immersion is given by the second Veronese embedding.
All complete parallel Kaehler submanifolds of a complex projective space were classified by H. Nakagawa and R. Tagaki [65] in 1976 (also [60] by M. Takeuchi in 1978). If M is reducible, then M is congruent to CP n1 × CP n2 with n = n 1 + n 2 and the embedding is given by the Segre embedding.
On the other hand, M. Kon [61] proved in 1974 the following result for parallel Kaehler submanifolds in complex hyperbolic spaces. Theorem 6.5. Every parallel Kaehler submanifold of CH m (−4) is totally geodesic.
6.3. Parallel Kaehler submanifolds of Hermitian symmetric spaces. Parallel submanifolds of Hermitian symmetric spaces were studied in 1985 by K. Tsukada [62] as follows.
Theorem 6.6. Let φ : M →M be a parallel Kaehler immersion of a connected complete Kaehler manifold M into a simply connected Hermitian symmetric spacẽ M . Then M is the direct product of a complex Euclidean space and semisimple Hermitian symmetric spaces. Moreover, φ = φ 2 • φ 1 , where φ 1 is a direct product of identity maps and (not totally geodesic) parallel Kaehler embeddings into complex projective spaces, and φ 2 is a totally geodesic Kaehler embedding.
All non-totally geodesic parallel Kaehler embeddings into complex projective spaces have been classified earlier by H. Nakagawa and R. Takagi [65] in 1976. More precisely, these are Veronese maps and Segre maps applied to complex projective spaces, and the first standard embeddings applied to rank two compact irreducible Hermitian symmetric spaces.
6.4. Parallel Kaehler manifolds in complex Grassmannian manifolds. Let G C (n, p) denote the complex Grassmannian manifold of complex p-planes in C n . We denote by S → G C (n, p) the tautological vector bundle over G C (n, p) (cf. e.g., [63]). Since the taulogical bundle S → G C (n, p) is a subbundle of a trivial bundle G C (n, p) × C n → G C (n, p), one has the quotient bundle Q → G C (n, p), which is called the universal quotient bundle.
The holomorphic tangent bundle T 1,0 (G C (n, p)) over G C (n, p) can be identified with the tensor product of holomorphic vector bundles S * and Q, where S * → G C (n, p) is the dual bundle of S → G C (n, p). If C n has a Hermitian inner product, S, Q have Hermitian metrics and Hermitian connections and so G C (n, p) has a Hermitian metric induced by the identification of T 1,0 (G C (n, p)) and S * ⊗ Q is called the standard metric on G C (n, p).
In [64], I. Koga and Y. Nagatomo proved the following result for parallel Kaehler manifolds in a complex Grassmannian manifold. Theorem 6.7. Let G C (n, p) be the complex Grassmannian manifold of complex pplanes in C n with the standard metric h Gr induced from a Hermitian inner product on C n and φ be a holomorphic isometric immersion of a compact Kaehler manifold (M, h M ) with a Hermitian metric h M into G C (n, p). We denote by Q → G C (n, p) the universal quotient bundle over G C (n, p) of rank n−p. Assume that the pull-back bundle of Q → G C (n, p) is projectively flat. Then φ has parallel second fundamental form if and only if the holomorphic sectional curvature of M is greater than or equal to 1. H. Naitoh [67] proved in 1981 that the classification of complete totally real parallel submanifolds in complex projective spaces is reduced to that of certain cubic forms of n-variables. Further, H. Naitoh and M. Takeuchi [68] classified in 1982 these submanifolds by the theory of symmetric bounded domains of tube type.
Theorem 7.2. A parallel totally real submanifold of a complex space formM n (c) with c = 0 is either a totally real submanifold which is contained in a totally real totally geodesic submanifold, or a totally real submanifold which is contained in a totally geodesic Kaehler submanifold whose dimension is twice of the dimension of the submanifold.
The classifications of Naitoh and Naitoh-Takeuchi given above rely heavily on the theory of Lie groups and symmetric spaces. Remark 1. Theorem 7.2 implies that the classification of complete parallel submanifolds of complex projective space CP m (c) is reduced to those of D. Ferus [29] and H. Naitoh and M. Takeuchi [68].

Remark 2.
For parallel totally real submanifolds in a complex hyperbolic space CH m , Theorem 7.2 implies that the classification reduces to those of M. Takeuchi [49].

7.2.
Parallel Lagrangian submanifolds of CP n . F. Dillen, H. Li, L. Vrancken and X. Wang gave in [71] explicitly and geometrically classification of parallel Lagrangian submanifolds in CP n (4) using a different method, which applies the warped products of Lagrangian immersions, called Calabi products, and the characterization of parallel Lagrangian submanifolds by Calabi products. For the definition of Calabi products and their characterization, see, e.g., [72,73].
The advantage of this classification given by Dillen et. al. is that it allows the study of details for these submanifolds, in particular, for their reduced cases. The classification theorem they obtained is as follows: Let M be a parallel Lagrangian submanifold in CP n (4). Then either M is totally geodesic, or (1) M is locally the Calabi product of a point with a lower-dimensional parallel Lagrangian submanifold; (2) M is locally the Calabi product of two lower-dimensional parallel Lagrangian submanifolds; or (3) M is congruent to one of the following symmetric spaces: 7.3. Parallel surfaces of CP 2 and CH 2 . For the explicit classification of parallel surfaces in CP 2 (see [74]).
Theorem 7.4. If M is a parallel surface in the complex projective plane CP 2 (4), then it is either holomorphic or Lagrangian in CP 2 (4).
(a) If M is holomorphic, then locally either (a.1) M is a totally geodesic complex projective line M is a flat surface and the immersion is congruent to π • L, where π : where a and b are real numbers with a = 0.
For parallel surfaces in CH 2 , we have the following result from [74].
7.4. Parallel totally real submanifolds in nearly Kaehler S 6 . Let O denote the Cayley numbers. E. Calabi [75] showed in 1958 that any oriented submanifold M 6 of the hyperplane Im O of the imaginary octonions carries a U (3)-structure, i.e., an almost Hermitian structure J.
The almost Hermitian structure J on S 6 (1) ⊂ Im O is a nearly Kaehler structure in the sense that the (2,1)-tensor field G on S 6 (1), defined by G(X, Y ) = ( ∇ X J)(Y ), is skew-symmetric, where ∇ is the Riemannian connection on S 6 (1). The group of automorphisms of this nearly Kähler structure is the exceptional simple Lie group G 2 which acts transitively on S 6 as a group of isometries.
In 1969, A. Gray proved in [76] the following.
N. Ejiri proved in [77] that a 3-dimensional totally real submanifold of the nearly Kaehler S 6 (1) is minimal and orientable It was proved by B. Opozda in [78] that every 3-dimensional parallel Lagrangian submanifold (respectively, a 2-dimensional totally real and minimal submanifold) of the nearly Kaehler S 6 (1) is totally geodesic (see also [79]). Opozda also proved in [78] that a 2-dimensional parallel totally real, minimal surface of the nearly Kaehler S 6 (1) is also totally geodesic. The same result holds for Lagrangian submanifolds of the nearly Kaehler S 3 × S 3 ; namely, a (3-dimensional) parallel Lagrangian submanifold of the nearly Kaehler S 3 × S 3 is totally geodesic (see, e.g., B. Dioos's PhD thesis [80]). 8. Parallel slant submanifolds of complex space forms 8.1. Basics on slant submanifolds. Besides Kaehler and totally real submanifolds in a Kaehler manifoldM , there is another important family of submanifolds, called slant submanifolds (cf. [81,82]).
Let N be a submanifold of a Kähler manifold (or an almost Hermitian manifold) (M, J, g). For any vector X tangent to M , we put where P X and F X denote the tangential and the normal components of JX, respectively. Then P is an endomorphism of the tangent bundle T N . For any nonzero vector X ∈ T p N at p ∈ N , the angle θ(X) between JX and the tangent space T p N is called the Wirtinger angle of X.
In 1990, the author [81] introduced the notion of slant submanifolds as follows. Complex submanifolds and totally real submanifolds are exactly θ-slant submanifolds with θ = 0 and θ = π 2 , respectively. A slant submanifold is called proper slant if it is neither complex nor totally real.
The following basic result on slant submanifolds was proved in [83] by Chen and Y. Tazawa. For higher dimensional parallel slant submanifolds, we have the following result by applying Theorem 6.7, the list of symmetric R-spaces and Ferus' Theorem. For further results on slant submanifolds, see, e.g., [6,82,85,86,87].
9. Parallel submanifolds of quaternionic space forms and Cayley plane 9.1. Parallel submanifolds of quaternionic space forms. K. Tsukada [88] classified in 1985 all parallel submanifolds of a quaternionic projective m-space HP m . Tsukada's results states that such submanifolds are either parallel totally real submanifolds in a totally real totally geodesic submanifold RP m , or parallel totally real submanifolds in a totally complex totally geodesic submanifold CP m , or parallel complex submanifolds in a totally complex totally geodesic submanifold CP m , or parallel totally complex submanifolds in a totally geodesic quaternionic submanifold HP k whose dimension is twice the dimension of the parallel submanifold. In [88], K. Tsukada also classified parallel submanifolds of the non-compact dual of HP m .

Parallel submanifolds of the Cayley plane.
A result of K. Tsukada [89] in 1985 states that parallel submanifolds of the Cayley plane OP 2 are contained either in a totally geodesic quaternion projective plane HP 2 as parallel submanifolds or in a totally geodesic 8-sphere as parallel submanifolds. Hence, all these immersions are completely known.
The non-compact case is treated in a similar way.

Parallel spatial submanifolds in pseudo-Euclidean spaces
The first classification result of parallel submanifolds in indefinite real space forms was given by M. A. Magid [90] in 1984 in which he classified parallel im- He showed that such immersions are either quadratic in nature, like the flat umbilical immersion with light-like mean curvature vector, or the product of the identity map and previously determined low dimensional maps. In this section we survey known results on parallel pseudo-Riemannian submanifolds in indefinite real space forms.
First we recall the next lemma which is an easy consequence of Erbacher-Magid's reduction theorem (see Lemma 3.1 of [92]).
where k is the dimension of the first normal spaces.
10.1. Marginally trapped surfaces. Now, we recall the notion of marginally trapped surfaces for later use.
The concept of trapped surfaces, introduced R. Penrose in [91] plays very important role in the theory of cosmic black holes. If there is a massive source inside the surface, then close enough to a massive enough source, the outgoing light rays may also be converging; a trapped surface. Everything inside is trapped. Nothing can escape, not even light. It is believed that there will be a marginally trapped surface, separating the trapped surfaces from the untrapped ones, where the outgoing light rays are instantaneously parallel. The surface of a black hole is the marginally trapped surface. As times develops, the marginally trapped surface generates a hypersurface in spacetime, a trapping horizon.
Spatial surfaces in pseudo-Riemannian manifolds play important roles in mathematics and physics, in particular in general relativity theory. For instance, a marginally trapped surface in a spacetime is a spatial surface with light-like mean curvature vector field.
In this article, we also call a Lorentzian surfaces in a pseudo-Riemannian manifold marginally trapped (or quasi-minimal ) if it has light-like mean curvature vector field. A nondegenerate surface in a pseudo-Riemannian manifold is called trapped (respectively, untrapped ) if it has time-like (respectively, space-like) mean curvature vector field.

10.2.
Classification of parallel spatial surfaces in E m s . In this subsection, we provide the classification of parallel spatial surfaces in indefinite space forms with arbitrary index and arbitrary dimension obtained by Chen in [92] as follows.
, or  (1) the Euclidean plane E 2 given by (0, u, v); Remark 3. The surfaces (1) is totally geodesic, the surfaces (2) is totally umbilical but not totally geodesic and surfaces (1) and (3) are products of parallel curves in totally geodesic subspaces.
11. Parallel spatial surfaces in S m s 11.1. Classification of parallel spatial surfaces in S m s . For parallel spatial surfaces in a pseudo-sphere S m s , we have the following classification theorem proved in [92]. (1) a totally geodesic 2-sphere s as r cosh u, 0, . . . , 0, r sinh u cos v, r sinh u sin v, 1 + r 2 , r, s > 0; , 0, . . . , 0, , 0, . . . , 0, . . , f 1 ), where φ is a surface given by (4), (5) or (9)-(17) from (A) and f 1 , . . . , f ℓ are polynomials of degree ≤ 2 in u, v, or 11.2. Special case: parallel spatial surfaces in S 3 1 . For parallel spatial surfaces in a de Sitter space-time S 3 1 , Theorem 11.1 implies the following. Corollary 11.1. If M is a parallel spatial surface in S 3 1 (1) ⊂ E 4 1 , then M is congruent to one of the following ten types of surfaces: (1) a totally umbilical sphere S 2 given locally by (a, b sin u, b cos u cos v, b cos u sin v), b 2 − a 2 = 1; (2) a totally umbilical hyperbolic plane H 2 given by (a cosh u cosh v, a cosh u sinh v, a sinh u, b) with b 2 − a 2 = 1; (3) a flat surface H 1 × S 1 given by (a cosh u, a sinh u, b cos v, b sin v) with a 2 + b 2 = 1.
12. Parallel spatial surfaces in H m s 12.1. Classification of parallel spatial surfaces in H m s . For parallel spatial surfaces in a pseudo-hyperbolic space H m s , we have the following classification theorem also proved in [92]. (1) a totally geodesic (4) a totally umbilical S 2 immersed in H m s (−1) ⊂ E m+1 s+1 as 1 + r 2 , 0, . . . , 0, r sin u, r cos u cos v, r cos u sin v , r > 0; with r < 1 and s ≥ 3, or . . , f ℓ are polynomials of degree ≤ 2 in u, v and φ is a surface given by (5), (7), (8) or (11)

A parallel spatial surfaces in
There is a famous minimal immersion of the 2-sphere S 2 ( 1 3 ) of curvature 1 3 into the unit 4-sphere S 4 (1), known as the Veronese surface, which is constructed by using spherical harmonic homogeneous polynomials of degree two defined as It is well known that the Veronese surface is the only minimal parallel surface lying fully in S 4 (1) (see, e.g., [93,94,95]). On the other hand, it was also known that there does not exist minimal surface of constant Gauss curvature lying fully in the hyperbolic 4-space H 4 (−1) (cf. [94,95,96]). Furthermore, it was known from [97] that there exist no minimal spatial parallel surfaces lying fully in H 4 1 (−1). B.-Y. Chen discovered in [98] a minimal immersion of the hyperbolic plane H 2 (− 1 3 ) of Gauss curvature − 1 3 into the unit neutral pseudo-hyperbolic 4-space H 4 2 (−1) as follows: The following map B : R 2 → E 5 3 : was introduced in [98]. It is direct to verify that the position vector field x of B satisfies x, x = −1 and the induced metric is given by g = ds 2 + e (1) a hyperbolic plane H 2 defined by (a, b cosh u cosh v, b cosh u sinh v, b sinh u), a 2 + b 2 = 1; (2) a surface H 1 × H 1 defined by (a cosh u, b cosh v, a sinh u, b sinh v), a 2 + b 2 = 1.

Parallel Lorentz surfaces in pseudo-Euclidean spaces
Lorentzian geometry is a vivid field that represents the mathematical foundation of the general theory of relativity, which is probably one of the most successful and beautiful theories of physics. An interesting phenomenon for Lorentzian surfaces in Lorentzian Kaehler surfaces states that Ricci equation is a consequence of Gauss and Codazzi equations (see [99]). This indicates that Lorentzian surfaces have many interesting properties which are different from surfaces in Riemannian manifolds. In particular, Lorentzian surfaces in indefinite real space forms behaved differently from surfaces in Riemannian space forms. For instance, the family of minimal surfaces in Euclidean spaces is huge (see, e.g., chapter 5 of [94]). In contrast, all Lorentzian minimal surfaces in a pseudo-Euclidean m-space E m s was completely classified in [100] (see also [101]) as the following. (1) a totally geodesic plane E 2 1 ⊂ E m s given by (x, y) ∈ E 2 1 ⊂ E m s ; (2) a totally umbilical de Sitter space S 2 1 in a totally geodesic E 3 1 ⊂ E m s given by (sinh x, cosh x cos y, cosh x sin y); (3) a flat cylinder E 1 1 ×S 1 in a totally geodesic E 3 1 ⊂ E m s given by x, cos y, sin y ; (4) a flat cylinder S 1 1 ×E 1 in a totally geodesic E 3 1 ⊂ E m s given by sinh x, cosh x, y ; (5) a flat minimal surface in a totally geodesic E 3 1 ⊂ E m s given by (6) a flat surface S 1 1 ×S 1 in a totally geodesic E 4 1 ⊂ E m s given by a sinh x, a cosh x, b cos y, b sin y , with a, b > 0; (7) an anti-de Sitter space H 2 1 in a totally geodesic E 3 2 ⊆ E m s given by (sin x, cos x cosh y, cos x sinh y); (8) a flat minimal surface in a totally geodesic E 3 2 ⊆ E m s defined by − y , a > 0; (9) a non-minimal flat surface in a totally geodesic E 3 2 ⊆ E m s defined by , a, b > 0; (10) a non-minimal flat surface in a totally geodesic E 3 2 ⊆ E m s defined by (14) a non-minimal flat surface in a totally geodesic E 4 3 ⊆ E m s defined by   (1), (3) and (4) are products of parallel curves in totally geodesic subspaces; the surface (5) is flat and minimal, but not totally geodesic.

Parallel surfaces in a light cone LC
The light cone LC of a pseudo-Euclidean (n + 1)-space E n+1 s is defined by A curve in a pseudo-Riemannian manifold is called a null curve if its velocity vector is a lightlike at each point.
14.1. Light cones in general relativity. In physics, a space-time is a timeoriented 4-dimensional Lorentz manifold. As with any time-oriented spacetime, the time-orientation is called the future, and its negative is called the past. A tangent vector in a future time-cone is called future-pointing. Similarly, a tangent vector in the past time-cone is called past-pointing.
Light cones play a very important role in general relativity. Since signals and other causal influences cannot travel faster than light, the light cone plays an essential role in defining the concept of causality: for a given event E, the set of events that lie on or inside the past light cone of E would also be the set of all events that could send a signal that would have time to reach E and influence it in some way. Likewise, the set of events that lie on or inside the future light cone of E would also be the set of events that could receive a signal sent out from the position and time of E, so the future light cone contains all the events that could potentially be causally influenced by E. Events which lie neither in the past or future light cone of E cannot influence or be influenced by E in relativity. (1) a totally umbilical surface of positive curvature given by a(1, cos u cos v, cos u sin v, sin u), a > 0; (2) totally umbilical surface of negative curvature given by a(cosh u cosh v, cosh u sinh v, sinh u, 1), a > 0; (3) a flat totally umbilical surface given by (1) a totally umbilical surface of positive curvature given by a(sinh u, 1, cosh u cos v, cosh u sin v), a > 0; (2) a totally umbilical surface of negative curvature given by a(sin u, cos u cosh v, 1, cos u sinh v), a > 0;

Parallel surfaces in
(3) a totally umbilical flat surface defined by The geometry of 4-dimensional space-time is much more complex than that of 3-dimensional space, due to the extra degree of freedom. Four-dimensional spacetimes play extremely important roles in the theory of relativity. In physics, spacetime is a mathematical model that combines space and time into a single continuum. Space-time is usually interpreted with space being three-dimensional and time playing the role of a fourth dimension. By combining space and time into a single manifold, physicists have significantly simplified a large number of physical theories, as well as described in a more uniform way the workings of the universe at both the super-galactic and subatomic levels.
In recent times, physics and astrophysics have played a central role in shaping the understanding of the universe through scientific observation and experiment. After Kaluza-Klein's theory, the term space-time has taken on a generalized meaning beyond treating space-time events with the normal 3+1 dimensions. It becomes the combination of space and time. Some proposed space-time theories include additional dimensions, normally spatial, but there exist some speculative theories that include additional temporal dimensions and even some that include dimensions that are neither temporal nor spatial. How many dimensions are needed to describe the Universe is still a big open question.
15.1. Classification of parallel spatial surfaces in de Sitter space-time S 4 1 . For parallel spatial surfaces in the de Sitter space-time S 4 1 (1), we have the following classification theorem proved by Chen and Van der Veken in [97].
, then M is congruent to one of the following ten types of surfaces: (1) a totally umbilical sphere S 2 given locally by (c, b cos u cos v, b cos u sin v, b sin u, a), a 2 + b 2 − c 2 = 1; (2) a totally umbilical hyperbolic plane H 2 given by (a cosh u cosh v, a cosh u sinh v, a sinh u, b, c) with b 2 + c 2 − a 2 = 1; (3) a torus S 1 ×S 1 given by (a, b cos u, b sin u, c cos v, c sin v) with b 2 +c 2 −a 2 = 1; (4) a flat surface H 1 × S 1 given by (b cosh u, b sinh u, c cos v, c sin v, a) with a 2 + c 2 − b 2 = 1; (5) a totally umbilical flat surface defined by (6) a flat surface defined by v 2 − 3 4 + a 2 , a cos u, a sin u, v, v 2 − 5 4 + a 2 , a > 0; (7) a flat surface defined by , u, v, a , a ∈ R; (8) a marginally trapped flat surface defined by 1 2 2u 2 − 1, 2u 2 − 2, 2u, sin v, cos v ; (9) a marginally trapped flat surface defined by (10) a marginally trapped flat surface defined by For parallel spatial surface in S 3 1 (1) ⊂ E 4 1 , Theorem 15.1 implies the following.  Theorem 16.1. If M is a parallel spatial surface in H 4 1 (−1) ⊂ E 5 2 , then M is congruent to one of the following ten types of surfaces: (1) a totally umbilical sphere S 2 given locally by (a, c, b sin u, b cos u cos v, b cos u sin v), a 2 − b 2 + c 2 = 1; (2) a totally umbilical hyperbolic plane H 2 given locally by (a, b cosh u cosh v, b cosh u sinh v, b sinh u, c) with by (a, b cosh u, b sinh u, c cos v, c sin v) with a 2 + b 2 − c 2 = 1; (4) a flat surface H 1 × H 1 given by (b cosh u, c cosh v, b sinh u, c sinh v, a) with b 2 + c 2 − a 2 = 1; (5) a totally umbilical flat surface defined by , u, v , a ∈ (0, 1); (6) a flat surface defined by , bu, bv , a 2 = 1 + b 2 > 1; (9) a marginally trapped flat surface defined by (10) a flat marginally trapped surface defined by Conversely, each surface of the ten types given above is spatial and parallel.
For parallel spatial surfaces in H 3 1 (−1), Theorem 16.1 implies the following. Corollary 16.1. If M is a parallel spatial surface in H 3 1 (−1) ⊂ E 4 1 , c > 0, then M is congruent to one of the following two types of surfaces: (1) a hyperbolic plane by (a cosh u, b cosh v, a sinh u, b sinh v), a 2 +b 2 = 1. (1) a totally umbilical de Sitter space S 2 1 given by (c, a sinh u cos v, a cosh u cos v, a cosh u sin b, b) with c 2 − a 2 − b 2 = 1;

Classification of parallel Lorentzian surfaces in anti de
(2) a totally umbilical anti-de Sitter space H 2 1 given by (a sin u, a cos u cosh v, a cos u sinh v, 0, b) with a 2 − b 2 = 1; (3) a flat surface S 1 1 × H 1 given by (c, a sinh u, a cosh u cos v, a cosh u sin v, b) with c 2 − a 2 − b 2 = 1; (4) a flat surface H 1 1 × S 1 given by (a cos u, a sin u, b cos v, b sin v, c) with a 2 + b 2 − c 2 = 1; (5) a flat surface S 1 1 × S 1 given by (a, b sinh u, b cosh u, c cos v, c sin v) with a 2 − b 2 − c 2 = 1; (6) a totally umbilical flat surface defined by with a 2 − b 2 = 1, cos k = 0; (10) a surface defined by (12) a surface defined by (3) an anti-de Sitter space H 2 1 defined by (a sin u, a cos u cosh v, a cos u sinh v, b) with a 2 − b 2 = 1; by (a cos u, a sin u, b cos v, b sin v) with a 2 − b 2 = 1; (6) a surface defined by cos u cosh v − tan k sin u sinh v, sec k sin u cosh v, cos u sinh v − tan k sin u cosh v, sec k sin u sinh v , cos k = 0; (7) the surface defined by 17. Parallel spatial surfaces in S 4 2 17.1. Four-dimensional manifolds with neutral metrics. The metrics of neutral signature (− − ++) appear in many geometric and physics problems in the last 25 years. It has been realized that the theory of integrable systems and the techniques from the Seiberg-Witten theory can be successfully used to study Kaehler-Einstein and self-dual metrics as well as the self-dual Yang-Mills equations in neutral signature. Riemannian manifolds with neutral signature are of special interest since it retains many interesting parallels with Riemannian geometry. Such parallels are particularly evident in four dimensions, where Hodge's star operator is involutory for both positive-definite and neutral signatures. Both signatures possess the decomposition of two-forms into self-dual and anti-self-dual parts without the need to complexify as in the Lorentzian case.
As an interplay between indefiniteness and parallels with Riemannian geometry for neutral signature, the curvature decomposition in four dimensions for the two signatures allows one to deduce a neutral analogue of the Thorpe-Hitchin inequality for compact Einstein 4-manifolds (cf. e.g., [103]). Also, the development of the geometry of neutral signature in the work of H. Ooguri and C. Vafa [104] showed that neutral signature arises naturally in string theory as well.
Para-Kaehler manifolds provide further interesting examples of metrics of neutral signature. Such manifolds play some important roles in super-symmetric field theories as well as in string theory (see, for instance, [105,106,107,108]  (2) a flat surface in a totally geodesic S 3 1 (1) ⊂ S 4 2 (1) defined by a 2 + b 2 − 1, a sinh u, a cosh u, b cos v, b sin v , a, b > 0, a 2 + b 2 ≥ 1; (3) a flat surface defined by flat surface defined by a cos u, a sin u, b cos v, b sin (7) a flat surface defined by (9) a marginally trapped surface of constant curvature one defined by (10) a flat surface defined by x + xy, y − xy, x − y + xy, 1 + xy, 0 ; (11) a surface of positive curvature c 2 defined by , xy + c 2 c 2 (x + y) , c 2 (x + y) − 2y c 2 (x + y) , 0 , c ∈ (0, 1), x + y = 0; (12) a surface of positive curvature c 2 defined by 0, xy − c 2 c 2 (x + y) , xy + c 2 c 2 (x + y) , , c > 1, x + y = 0; (13) a surface of negative curvature −c 2 defined by (4) A CMC flat surface in a totally geodesic H 3 2 (−1) given by , a, b, c > 0; (5) A non-minimal flat surface given by (6) A non-minimal flat surface given by bϕ Conversely, every parallel Lorentzian surface in H 4 3 (−1) is congruent to an open portion of one of the six families of surfaces described above. (c) → CP n 1 (4c) : z → z · C * with C * = C \ {0} is a submersion and there is a unique Lorentzian Kaehler metric on CP n 1 (4c) such that π is a Riemannian submersion. The space CP n 1 (4c) equipped with this metric is a Lorentzian Kaehler manifold of positive holomorphic sectional curvature 4c.

Parallel Lorentz surfaces in
Similarly, for any real number c < 0, the differentiable manifold with the induced metric, is an indefinite real space form of constant sectional curvature c < 0. The Hopf-fibration: π : H 2n+1 3 (c) → CH n 1 (4c) : z → z · C * is a submersion and there is a unique Lorentzian Kaehler metric on CH n 1 (4c) such that π is a Riemannian submersion. The space CH n 1 (4c) equipped with this metric is a Lorentzian Kaehler manifold of negative holomorphic sectional curvature 4c.
The manifolds C n 1 , CP n 1 (4c) and CH n 1 (4c) are called complex Lorentzian space forms. The Riemann curvature tensor of a complex Lorentzian space form of constant holomorphic sectional curvature 4c takes the form where X and Y are arbitrary tangent vectors at an arbitrary point and ∧ is defined by Remark 8. The mapping (1) a Lorentzian totally geodesic surface; (2) a Lorentzian product of parallel curves; (3) a complex circle, given by (a + ib) cos(x + iy), sin(x + iy) with a, b ∈ R, (a, b) = (0, 0); (4) a B-scroll over the null cubic in E 3 1 ⊆ C 2 1 ; (5) a B-scroll over the null cubic in E 3 2 ⊆ C 2 1 ; (6) a surface given by (7) a surface with light-like mean curvature vector given by (q(x, y), x, y, q(x, y)) with q(x, y) = ax 2 + bxy + cy 2 + dx + ey + f and a, b, c, d, e, f ∈ R; (8) a totally umbilical de Sitter space S 2 1 in E 3 1 ⊆ C 2 1 , given by a(0, sinh x, cosh x cos y, cosh x sin y) with a ∈ R \ {0}; (9) a totally umbilical anti-de Sitter space H 2 1 in E 3 2 ⊆ C 2 1 given by a(sin x, cos x cosh y, cos x sinh y, 0) with a ∈ R \ {0}. Conversely, each of the surfaces listed above is a Lorentzian surface with parallel second fundamental form in C 2 1 . 19.3. Classification of parallel Lorentzian surface in CP 2 1 . First we mention the following result from [74]. The next classification of parallel Lorentzian surface in CP 2 1 was obtained by Chen, Dillen and Van der Veken in [74]. (II) M is flat, and the immersion is congruent to π • L, where π : S 5 2 (1) → CP 2 1 (4) is the Hopf-fibration and L : M 2 1 → S 5 2 (1) ⊆ C 3 1 is locally one of the following twelve maps: ( ,

19.4.
Classification of parallel Lorentzian surface in CH 2 1 . It follows from Remark 8 that one obtains immediately the classification of parallel Lorentzian surfaces in CH 2 1 (−4) from Theorem 19.2 via the mapping: ψ : C 3 1 → C 3 2 : (z 1 , z 2 , z 3 ) → (z 3 , z 2 , z 1 ) since ψ gives rise to a conformal mapping with factor −1 between CP 2 1 (4) and of an open interval I and a Riemannian 3-manifold (R 3 (c), g c ) of constant curvature c, while the warping function f describes the expanding or contracting of our Universe (cf. [112,114]).
A Robertson-Walker space-time possesses two relevant geometrical features. On one hand, its fibers have constant curvature. Hence the space-time is spatially homogeneous. On the other hand, it has a time-like vector field K = f (t)∂ t which satisfies ∇ X K = f ′ (t)X for any X. In particular, we have L K g = 2f ′ g, where L K is the Lie derivative along K. Hence the canonical time-like vector field K is a conformal vector field. These properties of K show a certain symmetry on L n 1 (c, f ). One may also consider a higher dimensional Robertson-Walker space-time as where R n−1 (c) is a Riemannian (n − 1)-manifold of constant curvature c for n > 5.
A rest space or a space-like slice in L n 1 (c, f ) is a space-like hypersurface given by t constant. Thus a rest space in L n 1 (c, f ) is a fiber Hence a rest space S(t 0 ) in L n 1 (c, f ) is an (n − 1)-manifold of constant curvature whose metric tensor is given by f 2 (t 0 )g k .
A pseudo-Riemannian submanifold N of a Robertson-Walker space-time L n 1 (c, f ) is called transverse if it is contained in a rest space S(t 0 ) for some t 0 ∈ I. A pseudo-Riemannian submanifold N of L n 1 (c, f ) is called a H-submanifold if the tangent field ∂ ∂t , known as the comoving observer field, is tangent to N at each point on N .
20.2. Parallel submanifolds of Robertson-Walker space-times. For parallel submanifolds of L n 1 (c, f ), we have the next classification result from [113,114].
Similar result holds for submanifolds in a warped product I × f R n−1 (c) with the Riemannian warped product metric g = dt 2 + f 2 (t)g c (cf. [115,116]).

Thurston's eight three-dimensional model geometries
The uniformization theorem for 2-dimensional surfaces says that every simplyconnected Riemann surface is conformally equivalent to one of the three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. This result implies that every Riemann surface admits a Riemannian metric of constant curvature.
Roughly speaking, for closed 3-manifolds W. Thurston's Geometrization Conjecture states that every closed 3-manifold can be decomposed in a canonical way into pieces that each have one of eight types of geometric structure locally (see, [117]). In 2005, G. Perelman [118] provided a proof of Thurston's geometrization conjecture via Ricci flow with surgery.
The eight Thurston's 3-dimensional model geometries are the following.
(6) The geometry SL 2 (R). The 3-dimensional Lie group of all 2 × 2 real matrices with determinant one is denoted by SL 2 (R); and SL 2 (R) denotes its universal covering. SL 2 (R) is a unimodular Lie group with a special left invariant metric. Examples of these manifolds in this geometry include the manifold of unit vectors of the tangent bundle of a hyperbolic surface and, more generally, the Brieskorn homology spheres. We mentioned earlier in §1 that the complete classification of parallel surfaces in E 3 was obtained by V. F. Kagan; the complete classifications of parallel surfaces in S 3 and in H 3 were given in §5.4 and §5.5, respectively; the classifications of parallel surfaces in S 2 × R and in H 2 × R 3 were given in §20.
In this section, we will deal the classification of parallel surfaces in Sol 3 , SL 2 (R) and N il 3 in §22.2, §22.4 and §22.5, respectively.

Milnor's classification of 3-dimensional unimodular Lie groups.
A Lie group G is called unimodular if its left-invariant Haar measure is also rightinvariant. In [119], J. Milnor provides an infinitesimal reformulation of unimodularity for 3-dimensional Lie groups. We recall it briefly as follows: Let g be a 3-dimensional oriented Lie algebra equipped with an inner product , . Define the vector product operation × : g × g → g as the skew-symmetric bilinear map which is uniquely determined by the following three conditions: if X and Y are linearly independent, then det(X, Y, X × Y ) > 0, for all X, Y ∈ g. The Lie-bracket [ · , · ] on g is a skew-symmetric bilinear map. By comparing these two operations, one obtains a linear endomorphism L g which is uniquely determined by the formula If G is an oriented 3-dimensional Lie group equipped with a left-invariant Riemannian metric, then the metric induces an inner product on the Lie algebra g. With respect to the orientation on g induced from G, the endomorphism field L g is uniquely determined.
J. Milnor proved in [119] that the unimodularity of G is characterized as follows. Here E(1, 1) denotes the the group of orientation-preserving rigid motions of Minkowski plane, E(2) denotes the group of orientation-preserving rigid motions of Euclidean plane andẼ (2) is the universal covering of E(2).

22.4.
Parallel surfaces in the motion group E(2). The Euclidean motion group E(2) is given by the following matrix group: The universal covering group of E(2) is R 3 with multiplication (x, y, z) · (x ′ , y ′ , z ′ ) = (x + x ′ cos z − y ′ sin z , y + x ′ sin z + y ′ cos z , z + z ′ ).
We have the following result on E(2) from [120].
J. Inoguchi and J. Van der Veken classified parallel surfaces in E(2) in [121] as follows.
Theorem 22.4. The only parallel surfaces in E(2) are integral surfaces of the distribution spanned by {∂/∂x, ∂/∂y}. These surfaces are flat and minimal, but not totally geodesic. (2). The group SU (2) is diffeomorphic to S 3 , since
The next non-existence result was proved by J. Inoguchi and J. Van der Veken in [122]. 22.6. Parallel surfaces in the real special linear group SL(2, R). The group SL(2, R) is defined as the following subgroup of GL(2, R): This group is isomorphic to the following subgroup of GL(2, C): The Lie algebra of SU (1, 1) is explicitly given by We take the following split-quaternionic basis of the Lie algebra su(1, 1): Denote the left-translated vector fields of {j ′ , k ′ , i} by {E 1 , E 2 , E 3 } and choose strictly positive real constants λ 1 , λ 2 , λ 3 and define where {ω 1 , ω 2 , ω 3 } is the dual coframe field of {E 1 .E 2 , E 3 }. This three-parameter family of Riemannian metrics exhausts all left-invariant metrics on SL(2, R) as shown in the next proposition from [120]. We consider SL(2, R) equipped with a left-invariant metric such that the dimension of the isometry group is only 3. With the notations given above, we have that The following classification theorem for parallel surfaces in SL(2, R) was proved by J. Inoguchi and J. Van der Veken in [122].
The next was also proved by J. Inoguchi and J. Van der Veken in [122]. Remark 9. The oscillator group was introduced and first studied by R. F. Streater in [124] and owes its name to the fact that its Lie algebra coincides with the one generated by the differential operators associated to the harmonic oscillator problem. Generalizing this construction, oscillator groups have been defined in any even dimension greater or equal to four. Since their introduction, the oscillator groups have been intensively studied from several different points of view, both in differential geometry and in mathematical physics. Beside direct extensions with Euclidean groups, the oscillator groups are the only simply connected non-Abelian solvable Lie groups admitting a bi-invariant Lorentzian metric.
In [125] G. Calvaruso and J. Van der Veken obtained the complete classification and explicitly description of totally geodesic and parallel hypersurfaces of four-dimensional oscillator groups, equipped with a one-parameter family of leftinvariant Lorentzian metrics.

Parallel surfaces in three-dimensional Lorentzian Lie groups
Homogeneous Lorentzian 3-spaces (N, g) where classified by G. Calvaruso in [126]. Unless they are symmetric, they are Lie groups equipped with left-invariant Lorentzian metrics.
Theorem 23.1. Let (N, g) be a 3-dimensional connected, simply connected, complete homogeneous Lorentzian manifold. If (N, g) is not symmetric, then N = G is a 3-dimensional Lie group and g is left-invariant. Moreover, there exists a pseudoorthonormal frame field {e 1 , e 2 , e 3 }, with e 3 time-like, such that the Lie algebra of G is one of the following seven types. The following Table I lists all the Lie groups G which admit a Lie algebra g 3 , taking into account the different possibilities for α, β and γ:  The following Table II describes all Lie groups G admitting a Lie algebra  with α + δ = 0, αγ = 0. Lie algebras of types g 1 , g 2 , g 3 and g 4 correspond to unimodular groups, whereas Lie algebras of types g 5 , g 6 and g 7 correspond to non-unimodular groups.
G. Calvaruso determined in [127] those 3-dimensional Lorentzian Lie groups (G, g) which have constant sectional curvature and which are symmetric.
By a 3-dimensional Lorentzian Lie group G i we mean a connected, simply connected 3-dimensional Lie group G, equipped with a left-invariant Lorentzian metric g and having Lie algebra g i .

23.2.
Classification of parallel surfaces in three-dimensional Lorentzian Lie groups. Let (N, g) be a 3-dimensional homogeneous Lorentzian manifold and M a surface in N . We denote by ξ a fixed normal vector field on the surface, with ξ, ξ = ε. Here, either ε = −1 or ε = 1, according to the surface being either Riemannian or Lorentzian, respectively. We call ξ an ε-unit normal vector field. Parallel surfaces in 3-dimensional Lorentzian Lie groups were classified by G. Calvaruso and J. Van der Veken in [128]. More precisely, under the notations of Theorem 23.1, they proved the following.
Theorem 23.2. Let M be a parallel surface in a 3-dimensional Lorentzian Lie group G 1 . Then, β = 0, ξ = e 1 + be 2 + be 3 and the vector fields E 1 = (be 1 − e 2 )/ √ 1 + b 2 and E 2 = (be 1 + b 2 e 2 + (1 + b 2 )e 3 )/ √ 1 + b 2 form a pseudo-orthonormal basis for the tangent plane at every point. Moreover, the function b satisfies E 1 (b) = E 2 (b) and The surface is flat and parallel. Moreover, it is totally geodesic in the case that This case only occurs if α = β = 0. M is flat, but not necessarily totally geodesic. (c) M is an integral surface of the distribution spanned by The surface is flat and parallel. Moreover, it is totally geodesic in the special case that 24. Parallel surfaces in reducible three-spaces 24.1. Classification of parallel surfaces in reducible three-spaces. Parallel submanifolds of the a Robertson-Walker space-time I × f R n (c) have been treated in §20. In [137], G. Calvaruso and J. Van der Veken studied parallel surfaces in 3-dimensional reducible spaces M 2 × E 1 . More precisely, they proved the following results. The following is a consequence of Theorem 24.1.
Corollary 24.1. The pair (S 2 , E 3 ) is the only proper parallel surface in a reducible Riemannian 3-space.
For parallel surfaces in a reducible 3-dimensional Lorentzian manifold, G. Calvaruso and J. Van der Veken obtained the following.  24.2. Parallel surfaces in Walker three-manifolds. A particularly interesting class of pseudo-Riemannian manifolds are ones which admit a parallel null vector field. The study of such metrics in the 3-dimensional Lorentzian setting was initiated by M. Chaichi, E. García-Río and M. E. Vázquez-Abal in [138]. W. Batat and S. J. Hall named such manifolds as Walker manifolds in [139].
Complete classification of parallel surfaces of an arbitrary reducible 3-manifold, both in Riemannian and Lorentzian was obtained by G. Calvaruso and J. Van der Veken in [137]. It turns out that the Euclidean space E 3 and the Minkowski space E 3 1 are the only cases admitting parallel surfaces which are non-trivial, in the sense that they do not reflect the reducibility of the space itself. Since the reducibility of a pseudo-Riemannian manifold corresponds to the existence of a parallel non-null vector field, it is natural to study parallel surfaces in a Lorentzian 3-manifold which admits a parallel null vector field, i.e., in a Walker 3-manifold. G. Calvaruso and J. Van der Veken provided in [140] a complete classification of parallel surfaces in Walker 3-manifolds.
In [139], W. Batat and S. J. Hall proved that totally umbilical nondegenerate surfaces in a Walker 3-manifold with metric g = ǫdx 2 + f (x, y)dy 2 + 2dtdy with ǫ = ±1 and satisfying f xx = 0 are either one of a totally geodesic family described by G. Calvaruso and J. Van der Veken in [140] or the ambient manifold must be locally conformally flat (here the surface can also be totally geodesic).

Bianchi-Cartan-Vranceanu spaces
25.1. Basics on Bianchi-Cartan-Vranceanu spaces. The simply-connected homogeneous 3-manifolds are classified according to the dimension of their isometry group which is equal to 3, 4 or 6. If it is 6, one obtains the real space forms. The Bianchi-Cartan-Vranceanu spaces are homogeneous Riemannian 3-manifolds with isometry group of dimension 4 or 6. Such spaces, denoted by M 3 (λ, µ), are given by a two-parameter family of Riemannian 3-manifolds (M, g λ,µ ) where the underlying 3-manifolds M 3 are R 3 if µ ≥ 0; and The metricsg λ,µ on M 3 are given by The 2-parameter familyg λ,µ is called the Bianchi-Cartan-Vranceanu metrics. The metrics above are defined over the whole 3-space R 3 for µ > 0 and over the region x 2 + y 2 < −1/µ for µ < 0.
In the following, by a Hopf-cylinder we mean the inverse image of a curve in M 2 (µ) under π. By a leaf of the Hopf-fibration, we mean a surface which is everywhere orthogonal to the fibres. The family of Bianchi-Cartan-Vranceanu spaces M 3 (λ, µ) includes six of the eight Thurston's 3-dimensional geometries except Sol 3 and the hyperbolic space H 3 . The family of the Riemannian metrics given by (37) includes all 3-dimensional homogeneous metrics whose group of isometries has dimension 4 or 6, except for those with negative constant curvature.

Parallel surfaces in symmetric Lorentzian three-spaces
Symmetric spaces are one of the most important topics in Riemannian geometry. In the Lorentzian setting, their study goes back to the work of M. Cahen and N. Wallach [129] in the 1970s.
27.1. Symmetric Lorentzian three-spaces. It is well known that the curvature of a 3-dimensional pseudo-Riemannian manifold (N, g) is completely determined by the Ricci tensor, denoted by Ric, defined for any point p ∈ N and any X, Y ∈ T p N by (38) Ric where R is the Riemann curvature tensor, {e 1 , e 2 , e 3 } is a pseudo-orthonormal basis of T p N and ε i = g p (e i , e i ) = ±1 for all i. Throughout this section, if not stated otherwise, we shall assume that e 3 is time-like, i.e., ε 1 = ε 2 = −ε 3 = 1.
Due to the symmetries of the curvature tensor, the Ricci tensor Ric is symmetric [112]. Thus, the Ricci operator Q, defined by g(QX, Y ) = Ric(X, Y ), is self-adjoint. In the Riemannian case, there always exists an orthonormal basis diagonalizing Q, but in the Lorentzian case four different cases can occur [112], and there exists a pseudo-orthonormal basis {e 1 , e 2 , e 3 }, with e 3 time-like, such that Q takes one of the following canonical forms, called Segre types: When (N, g) is homogeneous, the Ricci operator Q has the same Segre type at any point p ∈ N and has constant eigenvalues. G. Calvaruso studied homogeneous Lorentzian 3-manifolds (N 3 , g) in [126,127]. For symmetric ones, he proved that 3-dimensional symmetric spaces can only occur for some Segre types of the Ricci operator Q. More precisely, he proved the following: I) For Segre type {11, 1}, (N, g) is symmetric if and only if (i) a = b = c. Then, (N, g) is an Einstein manifold and hence it has constant sectional curvature. If N is connected and simply connected, then (N, g) is isometric to one of the Lorentzian space forms: either S 3 1 , R 3 1 or H 3 1 . (ii) a = b = c. Then, N is reducible as a direct product M 2 × R 1 , where M 2 is a Riemannian surface of constant curvature. If N is connected and simply connected, (N, g) is then isometric to either S 2 × R 1 or H 2 × R 1 . (iii) a = b = c. Then, N is reducible as a direct product R × M 2 1 , where M 2 1 is a Lorentzian surface of constant sectional curvature. When N is connected and simply connected, (N, g) is isometric to either R × S 2 1 or R × H 2 1 .

27.2.
Classification of parallel surfaces in symmetric Lorentzian threespaces. Three-dimensional Lorentzian manifolds admitting a parallel null vector field were first studied in [138], in which the attention was focused on local properties. G. Calvaruso and J. Van der Veken described in [135] a global model carrying a metric described by (41)-(42) as follows. First they showed that the curvature components with respect to the pseudoorthonormal frame field {e 1 , e 2 , e 3 } for which (40) holds and then apply (38) to obtain its Ricci components. Since the Ricci operator must be of degenerate Segre type {21} (that is, with a = b − η), standard calculations lead to the following system of partial differential equations: Then they proved that, for any smooth function ω, with respect to the following new pseudo-orthonormal frame field (45) e ′ 1 = e 1 +ωe 2 −ωe 3 , e ′ 2 = −ωe 1 +(1− the Ricci operator still keeps the same components than with respect to {e 1 , e 2 , e 3 }. It follows from (40) and (45) that, with respect to {e ′ 1 , e ′ 2 , e ′ 3 }, the Levi Civita connection satisfies 2 )e 2 ω − ω 2 2 C − ω 2 2 e 3 ω, C ′ = Aω + ωe 1 ω + ω 2 2 B + ω 2 2 e 2 ω − (1 + ω 2 2 )C − (1 + ω 2 2 )e 3 ω. Thus, by choosing ω to be a solution of the system of differential equations (47) A + e 1 ω = k, de 2 ω − e 3 ω = C − B, where k is a real constant, we can always specify the pseudo-orthonormal frame field {e 1 , e 2 , e 3 } in such a way that A = k and B = C. In this case, system of equations (44) reduces to In [135], G. Calvaruso and J. Van der Veken proved the following.
Theorem 27.2. Let (N, g) be a connected, simply connected 3-dimensional Lorentzian manifold. Then the necessary and sufficient condition for (N, g) to be symmetric and to have a Ricci operator of (degenerate) Segre type {21}, is the existence of a global pseudo-orthonormal frame field {e 1 , e 2 , e 3 }, with e 3 timelike, a real constant k and a smooth function B, satisfying (48)- (49).
The following classification of parallel surfaces in a symmetric Lorentzian 3-space was also obtained by G. Calvaruso and J. Van der Veken in [135]. 28. Three natural extensions of parallel submanifolds 28.1. Submanifolds with parallel mean curvature vector. One natural extension of the class of parallel submanifolds (∇h = 0) is the class of submanifolds with parallel mean curvature vector, i.e.,∇(Tr h) = 0 or equivalently DH = 0. Trivially, both minimal submanifolds and parallel submanifolds have parallel mean curvature vector automatically. Further, a hypersurface of any Riemannian manifold has parallel mean curvature vector if and only if it has constant mean curvature.
Euclidean hypersurfaces with constant mean curvature are important since they are critical points of some natural functionals. In fact, a hypersurface of constant mean curvature in a Euclidean space is a solution to a variational problem. With respect to any volume-preserving variation of a domain D in a Euclidean space the mean curvature of M = ∂D is constant if and only if the volume of M is critical, where ∂D is the boundary of D.
The condition of submanifolds to have parallel mean curvature vector in higher dimensional Euclidean spaces is very interesting as well since it is equivalent to a critical points of being variational problem; namely, their Gauss maps are harmonic maps (see [141]).
During the last 50 years, there are many research done on submanifolds with parallel mean curvature vector. Among others, for submanifolds with parallel mean curvature vector in real space forms, see [142,143,144,145,146,147,148,149]; for surfaces with parallel mean curvature vector in complex space forms, see [150,151,152,154,155,156,153]; for surfaces with parallel mean curvature vector in indefinite space forms, see [157,158,159,160,161,162]; for surfaces with parallel mean curvature vector in homogeneous spaces or symmetric spaces, see [163,164]; for surfaces with parallel mean curvature vector in Sasakian space forms, see [165]; and for surfaces with parallel mean curvature vector in reducible manifolds, see [166,167,168,169]. For general references of submanifolds with parallel mean curvature vector, see [170].
28.2. Higher order parallel submanifolds. Higher order parallel submanifolds, i.e., submanifolds that satisfy∇ k h = 0 for some positive integer k, were first studied by D. Del-Pezzo in [171] and then investigated by several authors after Del-Pezzo (see J. A. Schouten and D. J. Struik's 1938 book [172] for details). This research topic was renewed in late 1980s by F. Dillen, V. Mirzoyan andÜ. Lumiste. Since then, this interesting research topic has been studied by several differential geometers.