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Low Cohomogeneity Actions and Positive Curvature

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Lie Groups and Geometric Aspects of Isometric Actions

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Abstract

In this chapter, we study special types of isometric group actions on compact Riemannian manifolds, namely those with low cohomogeneity. The cohomogeneity of an action is the codimension of its principal orbits, and can also be regarded as the dimension of its orbit space. Low cohomogeneity is an indication that there are few orbit types, and that the original space has many symmetries. In this situation, it is possible to study many geometric features that are not at reach in the general case. Throughout this chapter, we emphasize connections between the geometry and topology of manifolds with low cohomogeneity stemming from curvature positivity conditions, such as positive (\(\sec > 0\)) and nonnegative (\(\sec \geq 0\)) sectional curvature.

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Notes

  1. 1.

    Note that vertical/horizontal spaces for the submersion \(\rho: M \times G \rightarrow M\) are subspaces of \(T_{p}M \times \mathfrak{g}\), while vertical/horizontal spaces for the G-action on M are subspaces of T p M.

  2. 2.

    Recall Proposition  2.26 (iv).

  3. 3.

    Recall that N(H) is the largest subgroup of G where H is normal, see Exercise 1.61.

  4. 4.

    The reason why \(\mathbb{C}\mathrm{a}P^{2}\) can be constructed as a projectivization of \(\mathbb{C}\mathrm{a}^{3}\) is related to the fact that \(\mathbb{C}\mathrm{a}\) is 2-associative, i.e., any subalgebra generated by 2 elements is associative.

  5. 5.

    For instance, one can write the sphere as S 4n+3 = Sp(n + 1)∕Sp(n), but depending on what Sp(n + 1)-homogeneous metric we consider (e.g., the round metric), the full isometry group may be strictly larger than Sp(n + 1), see Example 6.16 and Remark 6.17.

  6. 6.

    Here, we assume G is connected. This entails no loss of generality, since the restriction to the identity component of a continuous transitive group action on a connected space is still transitive.

    Table 6.1 Transitive actions on spheres, G connected
    Full size table
  7. 7.

    Let us give a brief account of this analysis, see Ziller [235, Sec. 1] for details. On cases (i), (viii) and (ix), since the H-action on \(\mathfrak{m}\) is irreducible, the unique G-homogeneous metric is the round metric. On cases (ii) and (iii), the H-action on \(\mathfrak{m}_{1}\) is the standard U(n)-action on \(\mathbb{C}^{n}\). The metrics obtained in both cases are the same, yielding a family of U(n + 1)-homogeneous metrics on S 2n+1 that depends on two parameters. On case (iv), the H-action on \(\mathfrak{m}_{1}\) is the standard Sp(n)-action on \(\mathbb{H}^{n}\). This gives a family of Sp(n + 1)-homogeneous metrics on S 4n+3 that depends on seven parameters, six of which determine the restriction \(\langle \cdot,\cdot \rangle _{0}\) to \(\mathfrak{m}_{0}\). By appropriately diagonalizing \(A_{0} \in \mathrm{GL}(3, \mathbb{R})\), this family can be further reduced to one that depends on only four parameters. Cases (v) and (vi) are contained in case (iv). Finally, on case (vii), the H-action on the 7-dimensional space \(\mathfrak{m}_{1}\) and on the 8-dimensional space \(\mathfrak{m}_{2}\) are the only irreducible representations of Spin(7) in those dimensions. This gives a family of Spin(9)-homogeneous metrics on S 15 that depends on two parameters. All the above can be normalized so that a 1 = 1, reducing by one the number of parameters of each family.

  8. 8.

    Spheres equipped with the metric g λ are known as Berger spheres; though, historically, this term was mostly used referring to the case of (S 3, g λ ) first studied by Berger.

  9. 9.

    Note that, as no sphere fibers over the Cayley plane \(\mathbb{C}\mathrm{a}P^{2}\), there is no analogous interpretation for the Fubini-Study metric in this case.

  10. 10.

    Besides proving the above statement, this proof indicates that the moduli space of homogeneous metrics with \(\sec \geq 0\) is, in some sense, star-shaped with respect to normal homogeneous metrics.

  11. 11.

    More precisely, a Cheeger deformation of a bi-invariant metric on G with respect to the action of an intermediate subgroup \(H \subset K \subset G\).

  12. 12.

    Slices in this section are generally denoted by D, instead of S as in Chap. 3

  13. 13.

    Note that any cohomogeneity one action is polar, regardless of M∕G being a closed interval.

  14. 14.

    The possible pairs (K ±, H) with K ± connected for which K ±∕H is a sphere are listed in Table 6.1.

  15. 15.

    Recall that a map \(f: X \rightarrow Y\) is ℓ-connected if the induced homomorphisms \(f_{i}: \pi _{i}(X) \rightarrow \pi _{i}(Y )\) is an isomorphism for i < ℓ and onto for i = ℓ.

  16. 16.

    In order for M = F([0, R] × [0, 2π]) to be a smooth closed submanifold of \(\mathbb{R}^{3}\), some conditions on the derivatives of ϕ and ψ at the endpoints are needed. This illustrates the fact that the metrics g t in (6.27) must satisfy appropriate conditions at the corresponding endpoints t → ±1.

  17. 17.

    This is a horizontal curve in the sense that it is orthogonal to all circle orbits. However, in \(\mathbb{R}^{3}\), it sits in the plane y = 0, while the circle orbits are in planes z = const.

  18. 18.

    By definition, the symmetry rank of a manifold (M, g) with an isometric G-action is \(\mathop{\mathrm{rank}}\nolimits G\).

  19. 19.

    This is not a complete classification, as the candidate manifolds \(P_{k}^{7}\) and \(Q_{k}^{7}\) are not yet known to carry cohomogeneity one metrics with \(\sec > 0\), as we explain below.

  20. 20.

    For instance, given \(\sigma,\sigma ' \subset T_{p}M\), one can define dist(σ, σ′) as the Hausdorff distance between the great circles \(\sigma \cap S\) and \(\sigma ' \cap S\), where S is the unit sphere in T p M.

  21. 21.

    For the sake of simplifying notation, here we use the metric g to identify (0, k)-tensors and k-forms, that is, \(\varLambda ^{k}\mathit{TM}\mathop{\cong}\varLambda ^{k}\mathit{TM}^{{\ast}}\). Furthermore, we omit the parentheses, denoting Λ k V: = Λ k(V ).

  22. 22.

    Recall that positive-definiteness of the curvature operator is not preserved under Riemannian submersions. For instance, there are Riemannian submersions from the round sphere onto the projective spaces \(\mathbb{C}P^{n}\) and \(\mathbb{H}P^{n}\), but these only have positive-semidefinite curvature operator.

  23. 23.

    The Bianchi map \(\mathfrak{b}\) on symmetric operators on Λ 2 T p M can be seen as the orthogonal projection onto the subspace of operators induced by 4-forms via (6.46). The complement \(\ker \mathfrak{b}\) of this subspace consists of symmetric operators on \(\varLambda ^{2}T_{p}M\) that satisfy the first Bianchi identity, which are called algebraic curvature operators.

  24. 24.

    Moreover, a complete description of the moduli spaces of homogeneous metrics with strongly positive curvature on all Wallach flag manifolds was achieved in [35]. Namely, it is shown that a homogeneous metric on W 6 or W 12 has strongly positive curvature if and only if it has \(\sec > 0\), while a homogeneous metric on W 24 has strongly positive curvature if and only if it has \(\sec > 0\) and does not submerge onto \(\mathbb{C}\mathrm{a}P^{2}\) endowed with its standard metric.

  25. 25.

    We stress that the only topological obstructions to strongly positive curvature currently known are the obstructions for \(\sec > 0\).

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Authors and Affiliations

  1. Departamento de Matemática, Instituto de Matemática e Estatística Universidade de São Paulo, São Paulo, Brazil

    Marcos M. Alexandrino

  2. Department of Mathematics, University of Pennsylvania, Philadelphia, PA, USA

    Renato G. Bettiol

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  1. Marcos M. Alexandrino
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  2. Renato G. Bettiol
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Alexandrino, M.M., Bettiol, R.G. (2015). Low Cohomogeneity Actions and Positive Curvature. In: Lie Groups and Geometric Aspects of Isometric Actions. Springer, Cham. https://doi.org/10.1007/978-3-319-16613-1_6

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