Bounds on Sectional Curvature and Interactions with Topology

Recall that by Bonnet–Myers the fundamental group of \(M\) is finite under positive Ricci curvature, by Synge it is either 0 or \({\mathbb{Z}}_{2}\) in the even-dimensional positively curved case. (Note that this readily shows that \({\mathbb{R}}{\mathbf{P}}^{2}\times {\mathbb{R}}{\mathbf{P}}^{2}\) does not carry a metric of positive sectional curvature, whereas we have seen that the corresponding question for its finite cover \({\mathbb{S}}^{2}\times {\mathbb{S}}^{2}\) is completely open. This nicely illustrates that the simply-connected case is more cumbersome.) Work of Kapovitch–Petrunin–Tuschmann ([47]), Kapovitch–Wilking ([45]) and Breuillard–Green–Tao ([15]) sheds more light on the structure of the fundamental group of a manifold with non-negative Ricci curvature. In particular, the group is virtually nilpotent.

Due to this, we focus on the simply-connected case (for closed manifolds) in which there are just very few obstructions known; and the ones known are not very strong. Since positive sectional curvature implies positive scalar curvature, we have seen that the \(\alpha \)-invariant of a closed simply-connected positively curved spin manifold vanishes. Up to a deformation argument we may, without restriction, actually see that the vanishing of the \(\alpha \)-invariant indeed even obstructs the existence of a non-negatively curved metric on a closed manifold.

Gromov’s Betti number bound states that on a compact non-negatively curved manifold \(M^{n}\) there exists a constant \(c(n)\)—only depending on \(n=\dim M\)—which bounds all with arbitrary coefficient field . The number \(c(n)\) is explicit, yet very large even after improvements made by Abresch. In particular, this clearly yields an upper bound—yet usually tremendously large—for the Euler characteristic of \(M\). Nonetheless, this upper bound lets us draw the following conclusion.

Note that the general question when a connected sum or a product may carry non-negative or positive curvature remains open. However, clearly, any product of non-negatively curved metrics will be non-negatively curved.

These are the general known bounds to non-negative and at the same time positive sectional curvature. However, the Gauss–Bonnet theorem in dimension 2 lets us relate the Euler characteristic of a closed oriented surface to sectional curvature. Thus, the Euler characteristic is non-negative if and only if \(M\) carries a metric of non-negative curvature, it is positive if and only if \(M\) is positively curved, i.e. \({\mathbb{S}}^{2}\) or \({\mathbb{R}}{\mathbf{P}}^{2}\). Since the Euler characteristic of an orientable closed surface is related to its genus by \(\chi (S)=2-2g\), we see that no such surface of genus \(g\geq 2\) will admit a metric of non-negative curvature.

The situation in dimension 2 motivates some general conjectures.

We list several conjectures which connect non-negative and positive curvature to topology. Some of these are highly interdependent respectively follow from another.

Let us begin with conjectures involving cohomology. We have already seen one “Hopf conjecture”, there is a second one different in nature speculating about similar implications as we drew them from the Gauss–Bonnet theorem in dimension 2. Clearly, by Poincaré duality the Euler characteristic of a simply-connected closed odd-dimensional manifold vanishes.

The following upper bound was conjectured by Gromov when establishing his bound on Betti numbers.

By \(T^{n}={\mathbb{S}}^{1}\times \stackrel{(n)}{\ldots } \times { \mathbb{S}}^{1}\) we denote the \(n\)-torus. The Künneth formula lets us easily derive that \(b_{k}(T^{n})={n \choose k}\). Moreover, the torus admits a flat metric. Hence, the conjecture speculates that a general non-negatively curved space has topology bounded by the one of the torus.

Obvious consequences of this conjecture are that and, in particular, that

Certainly the strongest and most far-reaching conjecture in this context is the following. Recall that the homotopy groups \(\pi _{i}(M)\) of a topological space \(M\) are abelian for \(i\geq 2\). By \(\pi _{*}(M)\) we denote \(\bigoplus _{i} \pi _{i}(M)\). A simply-connected (or, more generally, nilpotent, i.e. the fundamental group is nilpotent acting nilpotently on higher homotopy groups) topological space is called rationally elliptic if both its rational cohomology algebra and its total rational homotopy \(\pi _{*}(M)\otimes {\mathbb{Q}}\) are finite dimensional. Clearly, on a closed manifold \(M\) this boils down to the question whether from some degree on all the homotopy groups are torsion or not.

Let us bring another classical property from rational homotopy into play: Recall that a topological space is called formal if its rational homotopy type is fully determined by its rational cohomology algebra. On smooth manifolds this is equivalent to the existence of a chain of quasi-isomorphisms (morphisms inducing isomorphisms in cohomology) from the commutative differential graded algebra of smooth differential forms \(\Omega (M)\) to the cohomology algebra (equipped with the zero-differential): (Clearly, we need to consider these chains, since any such quasi-isomorphism need not be invertible.) It is clear that such a weak equivalence with rational coefficients implies one with real coefficients. Although the real homotopy type does not determine the rational homotopy type in general, it is an astounding observation by Halperin–Stasheff that formality indeed does not depend on the extension field of ℚ.

In particular, formality implies the vanishing of Massey products. Formality is known for several examples (like compact Kähler manifolds, symmetric spaces or compact homogeneous spaces \(G/H\) with \({\operatorname{rk\,}}G={\operatorname{rk\,}}H\)) and is conjectured not only in special holonomy, but also in the following context.

Contrast this conjecture to results on “geometric formality” in the context of positive curvature. This will be discussed in Sect. 3.2.

We can further connect these conjectures to Cheeger’s theorem 2.4 above. First recall from Remark 2.13 that the connected sum of two (nilpotent) formal topological spaces is formal (the same is true for the direct product). The Euler characteristic of the connected sum of two compact rank one symmetric spaces is non-negative, which, however, is no longer true for \(\chi ({\mathbb{R}}{\mathbf{P}}^{4}\#{\mathbb{R}}{\mathbf{P}}^{4} \#{\mathbb{R}}{\mathbf{P}}^{4})=-1\).

One may be tempted to extend the Hopf conjecture on products to connected sums in the following way.

(Recall that the connected sum with a sphere does not change the diffeomorphism type of a manifold.) Again see [5] for results on this in the class of metrics invariant under torus actions.

Let us finally present a conjecture which comes from differential topology, more precisely from index theory. We refer the reader to [43] for an introduction to the latter. Here we are interested in how certain expressions in characteristic classes evaluate at the fundamental class of the manifold, i.e. when being integrated over the manifold. The values of certain such characteristic numbers then often can be related to indices of elliptic differential operators.

For us one important such combination characteristic of numbers is given by the elliptic genus \(\Phi (X)\). Recall its definition via on an oriented connected compact manifold \(X^{n}\) with \(n\equiv 0\ \mathrm{mod}\ 4\) (where \(T_{\mathbb{C}}\) denotes the complexified tangent bundle of \(X\) and \(\bigwedge _{t} T_{\mathbb{C}}=\sum _{k=0}^{\infty }(\bigwedge ^{k} T_{ \mathbb{C}})\cdot t^{k}\) is a formal power series over exterior powers and \(S_{t} T_{\mathbb{C}}=\sum _{k=0}^{\infty }(S^{k} T_{\mathbb{C}}) \cdot t^{k}\) one over symmetric powers of \(T_{\mathbb{C}}\). The elliptic genus is a Laurent series in the variable \(q\) with coefficients \(\hat{A}(X)\), \(\hat{A}(X,T_{\mathbb{C}}X)\), \(\hat{A}(X,\bigwedge ^{2} T_{\mathbb{C}}X)\), etc.

As we have seen the \(\hat{A}\)-genus vanishes under the existence of a metric of positive scalar curvature on spin manifolds (where it corresponds to the index of the Dirac operator, and the result follows from Lichnerowicz’ formula). In particular, under positive sectional curvature, the first non-trivial coefficient of the elliptic genus vanishes. The following conjecture speculates that actually all such coefficients, i.e. all of these twisted \(\hat{A}\)-genera should vanish under the existence of positive curvature leaving only a constant term.

Compare this to [42] where it is proved that the elliptic genus is constant on \(4k\)-dimensional spin homogeneous spaces. Expressing the elliptic genus as a power series of twisted signatures, it can be seen that in the latter case it equals the (usual) signature.

Let us finally present another conjecture which cannot be generalised from positive curvature to non-negative curvature.

We may suppose all positively curved manifolds to be compact. If \(M\) is not compact, Walschap proved that there are no non-trivial submersions from \(M\) (see [63, Theorem 2.1]). Then the submersion is a fibre bundle (with fiber \(F\)) according to Ehresmann and the conjecture can be stated as \(\dim B>\dim F\).

The conjecture was confirmed for the known simply-connected even-dimensional examples of positive curvature in [4] and for the known simply-connected odd-dimensional examples in [31]. It seems that this conjecture is very geometrical in nature lacking the connection to topology on which we focus in this survey. However, the proofs confirming the conjecture on the known examples draw essentially first on the structure of rational fibrations and second on integral computations in the Leray–Serre spectral sequence.

As announced this conjecture is again a “specialty” of positive curvature and is easily seen to fail in non-negative curvature. It suffices to consider the product metric on \(M\times N\)—both factors non-negatively curved and \(\dim M<\dim N\)—and the projection bundle \(N\overset{}{\hookrightarrow } M\times N\to M\).

Especially for this section we refer the reader to the much more elaborate corresponding section of [2]. As both examples and coincidences in the respective theories show, isometric group actions behave very well with positive curvature. In particular, representation and Lie theory then are extremely helpful. As for isometric torus actions there is another proof scheme which enters the stage: Since positive curvature (by the Weinstein fixed-point theorem) grants the existence of torus fixed-points in even dimensions, and since fixed-point components are totally geodesic, one may work inductively over (the dimension of) the fixed-point components. That is, a fixed-point component is again positively curved, and one would like to apply an induction assumption to it. The induction assumption usually is formulated connecting the rank of an isometrically acting torus to the dimension of the manifold. Hence, in order to be able to apply an induction assumption to such a fixed-point component, it remains to see that it is not fixed by too large a torus. In other words, one has to make sure that a certain, large enough subtorus, of the torus acting on the ambient manifold indeed acts effectively on the fixed-point component.

The basic proof scheme is as follows: Either the codimension of a fixed point component is “small” (such that the inductive assumption on the rank of the torus acting effectively cannot be maintained), or the codimension of the entire fixed-point set is “large” in the sense that the induction assumption may be applied to any of its components. In the first case one always needs a different argument to conclude something on the structure of the manifold.

Examples of such “inductive results” are the following: The classification of positively curved manifolds \(M^{n}\) admitting an effective isometric action of a torus \(T^{\lfloor (n+1)/2 \rfloor }\) respectively \(T^{\lfloor (n-1)/2 \rfloor }\) by Grove–Searle (see [36]) respectively Fang–Rong (see [22]), homotopy and cohomology classification results by Wilking under effective isometric \(T^{n/4+1}\) and \(T^{n/6+1}\) actions (see [65]), as well as Euler characteristic computations by Amann–Kennard under isometric torus actions—here the rank of the torus is either linear or logarithmic in the dimension of the manifold (see [3], [6]).

Exactly the same proof pattern can be applied to index theory. That is, due to a theorem by Hirzebruch–Slodowy (see [42, Corollary, p. 317]) one may apply again this inductive approach over fixed-point components.

Finally, as we already discussed, the Hopf conjecture on \({\mathbb{S}}^{2}\times {\mathbb{S}}^{2}\) is confirmed in the class of metrics invariant under an \({\mathbb{S}}^{1}\)-action.

Clearly, formulating the Hopf conjecture via metrics on \({\mathbb{S}}^{2}\times {\mathbb{S}}^{2}\), merely underlines our lack of knowledge on which metrics can appear on general products. One would extend it speculating that a fortiori on the product of less special manifolds there should not occur any positive curvature. Using symmetry the following theorem motivates this further.

Recall from Sect. 2.3 that a manifold is formal if its smooth differential forms \(\Omega (M)\) are connected by a chain of quasi-isomorphism to deRham cohomology \(H^{*}(\Omega (M))=H^{*}(M;{\mathbb{R}})\). Symmetric spaces are examples.

We consider \({\operatorname{d}}^{*}\colon \thinspace \Omega ^{*}(M)\to \Omega ^{*-1}(M)\) and \({\operatorname{d}}^{*}=(-1)^{np+n+1}\ast {\operatorname{d}}\ast \) where \(\ast \colon \thinspace \Omega ^{p} (M)\to \Omega ^{n-p}(M)\) is the Hodge-∗-operator. The main tool and main motivation in this section is the Hodge decomposition, which shows that harmonic forms \(\mathcal{H}^{*}\) represent cohomology classes. The starting point for the whole discussion of geometric formality is that this is true for the module structure of cohomology, however, harmonic forms seldom form a subalgebra of \(\Omega (M)\). Let us elaborate on this, starting with the

The inclusion of the module \(\mathcal{H}^{*}\) of harmonic forms hence is a quasi-isomorphism: Any non-trivial \({\operatorname{d}}\)-closed form lies in \(\mathcal{H}^{*}\oplus {\operatorname{im\,}}{\operatorname{d}}\), as a \({\operatorname{d}}\)-closed form in \({\operatorname{im\,}}{\operatorname{d}}^{*}\) is also \({\operatorname{d}}^{*}\)-closed, thus harmonic and hence equal to 0 by the direct sum decomposition. Hence, as a module, cohomology is isomorphic to harmonic forms.

In the case when harmonic forms are a subalgebra of the differential forms, the inclusion is a quasi-isomorphism of algebras and \(M\) is formal. This leads to the following definition (which, however, is much stronger than formality).

The formality of compact symmetric spaces and, indeed, their geometric formality follows from the fact that in this case harmonic forms are parallel whence their product is again parallel whence harmonic. Note that there are only very few examples of geometrically formal spaces; beside compact symmetric spaces, most prominently, one may find some geometrically formal Stiefel manifolds.

See [34, 48, 50‐53] for obstructions to geometric formality and properties of such manifolds. One of these is that the sum of the Betti numbers of a geometrically formal manifold \(M^{n}\) again satisfies that which we readily recognise as the very bound in Gromov’s conjecture respectively the bound resulting from rational ellipticity.

Let us summarise our observations: We conjectured positively curved manifolds to be formal. Moreover, this bound on Betti numbers would also be implied by the cited conjectures in positive curvature. Furthermore, at least on compact rank one symmetric spaces one may find a metric of positive curvature as well as a geometrically formal one. So it seems reasonable to try to relate geometric formality to positive curvature. This has been done in different settings.

However, for homogeneous spaces the relation between geometric formality and positive curvature is not stronger than what we already observed, i.e. we do not find new formal metrics.

In particular, even on a geometrically formal manifold, not every positively curved metric is formal—for example, this can be observed on \({\mathbb{C}}{\mathbf{P}}^{2n+1}\) where we may scale the metric according to shrinking the fibre of the twistor fibration \({\mathbb{S}}^{2}\to {\mathbb{C}}{\mathbf{P}}^{2n+1}\to { \mathbb{H}}{\mathbf{P}}^{n}\).

Nonetheless, as mentioned before, geometric formality provides a suitable setup to confirm the Hopf conjecture on \({\mathbb{S}}^{2}\times {\mathbb{S}}^{2}\).