Warped products and Spaces of Constant Curvature

Warped products and Spaces of Constant Curvature

Abstract

We will obtain the warped product decompositions of spaces of constant curvature (with arbitrary signature) in their natural models as subsets of pseudo-Euclidean space. This generalizes the corresponding result by \citeauthorNolker1996 in [Nolker1996] to arbitrary signatures, and has a similar level of detail. Although our derivation is complete in some sense, none is proven. Motivated by applications, we will give more information for the spaces with Euclidean and Lorentzian signatures. This is an expository article which is intended to be used as a reference. So we also give a review of the theory of circles and spheres in pseudo-Riemannian manifolds.

0] ]0 \addbibresource../../Math,../../Hamiltonan*Separation*Theory,../../Preprints*and*My*Journal \tensordelimiter? \newglossary[nlg]notationnotntnList of Notations \loadglsentriesHJ-theory-glossaries

\printglossaries

List of Results

1 Introduction

Warped products are ubiquitous in applications of pseudo-Riemannian geometry. Most of the separable coordinate systems in spaces of constant curvature are built up using them [Kalnins1986], and some exact solutions in general relativity are composed of them [Dobarro2005, Zeghib2011]. They can intuitively be thought of as a partial generalization of the spherical coordinate system to arbitrary pseudo-Riemannian manifolds. Indeed, it can be shown that all the spherical coordinate systems (on any space of constant curvature) can be constructed iteratively using warped products, and that they share several properties with these coordinate systems. Similarly the well known Schwarzschild metric in relativity can be constructed using warped products.

Various geometrical objects take canonical forms in warped products. For example, one can calculate general formulas for the Levi-Civita connection and the Riemann curvature tensor in a warped product [Meumertzheim1999]. These product manifolds can be used to construct geometrical objects with special properties. For example, it was shown in [Rajaratnam2014a], that one can use the warped product decompositions of a given space to construct Killing tensors and hence coordinates which separate the Hamilton-Jacobi equation. Thus it is only natural that we determine the warped products which are isometric to spaces of constant curvature.

We now describe more precisely the problem we solve, after introducing some definitions. A warped product is a product manifold of pseudo-Riemannian manifolds where for equipped with the metric

(1.1)

where are functions and are the canonical projection maps [Meumertzheim1999]. The warped product is denoted by . We say a warped product is a warped product decomposition of a pseudo-Riemannian manifold if it is isometric to some non-empty open subset of . In this article we will present an interesting class of warped product decompositions of spaces of constant curvature (with arbitrary signature).

Our solution follows that by \citeauthorNolker1996 in [Nolker1996], which is for the special case of Riemannian spaces of constant curvature. We make use of the observation that for a warped product , is a geodesic submanifold of and for each the manifold is a spherical submanifold of 2 [Meumertzheim1999]. Thus after characterizing all geodesic and spherical submanifolds of spaces of constant curvature, we generalize a formula given in [Nolker1996] to obtain an interesting class of warped product decompositions.

Our primary motivation for this work comes from [Rajaratnam2014a], where it was shown that warped products can be used to construct coordinates which separate the Hamilton-Jacobi equation. Based on this application, it will become clear (in a following article) that our work is “complete”. We are mainly interested in exposing these results for reference purposes. Since as of now, it is difficult to find any articles/books which can be used as a reference for our purposes. For a similar reason, we will present a review of the theory of circles and spheres in pseudo-Riemannian manifolds as well.

This article is mostly self-contained, so it can be used as a reference. However, we use some results from the theory of pseudo-Riemannian submanifolds in [chen2011pseudo], which are only necessary to understand certain proofs. We also assume the reader is familiar with [barrett1983semi], especially with the basic properties of pseudo-Euclidean vector spaces and (pseudo-)Riemannian submanifold theory. Familiarity with the article [Nolker1996] is useful but not necessary.

The article is organized as follows. After defining basic notations in Section 2, we summarize our results in Section 3. This summary should be sufficient for applications. The subsequent sections provide proofs and more details. In Section 4 we give a brief review of the theory of pseudo-Riemannian submanifolds/foliations. In Section 5 we apply these concepts by reviewing the theory of circles and spheres in pseudo-Riemannian manifolds. This section is optional but it gives a geometric interpretation of warped products and is included because there are relatively few reviews of this topic. In Sections 9, 8, 6 and 7 we review preliminary theory on the spherical submanifolds and warped products in spaces of constant curvature and warped products in general. We give the warped product decompositions of pseudo-Euclidean space in Section 10 and of spherical submanifolds of pseudo-Euclidean space in Section 12. Section 11 is another optional section which gives the isometry groups of spherical submanifolds of pseudo-Euclidean space, referring to [barrett1983semi] in the appropriate cases.

2 Notations and Conventions

All differentiable structures are assumed to be smooth (class ). Let be a pseudo-Riemannian manifold of dimension equipped with covariant metric . Unless specified otherwise, it is assumed that . The contravariant metric is usually denoted by and plays the role of the covariant and contravariant metric depending on the arguments. We denote by the set of functions from M to and denotes the set of vector fields over . If then we denote and .

Throughout this article we will be working in pseudo-Euclidean space, which is defined as follows. An -dimensional vector space equipped with metric of signature3 is denoted by \glseunn and called pseudo-Euclidean space. We obtain Euclidean space in the special case where . Also Minkowski space is obtained by taking . A subspace is called non-degenerate if the induced metric is non-degenerate and in this case we denote by the signature of the induced metric. Also note that since is a vector space, for any we identify vectors in with points in .

Given an open subset and , we denote by the central hyperquadric of contained in , which is defined by:

(2.1)

Usually and this is denoted \glseunnk. The notation represents a maximal connected component of . It is well known that is a pseudo-Riemannian manifold of dimension with signature and constant curvature4 [barrett1983semi]. Since , for any we identify vectors in with points in . Occasionally we use the following conventions: If , we set , if we set to be the light cone, i.e. the set of non-zero null vectors. We also use the following notations: If then , if then .

We define the parabolic embedding of in with mean curvature vector by [Tojeiro2007]

(2.2)

An explicit isometry with is obtained by choosing , i.e. is lightlike and . We let , note that , then for :

(2.3)

More details on the properties of will be given later on (see Proposition 7.18). Finally, we define the dilatational vector field in , , to be the vector field satisfying for any , .

3 Warped products in Spaces of Constant Curvature

In this section we will briefly describe the warped product decompositions of spaces of constant curvature, in a way which is useful for applications. The proofs of many of the assertions will come in the following sections. We will use the notation (where can be zero) to represent the general space of constant curvature. First we will need to know the spherical submanifolds of these spaces.

Theorem 3.1 (Spherical submanifolds of )

Let be arbitrary, a non-degenerate subspace with , and . Let , and . There is exactly one -dimensional connected and geodesically complete spherical submanifold with , and having mean curvature vector at , z. is an open submanifold of N; N is referred to as the spherical submanifold determined by , it is geodesic iff and is given as follows (where means isometric to):

  1. , in this case

    (3.1)
  2. is timelike, then and

  3. is spacelike, then
    For cases (b) and (c), let be the center of N, then N is given as follows:

    (3.2)
  4. is lightlike, then and

    (3.3)

Remark 3.2

If is lightlike, then is isometric to with mean curvature vector . Furthermore, let be a lightlike vector satisfying . Then the orthogonal projector onto , , induces an isometry of onto .

Remark 3.3

One can find more details on when is connected in the remarks following Theorems 12.38 and 10.25.

Proof

See Theorems 12.38 and 10.25.

With the knowledge of these spherical submanifolds, we can now specify how to construct warped products in . This construction depends on the following data: A point , a decomposition into non-trivial (hence non-degenerate) subspaces with , and vectors such that the vectors are pair-wise orthogonal and independent. We call the data , initial data for a (proper) warped product decomposition of . If , one can more generally let some of the be zero, this results in Cartesian products as done in [Nolker1996]. Since we assume the are non-zero, we sometimes use the additional qualifier “proper”.

With this initial data, for let be the sphere in determined by and . Let be the subset of the sphere in determined by where each . Then the data , induces a warped product decomposition (of ) given as follows:

(3.4)

We note that has the property that . Often the point doesn’t enter calculations, hence we will usually omit it. We note that the above formula generalizes one given in [Nolker1996].

For actual calculations, it will be more convenient to work with canonical forms. The following definition will be particularly useful.

Definition 3.4 (Canonical form for Warped products of )

We say that a proper warped product decomposition of determined by is in canonical form if: and .

Any proper warped product decomposition of can be brought into canonical form, see the discussion preceding Corollary 10.34 for details.

We will now give more information on standard warped product decompositions of in canonical form. Suppose the initial data is in canonical form, and let be the associated warped product decomposition given by Eq. 3.4. Denote and . We have two types of warped products:

non-null warped decomposition

If , let and .

null warped decomposition

If , then is lightlike, so fix another lightlike vector such that , let and .

For , let be the orthogonal projection. Then the following holds:

Theorem 3.5 (Standard Warped Products in [Nolker1996])

Let be the warped product decomposition of determined by the initial data given above. Then has the following form:

(3.5)

and

(3.6)

The map is an isometry onto the following set:

(3.7)

Furthermore, the following equation holds:

(3.8)

Proof

See Corollary 10.34.

In fact, for , has one of the following forms, first if is non-null:

(3.9)

where , and if is null:

(3.10)

The above forms are obtained from the equation for from the above theorem by expanding in an appropriate basis. We note that the warped products with multiple spherical factors can be obtained using the standard ones described above. Indeed, suppose is the warped product decomposition determined by as above. Since is pseudo-Euclidean, consider a warped product decomposition, , determined by with (hence ). Note that is the subspace from the above construction for . Let , then one can check that the map defined by:

(3.11)

is a warped product decomposition of satisfying Eq. 3.4. We illustrate this construction with an example.

Example 3.6 (Constructing multiply warped products)

Suppose and are given as follows:

(3.12)
(3.13)

Now observe that , which follows from the above equation for and the fact that . Then,

(3.14)
(3.15)
(3.16)

where is the orthogonal projector onto . A similar calculation shows that satisfies Eq. 3.4, since and each satisfy it.

This procedure can be repeated as many times as necessary to obtain the more general warped products given by Eq. 3.4. Hence the properties of the more general warped product decompositions of can be deduced from Theorem 3.5.

The following proposition shows that any proper warped product decomposition of in canonical form restricts to a warped product decomposition of where . Its proof is straightforward consequence of Eq. 3.8.

Theorem 3.7 (Restricting Warped products to )

Let be a proper warped product decomposition of associated with in canonical form. Suppose and let . Then defined by is a warped product decomposition of passing through .

Proof

See Theorem 12.41.

Hence the details of warped product decompositions of can be deduced from Theorem 3.5. More information on these decompositions can be found in the following sections. In particular, see Theorems 12.44 and 10.29. Some examples can be found in [Nolker1996] and also in a future article where we apply these results to construct coordinates which separate the Hamilton-Jacobi equation.

4 pseudo-Riemannian Submanifolds and Foliations

In this section we will summarize the theory of pseudo-Riemannian submanifolds and foliations that will be useful to us. We can conveniently treat this as a special case of the theory of pseudo-Riemannian distributions, so we will present this first. For more details on pseudo-Riemannian submanifolds see (for example) [barrett1983semi, Lee1997]. Similarly for pseudo-Riemannian foliations see [rovenskii1998foliations, tondeur1988foliations].

4.1 Brief outline of The Theory of Pseudo-Riemannian Distributions

The following brief exposition of the theory of pseudo-Riemannian distributions is a combination of that given in [Meumertzheim1999] and [Coll2006]. Suppose is an m-dimensional non-degenerate distribution defined on a pseudo-Riemannian manifold . Then we use the orthogonal splitting , , to define a tensor and a linear connection for by:

(4.1)

for all and . is called the generalized second fundamental form of and the above equation is referred to as the Gauss equation. One can also check that is metric compatible, i.e. for all and .

For the remainder of the discussion we set . For , we can further decompose into its anti-symmetric and symmetric parts

(4.2)
(4.3)
(4.4)
(4.5)

Since is torsion-free, , hence is integrable iff . is called the second fundamental form of . The second fundamental form can be decomposed in terms of its trace to get a further classification of as follows:

(4.6)
(4.7)

where is trace-less. is called the mean curvature normal of . is called minimal, umbilical or geodesic5 if , or respectively for all . We add the qualification “almost” to the three definitions above by replacing with ; this just drops the requirement that . For example is almost umbilical iff . We remark that when is one dimensional trivially, hence all one dimensional non-degenerate foliations and all one dimensional pseudo-Riemannian submanifolds are trivially umbilical. If is umbilical and for all then is called spherical. Finally if is spherical and is geodesic then is called Killing.

We also note here that and are not independent of each other:

Proposition 4.8

For and , the following holds:

(4.8)

Proof
(4.9)
(4.10)
(4.11)

4.2 Specialization to pseudo-Riemannian submanifolds

Suppose is a local embedding of (a pseudo-Riemannian submanifold) inside . Then for any point , it is known that there exist local coordinates on , such that the subset

(4.12)

for some can be identified with where is an open subset with . These coordinates induce a local foliation in a neighborhood of , with being a leaf given by the above equation. We will refer to such a foliation as a (local) foliation of associated with . Now suppose is an arbitrary foliation of associated with , and let be the induced distribution. Locally we can assume is a foliation by pseudo-Riemannian submanifolds of , hence is non-degenerate and the discussion in the previous section applies to it. Since is integrable, it follows that for any , that . Throughout this discussion, for any , we let denote the unique vector field such that for any , we have . Then for any we see that

(4.13)

Thus depends only on in .

Now denote by (resp. ) the Levi-Civita connection on (resp. ). By the uniqueness properties of the Levi-Civita connection on , it follows that for any we have for any that

(4.14)

Thus depends only on in . By also using the Gauss equation, we observe that for any , that depends only on and .

In consequence of these observations, it follows that the theory presented for pseudo-Riemannian distributions induces a similar one for pseudo-Riemannian manifolds. We now connect this with the standard notations [chen2011pseudo]; in effect this removes the appearance of the extraneous distribution, .

In this case and , then the Gauss equation becomes:

(4.15)

for all . We denote the set of normal vector fields over , i.e. the restriction of to by . The Gauss equation for is usually called the Weingarten equation and is only defined for and . This is because in this case, depends only on the values that and take on 6. Thus we can let , and the Gauss equation (for ) becomes:

(4.16)

for all and . Note that the properties of imply that is a connection7 on . In this notation, the relationship between and given in Proposition 4.8 becomes:

(4.17)

for all and .

Finally, we note that the definitions of minimal, umbilical, or geodesic foliations induces corresponding definitions for submanifolds. For example, a submanifold is geodesic if its second fundamental form vanishes identically.

In conclusion, we should mention that even though we have given a concise presentation of the theory, it’s not useful for practical calculations. For these, one will have to evaluate these quantities in terms of curves on . See for example, Proposition 4.8 in [barrett1983semi].

5 Circles and Spherical Submanifolds*

In this section we will briefly overview the theory of circles and spherical submanifolds of pseudo-Riemannian manifolds. Circles are covariantly defined using the Frenet formula, but the definition of a sphere requires more work [Nomizu1973]. This material is not necessary to read the rest of the article, although it gives an application of the general theory presented in the previous section, a geometric interpretation of spherical submanifolds, and gives some background for the results on the intrinsic properties of warped products to come. We also present this theory here because it’s not covered in standard references, in contrast with the corresponding theory for geodesic submanifolds (see [barrett1983semi]).

A proper circle8 in a pseudo-Riemannian manifold is defined using the Frenet formula as a unit speed curve whose first curvature is constant and non-zero and remaining curvatures vanish. To be precise, let be a unit speed curve in , i.e. . Let . Let be the (first) curvature of . Assuming , we define to be the unit vector field over derived from . I.e. satisfies the following equation

(5.1)

A proper circle is defined to be a curve which satisfies for some . We observe that

(5.2)
(5.3)

The above equation implies that where and . Thus a proper circle is defined by the equations

(5.4)
(5.5)

where is a constant. A proper circle satisfies the following third order ODE [Abe1990]:

(5.6)

Conversely we will see shortly that any unit speed curve satisfying the above equation with is a proper circle. We define a circle in a pseudo-Riemannian manifold to be a unit speed curve satisfying the above equation, hereafter called the circle equation. The following lemma shows that any pseudo-Riemannian manifold admits circles:

Lemma 5.9 (Existence and Uniqueness of Circles [Nomizu1974])

Consider the following initial conditions: , a unit vector and . There exists a unique locally defined unit speed curve in satisfying Eq. 5.6 and the initial conditions:

(5.7)
(5.8)
(5.9)

where and . Furthermore, is constant along any circle.

Proof

It follows by the existence and uniqueness theorem for ODEs that there exists a unique locally defined curve satisfying Eq. 5.6 with the above initial conditions. Then observe the following:

(5.10)
(5.11)
(5.12)
(5.13)

The above two equations define a system of ODEs for and , with initial values and . Thus by the uniqueness of the solutions, it follows that and wherever is defined. Hence is a unit speed curve.

Finally observe that

(5.14)
(5.15)
(5.16)

Hence is constant.

Note that in the above lemma is usually called the curvature of the circle. In Riemannian manifolds, circles are completely classified by their curvature, although this is not true for pseudo-Riemannian manifolds. Using the above lemma we can classify circles in a pseudo-Riemannian manifold as follows. Let be a circle in and suppose satisfies the initial conditions of the above lemma. Then can be classified as follows depending on :

Geodesic:

If .

Proper Circle:

If .

Null Circle:

If but , i.e. is lightlike, hence Eq. 5.6 reduces to .

Note that this classification is well defined globally since is a constant of a circle and the uniqueness theorem for ODEs forces any circle with to be a geodesic.

Example 5.10 (Geodesics in Spherical Submanifolds [Kassabov2010])

Let be a spherical submanifold of . Suppose is a unit speed geodesic on . We will show that is a circle in . By the Gauss equation, we have the following:

(5.17)

Then by the Weingarten equation and using the fact that where is the induced normal connection over , we have the following:

(5.18)
(5.19)
(5.20)
(5.21)

since for any , .

We note here that the above example in combination with Lemma 5.9 shows that the mean curvature vector field of a spherical submanifold is locally determined by its value at a single point. In Example 10.28 we will describe the circles in pseudo-Euclidean space after we have described the spherical submanifolds of the space.

We will now present some additional results that show how circles can be used to characterize spherical submanifolds. These results were first obtained for the Riemannian case by \citeauthorNomizu1974 in [Nomizu1974]. They were generalized to the Lorentzian case by \citeauthorIkawa1985 in [Ikawa1985] and to the pseudo-Riemannian case by \citeauthorAbe1990 in [Abe1990].

For the following theorems we denote a pseudo-Riemannian manifold with signature by . The following theorem characterizes spherical submanifolds in terms of circles, it is analogous to the corresponding theorem for geodesics and geodesic submanifolds (see [barrett1983semi, section 4.4]).

Theorem 5.11 (Circles and Spheres [Abe1990])

Let be an dimensional pseudo-Riemannian submanifold of . For any and satisfying and , the following are equivalent:

  1. Every circle in with and is a circle in .

  2. is a spherical submanifold of .

Proof

See [Abe1990].

More intuitively, the above theorem states that a spherical submanifold is precisely a submanifold in which all circles in are circles in the ambient space. Also note that the above theorem shows that a circle is precisely a spherical submanifold of dimension one. The following theorem is a variant of the above theorem which is known to hold (in full generality) only in the strictly pseudo-Riemannian case.

Theorem 5.12 (Circles and Spheres II [Abe1990])

Let be an dimensional () pseudo-Riemannian submanifold of having the same signature . For any , the following are equivalent:

  1. Every geodesic in with is a circle in .

  2. is a spherical submanifold of .

These results can be further generalized by considering more general types of curves such as helices (which we will not define here). See [Nakanishi1988] where a theorem analogous to Theorem 5.11 is proven characterizing helices in terms of geodesic submanifolds. Also in [Jun1994] results relating conformal circles to umbilical submanifolds are presented.

The following lemma describes how much information is required to specify a sphere. It is a partial generalization of the corresponding lemma for the Riemannian case proven in [Kassabov2010].

Lemma 5.13 (Uniqueness of Spheres)

Suppose that and are connected and geodesically complete spherical submanifolds of both satisfying the following condition: For some , and are tangent and have the same mean curvature vectors. Then .

Proof

Our proof is a generalization of the proof of lemma 4.14 in [barrett1983semi, P. 105].

Let be arbitrary and suppose that is a geodesic segment in running from to . Then observe that is a geodesic circle in with velocity and acceleration at where is the mean curvature vector field of . By the uniqueness of circles (see LABEL:{lem:circUniq}) and the hypothesis it follows that is also geodesic in which is defined everywhere since is geodesically complete. Note that this implies that mean curvature vector fields of and coincide over , so we denote this vector field by .

Now suppose and let be the parallel transport of over with respect to . Since parallel transport is an isometry, . Thus by the Gauss equation,

(5.22)
(5.23)

where is the Levi-Civita connection on and is the induced Levi-Civita connection on . Thus is also the parallel transport of over with respect to .

Thus the parallel transport of to on is equal to . Similarly the parallel transport of to on is equal to . Since the parallel transport on is uniquely determined, we deduce that . Since , we conclude that . Thus since is connected, one can apply this argument to an arbitrary broken geodesic (see [barrett1983semi]) to conclude that .

Finally by applying the argument for interchanged with , we see that .

Let be a space of constant curvature. We will show in this article that for every , non-degenerate subspace , and normal vector there exists a connected and geodesically complete spherical submanifold passing through with tangent space and mean curvature vector at . In the following theorem, we will show that this property characterizes Riemannian spaces of constant curvature. For the following theorem, we say a Riemannian manifold satisfies the axiom of -spheres if: for every and any dimensional subspace there exists a spherical submanifold passing through and tangent to .

Theorem 5.14 (Spheres in spaces of constant curvature [leung1971])

Let be a Riemannian manifold with dimension and fix