A splitting theorem for higher order parallel immersions

We consider isometric immersions into space forms having the second fundamental form parallel at order k. We show that this class of immersions consists of local products, in a suitably defined sense, of parallel immersions and normally flat immersions of flat spaces.


Introduction
A basic question in submanifold geometry is the study of isometric immersions of Riemannian manifolds having their second fundamental form subject to certain geometric or analytic properties. One of the best known examples is the class of immersions having parallel second fundamental form. Such immersions are called parallel. Their study is part of well established theories, for instance the Euclidean case has been fully described by Ferus [5,6,7]. Parallel immersions in space forms are the extrinsic counterpart of locally symmetric spaces [21].
In this spirit, a natural class to consider is that of k-parallel immersions, by requiring the second fundamental form be parallel at order k for k 1. At a pure analogy level, while each Riemannian manifold whose Riemannian curvature tensor is parallel at higher order is locally symmetric [20] the class of k-parallel immersions is sensibly larger than that of parallel ones. Curves such as the Cornu spiral provide simple 2-parallel, nonparallel examples.
Early work by Mirzoyan [15] on k-parallel immersions has been taken up by Dillen and Lumiste among others, see [14] for an overview. In the case of immersions into space forms classification results have been obtained for low dimension or codimension or for small k in [9,2,8,12,10,11,13]. A case by case inspection therein reveals that the immersion is a product and each factor is either a k-parallel curve, a k-parallel flat and normally flat surface, an affine subspace or a sphere.
That this reflects the general structure of k-parallel immersions into space forms is the main result of this note. We prove Theorem Any k-parallel isometric immersion of a Riemannian manifold (M, g) into a simply-connected space form is the local product of a parallel immersion and a k-parallel immersion of a flat space with flat normal bundle. This is Theorem 2.3 in the paper. Products of immersions are understood in the sense of Noelker [19], see Definition 2.4. Open manifolds are naturally considered since in the compact case integration shows that k-parallel immersions are parallel.
To prove the theorem we first note that having the ambient space locally symmetric forces the Riemann curvature be parallel at higher order hence parallel by [20]. We refine the natural local product decomposition of (M, g) into a flat space and a locally symmetric space with non-degenerate curvature. The key ingredient is to take into account the algebraic structure of the normal bundle reflecting the structure of higher order derivatives of the second fundamental form. It is used to modify the above mentioned splitting into one that satisfies the requirements in the generalised Moore splitting criterion [16,19].
Our theorem reduces the study of k-parallel immersions into space forms to that of parallel immersions and k-parallel normally flat immersions of flat space. This is an essential step towards the classification of k-parallel immersions of space forms. Indeed, parallel immersions of space forms are known. The Euclidean case was solved by Ferus as already mentioned above. If the target space is a sphere one can easily get a classification from the Euclidean case. For the hyperboloid as target a classification was achieved independently by Takeuchi [22] and Backes and Reckziegel [1]. Furthermore, in [13] Lumiste describes a general method (the so-called polynomial map method) for the investigation of k-parallel immersions in Euclidean spaces that are flat and normally flat; see also [18] where the latter are interpreted as certain integrable systems of hydrodynamic type. In conclusion, to obtain a full classification there remains to understand the structure of flat, normally flat k-parallel immersions in hyperbolic space.

Structure results
Let (N n , g) be a Riemannian manifold and let us consider an isometric immersion (M m , g) ֒→ (N, g), m n. We will denote by ∇ and ∇ N , respectively, the Levi-Civita connections and by R, R N the Riemann curvature tensors, with the convention that   A particular case thereof consists in the so-called parallel immersions, when one requires ∇α = 0.
In most of what follows we will look at immersions into standard spaces M n (c); that is simply connected, complete spaces of constant sectional curvature c and dimension n, realised by their standard models in R n+1 : the sphere, the hyperboloid and the hyperplane.
The main goal of this paper is to prove that the study of k-parallel immersions into standard spaces reduces to that of parallel immersions and k-parallel immersions of flat space. The latter class will be shown later on to be normally flat as well.
One of our main tools is Moore's decomposition criterion [17] and its generalisations to the case of standard spaces due to [16] (see also [19]). Its application requires setting up the notion of product immersion which we recall below in the Euclidean case, to begin with.
The next definition captures necessary conditions for an immersion to be a product. In the rest of this paper we will systematically use the natural identification Having decomposable second fundamental has been shown, for Euclidean target spaces, to yield a product structure in [17]; explicitly To present the generalisation of the above result to standard space target we first recall that a submanifold N in M n (c) is called spherical if its second fundamental form α = g ⊗ ζ for some parallel normal vector field, where g is the induced metric on N. For c = 0 this is equivalent with N being the intersection of the standard space and some affine subspace of R n+1 .
(i) N 1 , N 2 are isometric to standard spaces and admit isometric embeddings ϕ i : The curved counterpart of Moore's decomposition criterion in Theorem 2.1 is [16] (see also [19]) An isometric immersion f : where M 1 , M 2 are connected and f has decomposable second fundamental form is a product immersion.
Whereas Definition 2.2 is a particular case of Definition 2.4, the latter allows a broader treatment of the case when c = 0 e.g. when N 1 , N 2 are taken to be spheres.
The main result in this paper can now be formulated as follows.
It is the local product of a parallel immersion and a k-parallel immersion of a flat space with flat normal bundle. When (M, g) is simply connected and complete the splitting is global.
The proof requires a few technical steps we will outline below. When not specified otherwise we will work under the assumptions of Theorem 2.3.
We define at each point of M is locally symmetric. The following hold: Proof. (i) We differentiate the Gauss equation in directions tangent to M. Now, taking into account that ∇ N R N = 0 yields for all X 0 in T M. Now the Codazzi-Mainardi formula for ξ ∈ NM makes that, for instance, hence clearly ∇ 2k−1 R N = 0 and then ∇ 2k−1 R = 0. By [20] we have that ∇R = 0 whence the claim.
(ii) follows from the definition of E 0 . (iii) We exploit an argument used in [20,23] under slightly different assumptions. The function t = |α| 2 : M → R satisfies for all X, Y in T M; it follows that ∇ 2k−2 grad t = 0 along M, after differentiating at higher order. In particular ∇ 2k−2 X 1 = 0 where X 1 denotes the orthogonal projection of grad t onto the parallel distribution E 1 . Now we can use the following result due to Tanno. Suppose that at some point x of a Riemannian manifold and for some tangent vectors X, Y at x, R(X, Y ) is not singular. Then having ∇ k T = 0 for an arbitrary tensor T and for some k ≥ 1 implies ∇T = 0, see [23], Theorem 1. Applied to the integral manifolds of E 1 this yields ∇ U X 1 = 0 for all U in E 1 ; in particular R(E 1 , E 1 )X 1 = 0. Since, moreover, R(E 0 , T M) = 0 we get R(T M, T M)X 1 = 0, thus X 1 ∈ E 0 showing that X 1 = 0. In other words, grad t belongs to E 0 , which is parallel in T M. When taking X, Y in E 1 it follows that the right hand side of (2.4) vanishes and the claim follows by a positivity argument.
Under additional assumptions we have the following alternative, by using essentially that a Ricci flat, locally symmetric space is flat. In what follows the action of the curvature tensor on a tensor field Q in (T ⋆ M) l ⊗NM is given by whenever X, Y, Z i , 1 ≤ i ≤ l belong to T M. For s 0 we will use frequently the short hand notation Im ∇ s α to refer to span{∇ s X 1 ,...,Xs α} ⊆ NM where X i , 1 ≤ i s belong to T M. Proof. Because ∇ 2 (∇ k−2 α) = 0 it follows after anti-symmetrisation that the curvature operator R(X, Y ) acts trivially on ∇ k−2 α; since moreover R(E 0 , T M)T M = 0 we obtain that After taking scalar products with vectors in Im ∇ k−4 α a positivity argument shows that Continuing this procedure leads to R ⊥ (E 0 , T M)(Im ∇ k−2s α) = 0, s 0. Since k-parallel manifolds are also k + 1-parallel this equation holds also for k + 1 and (i) follows.
In particular, (i) gives R ⊥ (E 0 , T M)(Im α) = 0. We will now use the Ricci formula for X, Y in T M and ξ 1 , ξ 2 in NM, where A ξ X, Y = − α X Y, ξ is the shape operator. It shows that also (Im α) ⊥ = {ξ ∈ NM : A ξ = 0} is annihilated by R ⊥ (E 0 , T M) and (ii) follows.
Proposition 2.2. Let (M, g) be a k-parallel isometric immersion into a space form: (i) we have that (M, g) is semi-parallel, that is, R(X, Y )·α = 0 whenever X, Y belong to T M; (ii) and ∇R ⊥ = 0.
Proof. (i) From the lemma and the definition of E 0 it follows that the tensor X,Y α = 0 by (iii) in Proposition 2.1 and R(X, Y ) · α = 0 follows. (ii) By differentiation in the lemma above  Let (M, g) be a k-parallel isometric immersion into a space form. The following hold: (ii) We use (2.6) for the description of F 0 . Let X 1 , . . . , X l , Y, Z be in E 0 and 1 ≤ l ≤ k − 1. We have R(U, V ) · ∇ l α = 0 for all U, V ∈ T M by differentiating the equation R(U, V ) · α = 0, which we know from Proposition 2.2, (i). This gives Since E 0 is flat, the right hand side of the above equation vanishes.
Proof. (ii) Using again that (M, g) is semiparallel and that E 0 is flat we obtain for all U, V, X ∈ T M and Y ∈ E 0 . The claim now follows from E 1 = span{R(X, Y )Z : X, Y, Z ∈ T M}. (iii) follows from (ii). Indeed, After differentiation, using that ∇ E 1 α = 0 the Codazzi-Mainardi equation leads to α(Y 1 , Z 1 ), (∇ l X 1 ,...,X l α)(Y 0 , Z 0 ) = 0 for all 1 ≤ l ≤ k − 1 and X 1 , . . . , X l ∈ E 0 . Now it follows from (2.6) that α(E 1 , E 1 ) is orthogonal to F 0 . Now we will define a further decomposition of E 0 into subbundles E ′ 0 ⊕ E ′′ 0 such that Moore's criterion applies to D 0 := E 1 ⊕E ′ 0 and D 1 := E ′′ 0 . Let E ′′ 0 be spanned by elements of the form where ξ belongs to F 0 , X is in T M and U 1 , . . . , U l are in T M. Equivalently, it suffices to take X ∈ E 0 . Indeed, this follows from Lemma 2.2, (iv).
Obviously, E ′′ 0 ⊂ E 0 by Lemma 2.2, (iii). We will denote by E ′ 0 the orthogonal complement of E ′′ 0 in E 0 . Proposition 2.5. If (M, g) is a k-parallel isometric immersion into a space form, then (i) follows directly from having E 0 and F 0 parallel and ∇ k α = 0.
Since the distributions E 0 and hence E ′′ 0 are flat so is M 2 and the same argument as above shows that it is immersed in N 2 as a k-parallel immersion.
In absence of a factor of type E 1 it follows from (ii) in Lemma 2.1 that the normal bundle of M 2 is flat.