In this note I am going to introduce the notion of double forms on a Riemannian manifold and use it to define the so called -curvatures; finally I will prove an almost-Schur type theorem involving -curvatures. This extends one of the results in a previous post.
1. Preliminaries for double forms
1.1. Algebra structures
Let be a smooth Riemannian manifold. Let be the ring of differential forms on . Considering the tensor product over the space of smooth functions, we define . The ring of double forms is then defined as .
We define a multiplication on as follows. For and , we define
and extend it linearly.
Naturally, can be regarded as a multilinear form which is skew symmetric in the first arguments and also in the last arguments. Under this identification, we have
Here is the sign of the permutation and is the permutation group of elements. Whenever possible we will omit the dot and write as .
Naturally we can regard the Riemannian metric and regard the Riemannian curvature tensor . For example, if is an orthonormal coframe for , then .
The contraction operator is defined as follows. If , then if or , otherwise is defined by
where is a local orthonormal frame on . We then define inductively.
We can give an inner product on by declaring if and elements of the form to be an orthonormal basis for .
Remark 1 It should be noted that the inner product on is generally different from the inner product on the space of -tensors by a multiplicative constant. For example, using local orthonormal frame, where is the norm in the tensor algebra . However, these two inner products are the same on for . The only place where we will use is the proof of Theorem 6, and since in that proof the two inner products coincide, there will be no confusion.
Finally, the Hodge star operator is defined by
(the on the RHS is just the ordinary Hodge star operator) and extending it linearly.
The following basic but important identity will be used repeatedly ([Labbi1] Theorem 3.1):
Lemma 2 If is trace-free, i.e. , then
Proof: By [Labbi1] Lemma 2.1, we have . Thus
1.2. Curvature structures
Let denotes the symmetric elements of . For , we define its sectional curvature as follows. Let be a -dimensional plane in , then we define
where is an orthonormal basis of .
Definition 3 For and , the -curvature is defined as the sectional curvature of the following -curvature tensor:
Equivalently, is the sectional curvature of on the orthogonal complement of .
Remark 2 For , is called the -curvature. In particular, is half of the scalar curvature and is the sectional curvature of .
Up to a constant, is the Killing-Lipschitz curvature of , which are also called the -sectional curvatures as defined by Thorpe [Thorpe].
For , we have ([Labbi1] Theorem 4.1)
We define the -Einstein tensor by
The following result can be found, for example, in [Labbi2] p.179.
By iterating (1), we have for any
We conclude that
2. Main result
Theorem 6 Let () be a closed oriented Riemannian manifold with , and , then
Here is the average of .
The solution exists because . As and by Lemma 4, we have
Then (with Remark 1 in mind)
Here we have used the fact that the first eigenvalue . So (11) becomes
As , we have
Therefore (8) can be rephrased as