SOOCHOW JOURNAL OF MATHEMATICS Volume 30, No. 3, pp. 303-338, July 2004 WHATCANWEDOWITHNASH’SEMBEDDING THEOREM ? BY BANG-YEN CHEN Abstract. According to the celebrated embedding theorem of J. F. Nash, every Riemannian manifold can be isometrically embedded in some Euclidean spaces with sufficiently high codimension. An immediate problem concerning Nash’s theorem is the following: Problem: What can we do with Nash’s embedding theorem ? In other words, what can we do with arbitrary Euclidean submanifolds of arbitrary high codimension if no local or global assumption were imposed on the submanifold ? In this survey, we present some general optimal solutions to this and related prob- lems. We will also present many applications of the solutions to the theory of submanifolds as well as to Riemannian geometry. 1. What Can We Do with Nash’s Embedding Theorem ? According to the celebrated embedding theorem of J. F. Nash [65] published in 1956, every Riemannian manifold can be isometrically embedded in some Eu- clidean spaces with sufficiently high codimension. Received May 27, 2004. AMS Subject Classification. Primary 53C40, 53C42, 53B25; secondary 53A07, 53B21, 53C25, 53D12. Key words. Nash’s embedding theorem, δ-invariant, σ-invariant, warped product manifold, conformally flat submanifold, inequality, real space form, minimal immersion, Lagrangian im- mersion, Einstein manifold. This survey has been delivered at the International Workshop on “Theory of Submanifolds” held at Valenciennes, France from June 25th to 26th and at Leuven, Belgium from June 27th to 28th, 2003. The author would like to express his hearty thanks to Professors F. Dillen, L. Vrancken and L. Verstraelen for organizing this four days conference dedicated to his sixtieth birthday. 303 304 BANG-YEN CHEN An immediate question concerning Nash’s embedding theorem is the follow- ing. Problem 1. What can we do with Nash’s embedding theorem ? In other words, what can we do with arbitrary Euclidean submanifolds of arbitrary high codimension if “no local or global assumption” were imposed on the submanifold ? The Nash theorem was aimed for in the hope that if Riemannian manifolds could always be regarded as Riemannian submanifolds, this would then yield the opportunity to use extrinsic help. Till when observed as such by M. Gromov [52] this hope had not been materialized however. Themainreasonforthisisdue to the lack of controls of the extrinsic prop- erties of the submanifolds by the known intrinsic data (see [80]). In order to overcome such difficulty, one needs to introduce new type of Riemannian invariants and to establish general sharp relationships between ex- trinsic properties of the submanifolds with the new type of intrinsic Riemannian invariants on the submanifolds. In other words, one needs to establish some general optimal solutions to the following fundamental problem: Problem 2. isometric Sharp relationship between intrinsic ∀ M −−−−−−−−→ Em =⇒ ??? immersion and extrinsic invariants Notice that we do not impose any intrinsic or extrinsic assumption on the Riemannian submanifolds in Problem 2. In this survey, we will present optimal general solutions to these two fun- damental problems and also solutions to some closely related problems. Many immediate applications of these optimal general solutions will also be presented in this survey. Most of these will be presented in the first seven sections. In sections 8 and 9 we explain how to assign a number, called the con- tact number to an arbitrary Euclidean submanifold. The contact number of a Euclidean submanifold is either a natural number or +∞, which measures the degree of contact of geodesics and normal sections of the submanifold. In these NASH’S EMBEDDING THEOREM 305 two sections, we will explain the relationships between contact number with the notions of isotropic submanifolds, holomorphic curves and special Lagrangian surfaces. In section 10, we present an application of the study of contact number to pseudo-umbilical immersions; namely, to obtain the first non-trivial examples of pseudo-umbilical submanifolds in Euclidean spaces. In the last section, we present some sharp results for K¨ahlerian submanifolds which can be regarded as the K¨ahlerian version of a solution to Problem 2 for the Euclidean submanifolds given in earlier section. 2. Some Solutions to Problems 1 and 2 In order to provide some general optimal solutions to Problems 1 and 2, one needs to introduce a new type of scalar-valued curvature invariants, denoted by δ(n1,...,nk), which are obtained from the scalar curvature (which is the total sum of sectional curvatures) by deleting a portion of sectional curvatures. (See [15, 17] and also [11, 12] for details). This was done as follows: Let M be a Riemannian n-manifold. Denote by K(π) the sectional curvature of M associated with a plane section π ⊂ TpM, p ∈ M. For any orthonormal basis e1,...,en of the tangent space TpM, the scalar curvature τ at p is defined to be τ(p)= K(ei ∧ ej). (2.1) 1≤i<j≤n Let L be a subspace of TpM of dimension r ≥ 2and{e1,...,er} an orthonor- mal basis of L. The scalar curvature τ(L)ofther-plane section L is defined by τ(L)= K(eα ∧ eβ), 1 ≤ α, β ≤ r. (2.2) 1≤α<β≤r The scalar curvature τ(p)ofM at a point p ∈ M is nothing but the scalar curvature of the tangent space of M at p.AndifL is a 2-plane section, τ(L)is nothing but the sectional curvature K(L)ofL. In general, τ(L) with dim L ≥ 2 is nothing but the scalar curvature of the image expp(L)ofL at p under the exponential map at p. 306 BANG-YEN CHEN For any integer n ≥ 2, we denote by S(n) the finite set consisting of all k-tuples (n1,...,nk) of integers n1,...,nk ≥ 2withn1 + ···+ nk ≤ n,wherek is a non-negative integer. The cardinal number #S(n)ofS(n)isequaltop(n) − 1, where p(n) denotes the number of partition of n which increases quite rapidly with n. For instance, for n =2, 3, 4, 5, 6, 7, 8, 9, 10,...,50,...,100,...,200, the cardinal number #S(n) are given respectively by 1, 2, 4, 6, 10, 14, 21, 29, 41,...,204225,...,190569291,...,3972999029387. √ The asymptotic behavior of #S(n)isgivenby#S(n) ∼ (1/(4n 3) exp[π 2n/3]) as n →∞. For each k-tuple (n1,...,nk)inS(n), the author introduced a decade ago the Riemannian invariant δ(n1,...,nk)by δ(n1,...,nk)=τ − inf{τ(L1)+···+ τ(Lk)} (2.3) where L1,...,Lk run over all k mutually orthogonal subspaces of TpM such that dim Lj = nj,j=1,...,k. The following general optimal inequalities from [15, 17] provide sharp solu- tions to both Problem 1 and Problem 2. Theorem 1. For any Euclidean submanifold M and for any (n1,...,nk) ∈ S(n),n=dimM, we have the following inequalities: 2 δ(n1,...,nk) ≤ c(n1,...,nk)H , (2.4) 2 where H is the squared mean curvature and c(n1,...,nk) are positive numbers given by 2 n (n + k − 1 − nj) c(n1,...,nk)= . (2.5) 2(n + k − nj) The equality case of inequality (2.4) holds at a point p ∈ M if and only if, there exists an orthonormal basis e1,...,en at p, such that the shape operators of M with respect to e1,...,en at p take the following form: ξ 0 A1 .. Aξ = . , (2.6) ξ 0 Ak NASH’S EMBEDDING THEOREM 307 ξ k where {Aj }j=1 are symmetric nj × nj submatrices satisfying ξ ξ trace (A1)=···=trace(Ak)=µξ. (2.7) Inequalities (2.4) give prima controls on the most important extrinsic cur- vature, the squared mean curvature H2, by the initial intrinsic curvatures, the δ-invariants δ(n1,...,nk), of the Riemannian manifold. In general, there do not exist direct relationship between the newly defined δ-invariants δ(n1,...,nk). On the other hand, we have the following. Maximum Principle. Let M be an n-dimensional submanifold of a Eu- clidean m-space Em. If it satisfies the equality case of (2.4), i.e., it satisfies 2 H =∆(n1,...,nk) (2.8) for a k-tuple (n1,...,nk) ∈S(n),then ∆(n1,...,nk) ≥ ∆(m1,...,mj) (2.9) for any (m1,...,mj) ∈S(n),where δ(n1,...,nk) ∆(n1,...,nk)= . (2.10) c(n1,...,nk) For any isometric immersion x : M → Em of a Riemannian n-manifold M in Em. Theorem 1 yields 2 H (p) ≥ ∆ˆ 0(p), (2.11) where ∆ˆ 0 is the Riemannian invariant on M defined by ∆ˆ 0 =max{∆(n1,...,nk):(n1,...,nk) ∈S(n)}. Inequality (5.4) enables us to introduce the notion of ideal immersions as given in the following. Definition 1. An isometric immersion of a Riemannian n-manifold M in Em is called an ideal immersion if it satisfies the equality case of (2.11) identically. The above maximum principle yields the following important fact: 308 BANG-YEN CHEN If an isometric immersion x : M → Em satisfies equality (2.4) for a given k-tuple (n1,...,nk) ∈S(n), then it is an ideal immersion automatically. By applying the inequalities (2.4) and the theory of finite type submanifolds ([10, 14]), we can establish the following new intrinsic results concerning intrinsic spectral properties of homogeneous spaces via extrinsic data. Theorem 2. If M is a compact homogeneous Riemannian n-manifold with irreducible isotropy action, then the first nonzero eigenvalue λ1 of the Laplacian on M satisfies λ1 ≥ n ∆(n1,...,nk) (2.12) for any k-tuple (n1,...,nk) ∈S(n). The equality sign of (2.12) holds if and only if M admits a 1-type ideal im- mersion in a Euclidean space.