Isometric Embeddings between Space Forms David Brander A THESIS in Mathematics Presented to the Faculties of the University of Pennsylvania in Partial Ful¯llment of the Requirements for the Degree of Master of Philosophy 2003 Supervisor of Thesis Graduate Group Chairperson Acknowledgements Special thanks are due (in no particular order) to Professor Wolfgang Ziller for taking on the task of dissertation advisor and giving me advice whenever I needed it, to Dr. Ben Andrews for many helpful discussions, to Professor Chris Croke for helping me out with the ¯nal details of the dissertation, to Professor Dennis DeTurck for acting on my examination committee, to Professor David Harbater for always being willing to provide helpful advice during the time I spent at Penn, to all the administrative sta® in the Mathematics Department at Penn, who are always willing to help in any way they can, to all my friends among the graduate students at Penn (who know who they are) and especially to Cheni Wang for her love and support. ii ABSTRACT Isometric Embeddings between Space Forms David Brander Advisor: Wolfgang Ziller In 1901 D. Hilbert proved that there is no global isometric immersion of the hyperbolic plane into 3-dimensional Euclidean space, despite the fact that there is a local isometric embedding. Today it is known that there is a global isometric immersion of H2 into E5 and a global isometric embedding into E6, but it is an open problem whether these codimensions can be reduced. More generally, one can ask what the minimal codimensions are for a local embedding, complete immersion and n n+k global embedding of a simply connected space form Qc into another Qc~ , where the curvatures c and c~ are not equal. This thesis seeks to present the best results currently available with regard to this question. In particular, we will show that the problem is solved for all cases apart from global immersions and embeddings of negative extrinsic curvature (c < c~), when c = 0. For these unresolved cases we show 6 that the upper bounds for the minimal codimensions are quite small by constructing n explicit embeddings and immersions. The worst case is that of embedding H 1 into ¡ n+k 2 H c2 , for 1 < c < 0, where we only know that n 1 k 5n 5, and the best ¡ ¡ ¡ ¡ · · ¡ n n+k unresolved case is that of immersing a sphere SR into a smaller sphere Sr , where we know that the minimal codimension is either n or n + 1. We also present those non-existence results which are currently known. iii Contents 1 Introduction 1 1.1 Notation and Preliminaries . 3 2 Positive Extrinsic Curvature - Umbilic Hypersurfaces 5 2.1 0 < ~c < c: Spheres inside Larger Spheres . 6 2.2 ~c = 0 < c: Spheres into Euclidean Space . 6 2.3 ~c < 0; ~c < c : Hypersurfaces of Hn . 6 3 Negative Extrinsic Curvature: Local Results 10 3.1 A Non-Existence Result . 10 3.2 The Second Fundamental Form . 13 3.3 Explicit Local Embeddings . 20 3.3.1 0 < c < ~c: A Portion of a Sphere Embedded into a Smaller Sphere . 21 3.3.2 c = 0 < ~c: The Cli®ord Torus . 23 3.3.3 c < 0: Local Embeddings of Hyperbolic Space . 23 iv 4 Complete Immersions of Negative Extrinsic Curvature 26 4.1 0 = c < ~c: Euclidean into Spherical . 26 4.2 0 < c < ~c: Spheres into Spheres . 27 4.3 c < 0 : Immersing Hyperbolic Space - Some Constructions . 27 4.4 Non-Immersibility of the Hyperbolic Plane . 29 5 Global Embeddings of Negative Extrinsic Curvature 33 5.1 0 < c < ~c: Spheres inside Smaller Spheres . 33 5.2 c 0 < ~c : Hn and En into a Sphere . 36 · n 6n 6 5.3 Embedding H into E ¡ . 37 n 6n 5 5.4 Embedding H into H~c ¡ . 40 6 Conclusion 41 6.1 Open Problems . 42 A Smoothed Step Functions 44 v Chapter 1 Introduction A space form is a complete connected Riemannian manifold of dimension n 2 and ¸ constant sectional curvature c. Up to isometry, there is a unique simply connected n space form of dimension n and sectional curvature c, denoted here as Qc . The special cases where c is -1, 0 or 1 are the standard hyperbolic, Euclidean and spherical spaces, denoted Hn, En and Sn respectively. In this thesis we will address the question: when is it possible to isometrically embed, immerse or locally embed Qn into Qn+k, where c = c~. In particular, we would c c~ 6 like to know for each case what is the smallest codimension k for which this can be done. Naturally, this codimension must be at least one, because the two manifolds have di®erent sectional curvature. The class of di®erentiability we are considering is always C1. One can obtain isometric embeddings in very low codimensions of class C1 [14], but they do not preserve the di®erential geometric nature of the manifold 1 in question, because curvature comes from the second derivative of the metric. It is worthy of note that some of the global results we will give (speci¯cally the immersions and embeddings of the hyperbolic space with negative intrinsic curvature) are not real analytic. n n+k If Mc is a space form immersed into another Mc~ , then the di®erence in sectional curvatures, c c~, will be called the extrinsic curvature. It turns out that the problem ¡ of embeddings with positive extrinsic curvature is quite easy, and there is always a global embedding in codimension one. Therefore, the majority of this thesis is a discussion of the more problematic case of negative extrinsic curvature, where, in most cases, only the local problem has really been given a sharp answer. An example of the kind of thing that can happen is the following: the map f : ( ; 0) R E 3 ¡1 £ ! given by x f(x; y) = [ p1 e2tdt; exeiy]; Z0 ¡ where we have identi¯ed E3 with R C, is an immersion with constant sectional £ curvature 1. It looks like a horn enclosing the x axis in E3, with the wide end a ¡ 1 circle in the x -x plane and the pointed end at x = . It is a part of the hyperbolic 2 3 1 ¡1 plane, but not the whole, because it is not complete, being clearly incomplete as a metric subspace of E3. It is a famous theorem of Hilbert that there is in fact no complete immersion of H2 into E3 - even though there is a local embedding. It is also an interesting (and apparently still open) question whether the lowest codimension needed for a complete immersion is the same as that required for a global embedding. 2 It has been known for some time that there is a C 1 isometric embedding of any Riemannian manifold into En for su±ciently large n [15], [8]. Since, as will be shown below, En can be isometrically embedded into H n+1, and also realised as a one-to-one isometric immersion into a sphere of any radius of dimension 4n 1, this means that ¡ n+k any Riemannian manifold can be isometrically embedded into Qc , for any c, if k is large enough, with the caveat that if the manifold is not compact, and if c > 0, then \embedding" must be replaced with \one-to-one immersion". However, these general embedding theorems require a very large codimension, and it turns out that, for speci¯c cases, much better results are obtained by constructing explicit embeddings. Finally, why is it interesting to isometrically embed only simply connected space forms? The answer is that this is a natural ¯rst step to the general problem of n n embedding space forms, because any space form Mc has Qc as its universal cover. n n+k Thus, for example, an isometric embedding of Qc into Qc~ automatically gives an n n+k isometric immersion of Qc into any space form Mc~ . On the other hand, in order to n n+k have a chance of embedding Mc into Qc~ , it is necessary that there be an immersion n of Qc , which would be the lift of the desired embedding. 1.1 Notation and Preliminaries n Mc always stands for a Riemannian manifold of dimension n and constant sectional curvature c. In addition to the designations mentioned above, it is also common to n n use, for the sphere of radius r, the notation Sr instead of Q 1 , and when c < 0, r2 3 n n Hc = Qc . A metric is usually denoted by g and the second fundamental form by h. When a manifold is immersed in another manifold with the induced metric we will usually use g for the metric on both spaces. It is often convenient to use the notation ei® for the vector [cos ®; sin ®]. We will generally assume those de¯nitions and results commonly taught in ¯rst year graduate courses on Geometry and Topology. Assumptions more speci¯c to Riemannian Geometry can be found, for example, in [7]. The most important of these are the de¯nition of the curvature tensor R, which to every X and Y in the tangent space TpM assigns a mapping R(X; Y ) : TpM TpM given by ! R(X; Y )Z = X Y Z Y X Z Z; r r ¡ r r ¡ r[X;Y ] (where the expression does not depend on the extension of X, Y and Z to vector ¯elds in a neighbourhood of p), and Gauss' equation g(R(X; Y )Z; T ) = g(R~(X; Y )Z; T ) + g(h(X; T ); h(Y; Z)) g(h(Y; T ); h(X; Z)); ¡ which relates, through the second fundamental form h, the curvature tensor R~ of a manifold M~ to the curvature tensor R of a submanifold M.