Abstract

Hopf’s well-known conjecture states that there exists no metric of positive sectional curvature in the product manifold . In this paper, we show that the Hopf conjecture is true for conformal metrics to the product metric or doubly warped products on .

1. Introduction

A classical topic in Riemannian geometry is to study manifolds with positive sectional curvature. The most rigid classical curvature concept (consequently containing most information), which then gives rise to the other ones, is sectional curvature (see [1]). Let be a Riemannian manifold and let denote the tangent vector space of at . The sectional curvature is the most natural generalization to higher dimensions of the Gaussian curvature of a surface, given that it controls the behavior of geodesics (see [2]). We will say that has positive sectional curvature if for every point , the sectional curvature of every 2-plane is positive. An example of such manifolds are the -dimensional spheres of with the metric induced by . For 4-dimensional Riemannian manifolds, very few topological obstructions to positive sectional curvature are known, and many conjectures about this subject remain open, as, for example, the Hopf conjecture on , which is one of the oldest conjectures in global Riemannian geometry (see [3]).Conjecture (H. Hopf): does not admit a Riemannian metric of positive sectional curvature [4]

By Synge’s lemma (see [5]), one knows that on there is no Riemannian metric of positive sectional curvature. The only known examples of positively curved compact connected 4-manifolds are , , and (see [4]).

There have been various attempts to prove or disprove Hopf’s conjecture; one was to start with the standard product metric (which is nonnegatively curved) and try to deform it to a positively curved metric (see [1, 4, 68] for more details). Although one can make the curvature of mixed planes positive, there appear new planes of zero or even negative curvature; hence, this method seems not to answer the question. From work of Bourguignon [7], it is known that in the neighborhood of the product metric of , there is no metric of positive curvature. In [9], Weinstein shows that a metric of positive sectional curvature on cannot be induced by an immersion in . Now, it follows from a result of Hopf that an embedding of such a Riemannian structure in is not possible as then the manifold has to be a sphere.

Similarly, using the metric , the Riemannian metric is induced by on ; the authors of [10] gave a positive answer to the conjecture in the class of conformal metrics where is a smooth function on .

In [3, 11], Oscar Perdomo suggested an idea that may be useful for trying to prove Hopf’s conjecture. This idea relies on the main two theorems proven in [3]. The first one is a small variation of a theorem by Nash that states that any Riemannian manifold can be isometrically embedded in some . The second one states that if , in particular ( be two compact Riemannian surfaces with the metric induced by the euclidean spaces), then for any smooth function , the manifold with the metric induced by does not have positive sectional curvature. So, he makes the following remark:

Remark 1. If for any smooth mapthe manifoldwith the metric induced bydoes not have positive sectional curvature, then Hopf’s conjecture would be true.

The case is solved in [3].

More generally, one might ask whether any nontrivial product manifold admits a metric with positive curvature. It is easy to see that if and admit nonnegative curvature metrics, then with the product metric has nonnegative curvature. This generalized conjecture is open. Thus, in [12], Amann and Kennard have proven that no product manifold can carry a metric of positive sectional curvature admitting a certain degree of torus symmetry. Also, Hsiang and Kleiner proved that (see [13, 14]) does not admit a Riemannian metric with positive sectional curvature and an isometric circle action.

Singly warped products were first introduced by Bishop and O’Neil in their attempt to construct a class of Riemannian manifolds with negative curvature [15]. Warped products have many applications, in general relativity [16], in the studies related to solutions of Einstein’s equations (see [17] and references therein).

The study of relativity theory demands a wider class of manifold, and the idea of doubly twisted product as a generalization of warped product was introduced and studied by many authors (see [18] and references therein).

Let and be two Riemannian manifolds of dimensions and , respectively, and let and be the canonical projections. Also, let and be smooth functions. Then, the doubly twisted product ([18, 19]) of and with twisting functions and is defined to be the product manifold with metric tensor given by

We denote this Riemannian manifold by . In particular, if , then is called the twisted product of and with twisting function . Moreover, if only depends on the points of , then is called the warped product of and with warping function . Also, as a generalization of the warped product of the two Riemannian manifolds and , is called the doubly warped product of Riemannian manifolds and with the warping functions and if and and only depend on the points of and , respectively (see [18]).

The Hopf conjecture is true for the singly warped product manifold (see [20]). In this paper, we study the Hopf conjecture in the class of doubly warped products and conformal metrics to the product metrics on . Let be a doubly twisted product manifolds with twisting functions and . Next, we are going to prove the following results.

Theorem 1. A doubly warped product manifolddoes not have positive sectional curvature.

Theorem 2. A conformal metricto a product metricondoes not have positive sectional curvature.

The paper is organized as follows: After preliminaries, Section 2 contains the basic notions of doubly twisted products, and the proofs of the two theorems above will be given in Section 3.

2. Preliminaries

Let and be two Riemannian manifolds with Levi-Civita connections and , respectively. Denote by the Levi-Civita connection and the gradient of the doubly twisted products of and with twisting functions and . Let and . For a vector field of , the lift of to is the vector field whose value at each is the lift of to . Thus, the lift of is the unique vector field on , that is, -related to and -related to the zero vector field on (see [19]). We denote the set of lifts of vector fields on and to by and , respectively. The following two propositions are results from [18] (Proposition 1 and Proposition 2).

Proposition 1 (see [18]). If and, then

In a doubly twisted product , we can interchange the roles of the base manifold and the fiber manifold . In the above proposition, we have the expression of the Levi-Civita connection for vector fields in (2). Similarly, by making corresponding changes, one can easily see the expression of Levi-Civita connection for vector fields as in (2). Obviously, the analogue expressions hold for vector fields in by making corresponding changes [18].

We will denote by , , and the curvature tensors of , , and , respectively.

For , define . First note that if and , we have and by [18] the Hessian form of on satisfies

Then, we have the following relations.

Proposition 2 (see [18]). Let and. Then,where is the Hessian tensor of on , i.e., .

By using Propositions 1 and 2, we get the following lemma for later use.

Lemma 1. Letbe a doubly twisted product manifold with metric tensordenoted by. Forsuch thatandare orthonormal on some neighborhoodof a point for the product metric. Then, at any pointof, the sectional curvature of the mixed 2-plane spanned byandis given by

Proof. By definition,By using Proposition 2, we have thatSinceand by using (4), we haveThen, we haveMoreover, we haveThus, one obtains the result.

3. The Main Results

In this section, we give the proof of Theorem 1 and Theorem 2.

3.1. Proof of Theorem 1

Proof. We will assume that the sectional curvature of the manifold is everywhere positive. We consider the case where and are the standard unit sphere and the doubly warped product manifolds with the warping functions and . We keep the hypotheses and notations of Lemma 1. Since is a doubly warped product manifolds, thenBy using the above relations with (6) and the fact that the mixed sectional curvature is positive, we haveSince (respectively, ) depends only of the points of the left sphere of (respectively, depends only of the points of the right sphere of ), we can define a smooth function in (respectively, a smooth function in ) such that and . So, we have the following remarks:(1)If a point is a maximum for , then for any , is a maximum of (2)If a point is a maximum for , then for any , is a maximum of (3)Consequently, if is a maximum for and a maximum for , then is a maximum for and Then, at the critical point of and , we haveSince the critical point of and is a maximum, then we have thatUsing (15)–(17) in (14), we get a contradiction.

3.2. Proof of Theorem 2

Proof. We take and . If , is a doubly twisted product with the twisting function and . Then, the tensor metric is a conformal metric to the product metric on .
Now, for such that and are orthonormal with respect to the product metric on some neighborhood of a point, then at any point of the sectional curvature of the mixed 2-plane spanned by and is given byFinally, Since is a compact manifold, the smooth function has a maximum . At the critical point of , we have . And the sectional curvature of the mixed 2-plane spanned by and is given bySince the critical point of is a maximum then, we have thatwhere denotes the Hessian of with respect to the metric . In this case, the sectional curvature at is not positive [21, 22].

Data Availability

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Conflicts of Interest

The authors declare that there are no conflicts of interest or personal relationships that could have appeared to influence the work reported in this paper.