Isotropic submanifolds of pseudo-Riemannian spaces

ISOTROPIC SUBMANIFOLDS OF PSEUDO-RIEMANNIAN SPACES J. L. Cabrerizoa,1, M. Fernándeza, J. S. Gómeza aDepartment of Geometry and Topology, University of Seville, Ap. 1160, 41080-Seville, Spain ABSTRACT The family of all the submanifolds of a given Riemannian or pseudo-Riemannian manifold is large enough to classify them into some interesting subfamilies such as minimal (maximal), totally geodesic, Einstein, etc. Most of these have been extensively studied by many authors, but as far as we know, no paper has hitherto been published on the class of isotropic submanifolds. The purpose of this paper is therefore to gain a better understanding of this interesting class of submanifolds that arise naturally in mathematics and physics by studying their relationships with other closely distinguised families. Mathematics Subject Classification: 53C25, 53C40 Keywords: Pseudo-Riemannian manifold, isometric immersion, isotropic submanifold, spacelike submanifold, Lorentzian submanifold. 1 Introduction A submanifold Nn sof a pseudo-Riemannian space Mm νis said to be totally geodesic if the geodesic lines of Nn sare also geodesic lines in Mm ν.A totally geodesic submanifold is characterized by the fact that for every normal vector of the corresponding second fundamental form hvanishes: h(X, Y )=0for any vectors fields X, Y. The existence of totally geodesic submanifolds in a general Riemannian manifold is exceptional. Conversely, the existence of many such totally-geodesic submanifolds characterize some special manifolds, e.g. symmetric spaces. The submanifold Nn sis called totally umbilical if the second fundamental form his proportional to the first, h(X, Y ) = g(X, Y )H,where H= (1/n)trace(h) is the mean curvature vector, and is pseudo-umbilical if g(h(X, Y ),H) = ρ g(X, Y ).Obviously every totally umbilical submanifold is pseudo-umbilical. An Einstein manifold is a Riemannian or pseudo-Riemannian manifold whose Ricci tensor is proportional to the metric. Aminimal submanifold is a submanifold with H≡0,therefore any totally geodesic submanifold is minimal. The analogous submanifolds in spacetimes are spacelike maximal submanifolds. They are all pseudo-umbilical, and have been used to prove positivity of 1Corresponding author. Tel: +34 954557969; fax: +34 954557970. E-mail addresses: [email protected] (J.L. Cabrerizo), [email protected] (M. Fernández), jsalv[email protected] (J.S. Gómez) 1 mass, analyse the space of solutions of Einstein equations and in numerical integration schemes for Einstein equations ([3], p. 155). In the pseudo-Euclidean space Rm ν, maximal submanifolds are the harmonic submanifolds. In this article we are interested in a distinguised family of submanifolds of a pseudo- Riemannian manifold: the isotropic submanifolds. Isotropy is a notion that naturally appear in Particle Physics, Kinetic theory, Electromagnetics, Optics, Cosmology, Materials science, etc. In the sixties, the fascinating aspect of unexpected features in Einstein’s theory came to development namely the existence of singularities in solutions. An outside observer does not perceive the singularity itself but an event horizon. Conversely light cannot escape the singularity, it becomes infinitely redshifted as the horizon is approached. Then, modulo some reservations it has been shown that horizons are isotropic hypersurfaces [16]. Hence theoretical physicists have widely used isotropic hypersurfaces with their degenerate metric induced by the embedding into the Lorentzian spacetime. The concept of isotropic submanifold of a Riemannian manifold was introduced by B. O’ Neill [20], who studied the general properties of such a class of submanifold. A pseudo-Riemannian submanifold is isotropic if, roughly speaking, the geometry of the submanifold is the same regardless of direction. This notion remind us of the cosmological principle that no matter where we look in the Universe, we still see the same distribution of objetcs (see [8] p. 835 and Prop. 5.1). The isotropic submanifolds can be considered as a generalization of both, totally geodesic and totally umbilical submanifolds. In a previous paper ([10]) we studied spacelike isotropic surfaces in a Lorentzian manifold and, in particular in the standard four-dimensional model spaces: Minkowski, De Sitter and anti De Sitter spaces. Now, the aim of the present paper is to place the notion of isotropic submanifold in a wider setting by studying its interrelationships with other distinguised families of submanifold such as minimal, pseudo-umbilical, Einstein, etc., into a pseudo-Riemannian manifold Mm ν.For this end, we shall use the advantageous language of immersion theory. Thus, we first introduce in Section 2 the necessary notation and recall some useful facts concerning an isometric immersion of a pseudo-Riemannian manifold Nn s(n≥2) into the pseudo-Riemannian manifold Mm ν.We develop in Section 3 the notion of isotropic submanifold. In particular, every totally umbilical submanifold is obviously an isotropic submanifold. We give some characterizations of isotropic submanifolds and include an example showing that isotropy in a Riemannian or in a pseudo-Riemannian environment may have different consequences. The class of constant discriminant isotropic immersions (for instance, into a sphere) seems to be too large to classify them. We first show that in such case an isotropic submanifold is also pseudo-umbilical. We give some examples of isometric embeddings between Riemannian or pseudo-Riemannian manifolds. As we remarked in Section 3, every totally umbilical submanifold is isotropic, and now we prove that the converse is false. Actually we can prove a slightly stronger result. In fact, we give an example of a submanifold with dimension n≥3which is isotropic but not pseudo-umbilical. As a consequence, we shall sometimes assume the submanifold to be isotropic and pseudoumbilical. In Section 5 we consider the Ricci tensor of an isotropic pseudo-Riemannian submanifold of a pseudo-Riemanian space form of constant sectional curvature c. An useful formula is obtained for this tensor which allows us to obtain several applications. Thus, if the sub- 2 manifold is also maximal and its dimension is n≥3,then it is an Einstein manifold (when n= 2 this result is false). We include an example that exhibits how different things may happen depending on the surronding space being Riemannian or pseudo-Riemannian. In Section 6 we consider isotropic spacelike submanifolds Nn(n≥3) of a pseudo- Riemannian space of constant curvature c, Mm ν(c).When the isotropy function λis approppiately bounded above then strong topological conditions are obtained for Nn. In particular, if the background space is Rm ν,the hyperbolic space Hm νor the pseudo-sphere Sm ν,aditional information is derived. Next we use Bochner’s technique to show that when Nnis compact and admits a nontrivial conformal vector field, the isotropy function is bounded as desired. On the contrary, when the isotropy function is bounded below by the same function as before, then Nnis non-compact. We give an example to illustrate this result. Finally, in Section 7 we specialize in the case when the index of the submanifold s= 1, that is, a Lorentzian submanifold Nn 1of a pseudo-Riemannian space form Mm ν(c).When Nn 1is compact and admits a spatially conformal reference frame (or a projective vector field), then we prove that the isotropy function λis bounded below by the same function as in Section 6. If in addition Nn 1is a maximal submanifold, then λis a constant. By changing the hypothesis on the existence of the spatially conformal reference frame by the existence of a timelike conformal vector field, we get the same bound for λ. We close this Section with some further consequences obtained when we assume that λreaches the bound. 2 Preliminaries Let Mm νbe a m-dimensional pseudo-Riemannian manifold with metric tensor gof signature (ν, m −ν). Let φ:Nn s→Mm νbe an isometric immersion of a connected pseudo-Riemannian manifold Nn sof dimension n≥2and signature (s, n −s). For all local formulae and computations we may assume φis an imbedding and thus we shall often identify p∈Nn swith φ(p)∈Mm ν. The tangent space TpNn sis identified with the subspace φ∗(TpNn s)of TpMm ν,and the normal space is denoted by T⊥ pNn s. We will use letters X, Y, Z (resp. ξ, η, ζ) to denote vectors fields tangent (resp. normal) to Nn s.Let ˜ ∇and ∇ be the Levi-Civita connections of Mm νand Nn s, respectively. Then, the Gauss-Weingarten formulas are given by ˜ ∇XY=∇XY+h(X, Y ),(1) ˜ ∇Xξ=−AξX+DXξ, (2) where hdenotes the second fundamental form of φ, Aξthe shape operator, and Dis the normal connection. The shape operator and the second fundamental form are related by g(AξX, Y ) = g(h(X, Y ), ξ). A point p∈Nn sis umbilic [21] provided there exists a vector ξp∈T⊥ pNn ssuch that for all u, v ∈TpNn sthen h(u, v) = g(u, v)ξp.When every point of Nn sis umbilic the immersion φis called totally umbilical. In such case, it is well known that h(X, Y ) = g(X, Y )Hwhere H= (1/n)traceg(h)is the mean curvature vector of φ. We also recall that a point p∈Nn s is flat if h= 0 at p, and the immersion φis said to be totally geodesic provided every point of Nn sis flat. 3 The immersion φis called pseudo-umbilical if its second fundamental form satisfies g(h(X, Y ),H) = ρg(X, Y )for some function ρ. Necessarily this function ρis given by ρ=g(H,H). Clearly, any totally umbilical immersion is pseudo-umbilical. Denote by ˜ Rand Rthe curvature tensor of Mm νand Nn s, respectively. The equation of Gauss is given by (˜ R(X, Y )Z)>=R(X, Y )Z+Ah(X,Z)Y−Ah(Y,Z)X, , (3) where (˜ R(X, Y )Z)>denotes the tangential component of ˜ R(X, Y )Z. The notion of discriminant Dof the immersion φfor a non-degenerate plane can be introduced in a similar way to the case of an immersion between Riemannian manifolds [20]. In fact, the discriminant Dpof φat p∈Nn sis Dp(Π) = K(Π) −˜ K(Π),where K(Π) and ˜ K(Π) denote the sectional curvature of Nn sand Mm ν,respectively, on a nondegenerate plane Πof TpNn s. The immersion φis said to be of constant discriminant at p∈Nn sif Dp(Π) is a constant for all non-degenerate planes Πof TpNn s. Notice that if φ is of constant discriminant at each point of Nn s, then the discriminant can be viewed as a function D:Nn s→R. From the view-point of infinitesimal transformations, a vector field Xis said to preserve a certain geometric quantity if the Lie derivative LXof this quantity vanishes. On a pseudo- Riemannian manifold (M, g)a vector field Xis called isometric or Killing if it preserves the metric in the sense of LXg= 0.The vector field Xis called conformal if it preserves the conformal class of the metric in the sense that LXg= 2ρg holds for some function ρ:M−→ R. Aprojective vector field Uis a smooth vector field whose flow preserves the geodesic structure of Mwithout necessarily preserving the affine parameter of any geodesic. An affine vector field is a projective vector field preserving geodesics and preserving the affine parameter. It is known that Uis projective if and only if there exists a 1-form ωon Msuch that LU∇(X, Y ) = ω(X)Y+ω(Y)Xfor all X, Y ∈X(M), and ω= (1/(n+ 1))d(div(U)) (see [22, Props. 5.27, 5.28] for instance). Note that Uis affine if and only if ω= 0. Let Rm νbe the n-dimensional pseudo-Euclidean space with inner product of signature νgiven by hx, yi=− ν X i=1 xiyi+ m X i=ν+1 xiyi where x= (x1, . . . , xm),y= (y1, . . . , ym). For a positive number c, the standard space form Sm ν(c)is the hypersurface Sm ν(c) = {x∈Rm+1 ν:hx, xi= 1/c}with the induced metric of signature νand constat curvature cand it is called a pseudo-sphere. For a negative number c, the standard space form Hm ν(c)is the hypersurface Hm ν(c) = {x∈Rm+1 ν+1 :hx, xi= 1/c} with the induced metric of signature νand curvature c < 0.This is called the pseudohyperbolic space. We simply denote Sm ν(1) and Hm ν(−1) by Sm νand Hm ν,respectively. For short we shall write Mm ν(c)to indicate Sm ν(c),Rm νor Hm ν(c), according to c > 0,c= 0 or c < 0, respectively. 4 3 Isotropic immersion We first recall that an isometric immersion φ:Nn s→Mm νis called isotropic at p∈Nn s ([10, 19]) if g(h(u, u), h(u, u)) = λ(p)∈R(4) does not depends on the choice of the unit tangent vector u∈TpNn s, and φis said to be isotropic if φis isotropic at each point of Nn s. In such a case, the smooth function λ:Nn s→Rdefined by equation (4) is called the isotropy function, and the isometric immersion φis said to be λ-isotropic. In particular, if λis a constant function the immersion is called constant isotropic. Remark 3.1 (a) It is clear that every totally umbilical immersion φ:Nn s→Mm νis an isotropic immersion with λ=g(H,H). Notice that the function g(H,H)is constant when φis totally umbilical and Mm νis of constant sectional curvature c[13, p. 11]. Thus, pseudo-spheres Sn s(c)and pseudo-hyperbolic spaces Hn s(c)into a pseudo-Euclidean space are constant isotropic submanifolds with λ=g(H,H) = c. (b) Every isotropic immersion φ:Nn s→Mm νwith 0< s < n and s=ν, or ν−s=m−n, is totally umbilical. In fact, in both cases T⊥ pNn sis a defined metric space. Thus, from isotropy condition (4) we obtain h(x, x) = 0 for any lightlike tangent vector x, and this means that our immersion is totally umbilical [15]. In particular, every isotropic immersion between Lorentzian manifolds is totally umbilical. (c) Any isotropic hypersurface φ:Nn s→Mn+1 νis totally umbilical. Notice that necessarily s=νor s=ν−1. Thus, if the index ssatisfies 0<s<n, the hypersurface is totally umbilical as before. It is easy to prove the defined cases s= 0 and s=n(see [9]). (d) The composition of isotropic immersions is an isotropic immersion, and the isotropy function of the composition is the sum of the corresponding isotropy functions. Now we provide a characterization of isotropic immersions [8]. Lemma 3.2 Let φ:Nn s→Mm νbe an isometric immersion. Then, the following conditions are equivalent: 1. φis isotropic. 2. There exists a smooth function λ:Nn s→Rsuch that Ah(X,X)X=λg(X, X)X for all X∈X(Nn s). 3. There exists a smooth function λ:Nn s→Rsuch that Ah(X,Y )Z+Ah(Y,Z)X+Ah(Z,X)Y=λ{g(X, Y )Z+g(Y, Z)X+g(Z, X)Y} for all X, Y, Z ∈X(Nn s). As a consequence of this lemma we have the following. Theorem 3.3 Let φ:Nn s→Mm νbe an isometric immersion. Then, the following conditions are equivalent. 5 1. φis isotropic. 2. There exists a smooth funtion λ:Nn s→Rsuch that 3Ah(X,Y )Z=λ{g(X, Y )Z+g(Y, Z)X+g(X, Z)Y} +R(Z, X)Y−(˜ R(Z, X)Y)> +R(Z, Y )X−(˜ R(Z, Y )X)>(5) for any X, Y, Z, W ∈X(Nn s). Furthermore, the condition 2 holds with λa constant function if and only if φis constant isotropic. Proof. It follows from Gauss equation (3) that 3Ah(X,Y )Z=Ah(X,Y )Z+Ah(Y,Z)X+Ah(Z,X)Y +R(Z, X)Y−(˜ R(Z, X)Y)> +R(Z, Y )X−(˜ R(Z, Y )X)>. Now, by statement 3of Lemma 3.2, we find equation (5). ¤ Remark 3.4 From Theorem 3.3 one can easily prove that every isotropic immersion φ: Nn s→Mm νsatisfies n2g(H,H) = (n+ 2)nλ −2g(h, h)(6) where g(h, h)is defined by g(h, h) = Pn i,j=1 εiεjg(h(ei, ej), h(ei, ej)) for a local orthonormal frame {e1, . . . , en}on Nn s,εi=g(ei, ei). As a consequence of equation (6), a λ-isotropic immersion between Riemannian manifolds is totally-umbilical if and only if λ=g(H,H). However, in the indefinite case the corresponding result is false. In fact, let f1, . . . , f`: Rn s→Rbe smooth functions and consider the isometric immersion φ:Rn s→Rn+2` s+`defined by φ(x) = (f1(x), . . . , f`(x), x, f1(x), . . . , f`(x)).It is easy to check that φis 0-isotropic with g(H,H) = 0. Notice that φis totally umbilical if and only if fjis given by (see [8]) fj(x1, . . . , xn) = ajÃ− s X i=1 x2 i+ n X i=s+1 x2 i!+ n X i=1 bi jxi+cj, with aj, b1 j, . . . , bn j, cj∈R, j = 1, . . . `. 4 The constant discriminant case We need the following result for later use. Lemma 4.1 Let φ:Nn s→Mm νbe an isometric immersion. Then, φis of constant discriminant at each point of Nn sif and only if there exists a smooth function D:Nn s→R such that Ah(Y,Z)X−Ah(X,Z)Y=D{g(Y, Z)X−g(X, Z)Y}(7) for any X, Y, Z ∈X(Nn s). 6 Proof. Suppose that φis of constant discriminant Dpat p∈Nn s. Let Tbe the 4-covariant tensor on TpNn sdefined by T(x, y, z, w) = g(h(x, w), h(y, z)) −g(h(x, z), h(y, w)) −Dp{g(x, w)g(z, y)−g(x, z)g(y, w)}.(8) Then, Tis a curvaturelike function, i. e., Thas the following symmetries: (i) T(x, y, z, w) + T(y, z, x, w) + T(z, x, y, w) = 0, (ii) T(y, x, z, w) = −T(x, y, z, w), (iii) T(x, y, w, z) = −T(x, y, z, w), (iv) T(z, w, x, y) = T(x, y, z, w). From Gauss equation (3) we have Dp=g(h(x, x), h(y, y)) −g(h(x, y), h(x, y)) g(x, x)g(y, y)−g(x, y)2, for all x, y ∈TpNn ssuch that g(x, x)g(y, y)−g(x, y)26= 0. Therefore T(x, y, y, x) = 0 and this means that T= 0 [21, p. 79]. This concludes the proof. ¤ Theorem 4.2 Let φ:Nn s→Mm νbe a λ-isotropic immersion with constant discriminant Dat each point of Nn s. Then, φis pseudo-umbilical and 3ng(H,H) = (n+ 2)λ+ 2(n−1)D.(9) In particular, we have (a) λ=Dif and only if λ=g(H,H). (b) λ=−{2(n−1)/(n+ 2)}D if and only if g(H,H) = 0. Proof. From Theorem 5 and equation (7), we have 3Ah(X,Y )Z= (λ− D){g(Y, Z)X+g(X, Z)Y} + (λ+ 2D)g(X, Y )Z. (10) Consider an orthonormal reference frame {e1, . . . , en}on Nn s. From Eq. (10) we obtain g(H, h(ei, ej)) = (εiρif i=j, 0if i6=j, where ρ= ((n+ 2)λ+ (2n−2)D)/3n, and then the immersion is pseudo-umbilical with 3ng(H,H) = (n+ 2)λ+ 2(n−1)D. The other statements are trivial. ¤ As a immediate consequence we have the following. 7 Corollary 4.3 Let φ:Nn s→Mm νbe a λ-isotropic immersion of a pseudo-Riemannian manifold Nn sof constant sectional curvature kinto a pseudo-Riemannian manifold Mm νof constant sectional curvature c. Then, φis pseudo-umbilical and 3ng(H,H) = (n+ 2)λ+ 2(n−1)(k−c).(11) In particular, if k=cthen φis 0-isotropic if and only if g(H,H) = 0. Remark 4.4 It is well known that an isometric immersion between two Riemannian spaces of the same constant sectional curvature is totally geodesic. In this case, this result is irrelevant when k=c. However, in the following example we obtain an isotropic immersion of Snin Sn+2 1with lightlike mean curvature vector, but the immersion is not totally geodesic. Example 4.5 Let f:Sn→Rbe a smooth function. The isometric immersion ψ:Sn→ Sn+2 1defined by ψ(x) = (f(x), ι(x), f(x)) is 0-isotropic, where ι:Sn→Rn+1 denotes the canonical inclusion. In fact, a straightforward computation shows that the second fundamental form hof ψis given by ([14]) h(X, Y ) = ¡(∇2f)(X, Y ) + fg(X, Y )¢η, where ηdenotes the lightlike vector (1,0, . . . , 0,1) ∈Rn+3 1and ∇2fis the Hessian of fin Sn. Thus, the immersion ψis 0-isotropic with mean curvature vector H=1 n(∆f+nf)η, where ∆ = traceg(∇2)is the Laplacian operator. On the other hand, an eigenfunction of −∆corresponding to the eigenvalue nis the restriction to Snof a homogeneous polynomial of degree 1 ([4]), and therefore the immersion ψis totally umbilical if and only if fis an affine function. Notice that Snis compact, Iso(Sn)is finite and the first Betti number of Snis zero. Corollary 4.6 Let φ:N2 s→Mm νbe a λ-isotropic immersion. Then, φis pseudo-umbilical and 3g(H,H) = 2λ+D. Proof. This follows immediately from Theorem 4.2. Notice that every surface has constant discriminant. ¤ Recall that we noted in Remark 3.1 (a) in Section 3 that every totally umbilical immersion φ:Nn s→Mm νis an isotropic immersion. The converse is not true in general (Remark 3.4). Here we can prove a slightly sharper result: an isotropic immersion φ: Nn s→Mm νwith n≥3does not need to be even pseudo-umbilical. In fact, we have the following counterexample. Example 4.7 Consider the complex Euclidean space Cn,n≥3and let γ:R→Cbe the smooth complex curve defined by γ(t) = pcosh(2t) exp{iarctan(tanh 2t)}. 8 It is easy to prove that φ:R×Sn−1→Cn,φ(t, x1, . . . , xn) = γ(t)(x1, . . . , xn)is an isometric immersion when R×Sn−1is endowed with the induced metric g= cosh 2t(dt2+g0),where g0denotes the standard Euclidean induced metric of Sn−1. Let {˜e2, . . . , ˜en}be a local orthonormal basis of Sn−1with respect to the metric g0. Now we define e1=1 |γ|(∂t,0), ei=1 |γ|(0,˜ei),2≤i≤n, and then we get a local orthonormal basis {ei}of φand {ei, Jei}a local orthonomal basis of Cn. We apply Lemma 4.2 in reference [17] and we find that the second fundamental form hof φis given by ½h(e1, e1) = aJe1h(e2, e2) = · · · =h(en, en) = −aJe1, h(e1, ej) = −aJejh(ej, ek) = 0,2≤j6=k≤n, where a=−(cosh 2t)−3/2.A straightforward computation gives that φis an isotropic immersion with isotropy function λ=a2which satisfies g(h(e1, e1),H) = −g(h(ej, ej),H) = a(2 −n)/n, 2≤j≤n. Thus, as n≥3we have that φis non-pseudo-umbilical. 5 The Ricci tensor of an isotropic submanifold Recall that the Ricci tensor of a pseudo-Riemannian manifold Nn swith metric tensor gis defined by Ric(X, Y ) = traceg(Z7→ R(Z, X)Y). A pseudo-Riemannian manifold is an Einstein manifold when the Ricci tensor satisfies Ric = µg for some constant µ∈ R. Pseudo-Riemannian manifolds Nn swith constant sectional curvature care simplest examples of Einstein manifolds, where µ=c(n−1). Conversely, for n= 2 or n= 3, every Einstein manifold has constant sectional curvature. However, there are 4-dimensional Einstein manifolds with non-constant sectional curvature. It is also well known that a connected manifold Nn swith n≥3and Ric = µg for some function µ:Nn s→R,is Einstein. For a pseudo-Riemannian space form Mm ν(c)with constant sectional curvature cwe have the following result. Theorem 5.1 Let φ:Nn s→Mm ν(c)be a λ-isotropic immersion. Then, (a) The Ricci tensor of Nn sis given by Ric(X, Y ) = {c(n−1) −λ(n+ 2)/2}g(X, Y ) +3ng(h(X, Y ),H)/2.(12) (b) If n≥3,φis pseudo-umbilical if and only if Nn sis an Einstein manifold. Proof. (a) Using the Gauss equation (3) we have Ric(X, Y ) = c(n−1)g(X, Y ) + ng(h(X, Y ),H)−b(X, Y ),(13) 9 (3) n= 4, holds, then Nn 1is isometric (up to a finite covering in the cases (2) and (3)) to a flat n-dimensional Lorentzian torus. Remark 7.4 It is well known that in the Riemannian case a homogeneous Ricci-flat manifold is flat, but in the Lorentzian case the corresponding result is not true. 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