Constant angle null hypersurfaces

The classification of submanifolds subject to certain geometric restrictions has been one of the main driving forces in semi-Riemannian geometry. The characterization of totally geodesic, totally umbilical, minimal or isoparametric submanifolds of specific classes of ambient spaces are active areas of research up to this day (for detailed accounts refer to [7, 9, 12]). Historically, among the first such objects to be explored are the so-called constant angle submanifolds, its study dating back to the works of J. Bernoulli on spirals and rhumb lines [38].

In the Riemannian case, given an oriented immersed hypersurface $M$ of a Riemannian manifold $(\bar{M},\bar{g})$ and a distinguished vector field $V\in\Gamma(T\bar{M})$, $V\neq 0$, the angle $\theta$ of $M$ with respect to $V$ is defined by

where $\mathbf{n}$ is a unit normal vector field to $M$. Thus, a constant angle hypersurface is characterized by the constancy of $\theta$. This notion can be straightforwardly generalized to the semi-Riemannian context by requiring the right-hand size in Eq. (1) to be constant along $M$. Notice that this latter expression carries a distinct geometrical meaning in some specific settings. For instance, in the Lorentzian case, it can be interpreted in terms of a hyperbolic cosine, when $M$ is spacelike and $V$ is timelike [35]. From the geometrical point of view, the most meaningful properties arise in the cases in which the vector field $V$ carries some relevant geometrical information. Classical examples of such fields include those that arise as infinitesimal generators of geometrical transformations like isometries (Killing fields) or conformal maps (closed conformal or concircular fields). It is worthwhile mentioning that the latter class include parallel, radial and gradient vector fields [27].

Several classification results for constant angle semi-Riemannian (non-degenerate) hypersurfaces have been established recently in ambient spaces such as cartesian products [10, 13, 16, 19], warped products [15, 22], spaceforms [20, 28, 30, 34] and other geometrically relevant spaces [1, 36, 37]. One remarkable relation between the distinguished vector field $V$ and the geometry of a constant angle hypersurface $M$ is encapsulated in the notion of canonical principal direction: the tangent component $V^{\top}\in\Gamma(TM)$ is a principal direction of $M$ [11, 21, 23, 29].

We notice that the above discussion deals exclusively with non-degenerate hypersurfaces. Nevertheless, some of the most studied submanifolds in Lorentzian geometry and mathematical relativity —such as event and Killing horizons— are null, that is, degenerate [26, 25, 40]. In spite of its relevance in theoretical physics, a framework for the study of null submanifolds in a spirit close to the classical Riemannian submanifold theory was missing until the 1980’s. The main difficulty to be surmounted in this effort lies in the fact that in the realm of Lorentzian geometry the vector fields orthogonal to a null hypersurface $M\subset\bar{M}$ are also tangent to it. Hence, there is no canonical orthogonal splitting of $T\bar{M}$ into normal and tangent components. In [17] K. Duggal and A. Bejancu established the foundations for a null submanifold theory based on the choice of an additional structure, the so called screen distribution $S(TM)$. A considerable amount of research has developed following this approach (see for instance [18] and references therein). In particular, the study of null hypersurfaces in generalized Robertson-Walker spacetimes is an active area of research to this day [31].

Following [17, 18], we denote by $(\bar{M},\bar{g})$ an $(n+2)$-dimensional Lorentzian manifold with metric $\bar{g}$ and by $M$ an $(n+1)$-manifold immersed in $\bar{M}$ with degenerate induced metric $g$. Equivalently, there exists a vector field $\xi\neq 0$ tangent to $M$ such that

Since $\dim T_{p}M+\dim T_{p}M^{\perp}=\dim T_{p}\bar{M}$, the radical space $\mathrm{rad}(T_{p}M)=T_{p}M\cap T_{p}M^{\perp}$ at each $p\in M$ satisfies

The radical bundle $\mathrm{rad}(TM)$ is given by

A screen distribution on $M$ is a vector sub-bundle $S(TM)$ of $TM$ such that

thus, $S(TM)$ is complementary to $\mathrm{rad}(TM)$ in $TM$.

Notice that due to the Lorentzian character of $\bar{g}$, the metric restricted to $S(TM)$ is positive definite. Intuitively, an adequate choice of $S(TM)$ would enable us to relate the geometry of $(M,g)$ to a more familiar (Riemannian) geometry in $S(TM)$. Henceforth, by a null hypersurface we mean a triple $(M,g,S(TM))$.

The last main ingredient in this framework is the so called transversal bundle $\mathrm{tr}(TM)$, which is the unique rank $1$ vector bundle with the following property: given $\xi\in\Gamma(\mathrm{rad}(TM))$ there exists a unique $N\in\Gamma(\mathrm{tr}(TM))$ such that

Thus, we can split the tangent bundle of the ambient manifold as

In virtue of decompositions (2) and (4) we can introduce the notions of induced connections, shape operators and second fundamental forms, as well as the fundamental equations relating them.

Let $\bar{\nabla}$ be the Levi-Civita connection of $(\bar{M},\bar{g})$ and $V\in\Gamma(\mathrm{tr}(TM))$. Given $X,Y\in\Gamma(TM)$, we can write the Gauss and Weingarten formulae related to decomposition (4) as

where $\nabla_{X}Y,A_{V}X\in\Gamma(TM)$, and $h(X,Y),\nabla_{X}^{t}V\in\Gamma(\mathrm{tr}(TM))$. The above induces linear connections $\nabla$ and $\nabla^{t}$ on $TM$ and $\mathrm{tr}(TM)$, respectively. Notice however that $\nabla$ is not a metric connection, though it is torsion-free. Further, $A_{V}:\Gamma(TM)\to\Gamma(TM)$ and $h$ is a symmetric section of $\mathrm{Hom}^{2}(TM,TM;\mathrm{tr}(TM))$ that satisfies

for all $X,Y,Z\in\Gamma(TM)$. $\nabla$, $A_{V}$ and $h$ are called the induced connection, shape operator and second fundamental form of $M$, respectively.

On the other hand, relative to decomposition (2) we have another set of Gauss-Weingarten formulae for $TM$:

where $U\in\Gamma(\mathrm{rad}(TM))$, $P$ denotes the projection of $TM$ onto $S(TM)$; $\nabla^{*}$ and $\nabla^{*t}$ are connections on $S(TM)$ and $\mathrm{rad}(TM)$, respectively. Here $A_{U}^{*}:\Gamma(TM)\to\Gamma(S(TM))$ and $h^{*}\in\mathrm{Hom}^{2}(TM,S(TM),\mathrm{rad}(TM))$ are the screen shape operator and screen second fundamental form of $S(TM)$, respectively. A straightforward computation shows that $\nabla^{*}$ is a metric connection with respect to $\bar{g}|_{S(TM)}$.

For further reference, we cite some of the standard results pertaining null hypersurfaces that will be used throughout this work. For complete proofs, refer to Proposition 2.1.2 and 2.2.6 in [17] and [18], respectively.

For $Y\in\Gamma(T\bar{M})$, applying decompositions (2) and (4) leads to

where

and $\overset{*}{Z}$ is the projection of $Y$ on $S(TM)$. We now establish the basic relations arising from coupling the above relation to the Gauss and Weingarten formulae when $X\in\Gamma(S(TM))$.

In summary

Let us now consider a null frame $\{\xi,N\}$ as in Eq. (3) and denote by $B$, $C$ the bilinear forms associated to $h$ and $h^{*}$, respectively. That is

Thus, in virtue of equations (6) and (8) we have

for all $X,Y\in\Gamma(TM)$. It then follows that

Notice that since $h$ is symmetric, then by Eq. (14) $A_{N}$ is symmetric with respect to $g$. Hence Proposition 2.3 yields that the same is true for $A_{\xi}$ provided $S(TM)$ is integrable.

The null mean curvature $H$ is the trace of the shape operator $A^{*}_{\xi}$. Due to Eq. (15) we have

where $\{e_{1},\ldots,e_{n}\}$ denotes an orthonormal frame field in $S(TM)$.

Furthermore, note that $\bar{g}(\nabla^{t}_{X}N,\xi)=-\bar{g}(\nabla_{X}^{*t}\xi,N)$. Hence, in terms of the one form $\tau$ defined by

the Weingarten formulae read as

The curvature operators $\bar{R}$, $R$, $R^{*}$ associated to the linear connections $\bar{\nabla}$, $\nabla$, $\nabla^{*}$ are intertwined in a series of fundamental equations that resemble the standard Gauss-Codazzi structure equations in the non-degenerate case. We only present one of them, that will be used in Theorem 6.4 and refer to it as the Codazzi equation of $M$. For all $X,Y,Z\in\Gamma(TM)$ and $U\in\Gamma(\textrm{rad}(TM))$ we have

Thus, once the null fields $\{\xi,N\}$ are fixed,

where Eq. 7 translates to

Many of the most significant geometrical restrictions widely studied in the semi-Riemannian (non degenerate) theory of submanifolds have analogues in the null setting. For instance, $M$ is totally umbilical (resp. totally geodesic) if there exists a smooth function $\lambda$ such that $h(X,Y)=\lambda g(X,Y)$ (resp. $h\equiv 0$). Similarly, $S(TM)$ is totally umbilical (resp. totally geodesic) if there exists a smooth function $\lambda$ such that $h^{*}(X,Y)=\lambda g(X,Y)$ (resp. $h^{*}\equiv 0$). In terms of the shape operators, we have that $M$ is totally umbilical (resp. totally geodesic) if $A_{\xi}^{*}PX=\lambda PX$ ($A_{\xi}^{*}\equiv 0$) and likewise, $S(TM)$ is totally umbilical (resp. totally geodesic) if $A_{N}X=\lambda PX$ ($A_{N}\equiv 0$) for all $X\in\Gamma(TM)$. Notice, that all these conditions, as well as the vanishing of the null mean curvature ($H\equiv 0$) do not depend on the choice of the null frame $\{\xi,N\}$.

As an illustrative example, let us focus on the case of generalized Robertson-Walker spacetimes, or GRW spacetimes for short. Let us recall that they are defined as Lorentzian warped products of the form $\bar{M}=-I\times_{\varrho}F$, with $(F,g_{F})$ a Riemannian manifold and $\varrho$ a positive smooth function defined on the real interval $I$. Thus we have

where $\sigma$ and $\pi$ are the projections of $\bar{M}$ over $I$ and $F$, respectively. If the fiber $F$ is a Riemannian manifold of constant sectional curvature suitable choices of the warping function deliver open sets of all Lorentzian spaceforms as follows:

• •

  Lorentz-Minkowski space: $\mathbb{L}^{n+2}=-\mathbb{R}\times_{\text{id}}\mathbb{R}^{n+1}$.

• •

  de Sitter space: $\mathbb{S}_{1}^{n+2}=-\mathbb{R}\times_{\cosh}\mathbb{S}^{n+1}$.

• •

  Anti de Sitter space $\mathbb{H}_{1}^{n+2}=-\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\times_{\cos}\mathbb{H}^{n+1}$.

The above example serves as motivation for the concept of screen quasi-conformal hypersurface, which will be studied in the next section. For a detailed account, see [33, 32].

One key question that arises naturally is whether the geometry of $S(TM)$ as a subbundle of $TM$ is related in a natural way to its geometry as a subbundle of $T\bar{M}$. In [3] K. Duggal and C. Atindogbe define the notion of screen conformal null hypersurface and show that the answer is affirmative for this class of hypersurfaces. In precise terms, a null hypersurface $(M,g,S(TM))$ is screen conformal if its shape operators are linearly dependent, so there exists $\phi\in\mathcal{C}^{\infty}(M)$ such that

Numerous classification results exist for such hypersurfaces when the ambient manifold $(\bar{M},\bar{g})$ has constant curvature [18]. However, in virtue of Example 2.6 an important class of space-times does not fit in this context. Thus, in order to incorporate such examples, the notion of screen quasi-conformal null hypersurface was introduced in [32].

Screen quasi-conformal null hypersurfaces satisfying classical geometric restrictions (totally geodesic, totally umbilical, isoparametric, Einstein) have been studied recently (see [31] and references therein). Moreover, if $(M,g,S(TM))$ is screen quasi-conformal, then $S(TM)$ is integrable (see Theorem 3.7 in [32]). As can be readily checked, the null hypersurfaces $(M,g,S^{*}(TM))$ depicted in Example 2.6 are screen quasi-conformal with quasi-conformal pair $(1,\sqrt{2}\varrho^{\prime}/\varrho)$. ²²2Notice that in this case the following additional conditions hold: (a) $\phi$ and $\psi$ are constant along the leaves of $S(TM)$, and (b) $\tau(X)=0$ for all $X\in\Gamma(S(TM))$.

At this point it is natural to ask under which circumstances we can endow a degenerate hyperfurface with a screen distribution in a way that the corresponding null hyperfurface $(M,g,S(TM))$ is screen conformal. In [18], Theorem 2.3.5 establishes that if the ambient space admits a parallel vector field, then it induces a screen conformal structure in any null hyperfurface. As the next result shows, there is an analogous connection between closed and conformal vector fields and screen quasi-conformal distributions, thus generalizing the aforementioned result.