Symmetry Rank Research Articles

Positive Intermediate Ricci Curvature with Maximal Symmetry Rank

Positive Intermediate Ricci Curvature with Maximal Symmetry Rank

Free rank of symmetry of products of Dold manifolds

AbstractDold manifolds $P(m,n)$ are certain twisted complex projective space bundles over real projective spaces and serve as generators for the unoriented cobordism algebra of smooth manifolds. The paper investigates the structure of finite groups that act freely on products of Dold manifolds. It is proved that if a finite group G acts freely and $ \mathbb{Z}_2 $ cohomologically trivially on a finite CW-complex homotopy equivalent to ${\prod_{i=1}^{k} P(2m_i,n_i)}$, then $G\cong (\mathbb{Z}_2)^l$ for some $l\leq k$ (see Theorem A for the exact bound). We also determine some bounds in the case when for each i, ni is even and mi is arbitrary. As a consequence, the free rank of symmetry of these manifolds is determined for cohomologically trivial actions.

Torus actions on manifolds with positive intermediate Ricci curvature

We study closed, simply connected manifolds with positive $2^\mathrm{nd}$-intermediate Ricci curvature and large symmetry rank. In odd dimensions, we show that they are spheres. In even dimensions other than $6$, we show that they must have positive Euler characteristic. Under stronger assumptions on the symmetry rank, we show that such even dimensional manifolds must have trivial odd degree integral cohomology, and if the second Betti number is no more than $1$, they are either spheres or complex projective spaces. In the process, we establish new tools for studying isometric actions on closed manifolds with positive $k^\mathrm{th}$-intermediate Ricci for values of $k \geq 2$. These tools include generalizations of the isotropy rank lemma, symmetry rank bound, and connectedness principle from the setting of positive sectional curvature.

Local symmetry rank bound for positive intermediate Ricci curvatures

We use a local argument to prove if an $r$-dimensional torus acts isometrically and effectively on a connected $n$-dimensional manifold which has positive $k^\mathrm{th}$-intermediate Ricci curvature at some point, then $r \leq \lfloor \frac{n+k}{2} \rfloor$. This symmetry rank bound generalizes those established by Grove and Searle for positive sectional curvature and Wilking for quasipositive curvature. As a consequence, we show that the symmetry rank bound in the Maximal Symmetry Rank Conjecture for manifolds of non-negative sectional curvature holds for those which also have positive intermediate Ricci curvature at some point. In the process of proving our symmetry rank bound, we also obtain an optimal dimensional restriction on isometric immersions of manifolds with non-positive intermediate Ricci curvature into manifolds with positive intermediate Ricci curvature, generalizing a result by Otsuki.

Torus actions, maximality, and non-negative curvature

Abstract Let ℳ 0 n {\mathcal{M}_{0}^{n}} be the class of closed, simply connected, non-negatively curved Riemannian n-manifolds admitting an isometric, effective, isotropy-maximal torus action. We prove that if M ∈ ℳ 0 n {M\in\mathcal{M}_{0}^{n}} , then M is equivariantly diffeomorphic to the free, linear quotient by a torus of a product of spheres of dimensions greater than or equal to 3. As a special case, we then prove the Maximal Symmetry Rank Conjecture for all M ∈ ℳ 0 n {M\in\mathcal{M}_{0}^{n}} . Finally, we show the Maximal Symmetry Rank Conjecture for simply connected, non-negatively curved manifolds holds for dimensions less than or equal to 9 without additional assumptions on the torus action.

Polarization tensor in de Sitter gauge gravity

The gauge theory of the de Sitter group, [Formula: see text], in the ambient space formalism has been considered in this paper. This method is important to construction of the de Sitter super-conformal gravity and Quantum gravity. [Formula: see text] gauge vector fields are needed which correspond to [Formula: see text] generators of the de Sitter group. Using the gauge-invariant Lagrangian, the field equations of these vector fields have been obtained. The gauge vector field solutions are recalled. By using these solutions, the spin-[Formula: see text] gauge potentials has been constructed. There are two possibilities for presenting this tensor field: rank-[Formula: see text] symmetric and mixed symmetry rank-[Formula: see text] tensor fields. To preserve the conformal transformation, a spin-[Formula: see text] field must be represented by a mixed symmetry rank-[Formula: see text] tensor field, [Formula: see text]. This tensor field has been rewritten in terms of a generalized polarization tensor field and a de Sitter plane wave. This generalized polarization tensor field has been calculated as a combination of vector polarization, [Formula: see text], and tensor polarization of rank-2, [Formula: see text], which can be used in the gravitational wave consideration. There is a certain extent of arbitrariness in the choice of this tensor and we fix it in such a way that, in the limit, [Formula: see text], one obtains the polarization tensor in Minkowski spacetime. It has been shown that under some simple conditions, the spin-[Formula: see text] mixed symmetry rank-[Formula: see text] tensor field can be simultaneously transformed by unitary irreducible representation of de Sitter and conformal groups ([Formula: see text]).

Symmetrical Rank-Three Vectorized Loading Scores Quasi-Newton for Identification of Hydrogeological Parameters and Spatiotemporal Recharges

In a multi-layered groundwater model, achieving accurate spatiotemporal identification and solving the ill-posed problem is the vital topic for model calibration. This study proposes a symmetry rank three vectorized loading scores (SR3 VLS) quasi-Newton algorithm by modifying the Levenberg–Marquardt algorithm and importing a rank three structure from Broyden–Fletcher–Goldfarb–Shanno algorithm for identification of hydrogeological parameters and spatiotemporal recharge simultaneously. To accelerate directional convergence and approach a global optimum, this study uses a vectorized limited switchable step size in the transmissive groundwater inverse problem. The Hessian approximation rank three uses high and low-rank factor loading scores analyzed from simulated storage fluctuation between adjacent iterations for calculation and matrix correction. Two numerical experiments were designed to validate the proposing algorithm, showing the SR3 VLS quasi-Newton reduced the error percentages of the identified parameters by 1.63% and 9.65% compared to the Jacobian quasi-Newton. The proposing method is applied to the Chou-Shui River alluvial fan groundwater system in Taiwan. Results show that the simulated storage error decreased rapidly in six iterations, and has good head convergence as small as 0.11% with a root-mean-square-error (RMSE) of 0.134 m, indicating that the proposing algorithm reduces the computational cost to converge to the true solution.

Non-negatively Curved 6-Manifolds with Almost Maximal Symmetry Rank

We classify closed, simply connected, non-negatively curved 6-manifolds of almost maximal symmetry rank up to equivariant diffeomorphism.

Positive weighted sectional curvature

In this paper, we give a new generalization of positive sectional curvature called positive weighted sectional curvature. It depends on a choice of Riemannian metric and a smooth vector field. We give several simple examples of Riemannian metrics which do not have positive sectional curvature but support a vector field that gives them positive weighted curvature. On the other hand, we generalize a number of the foundational results for compact manifolds with positive sectional curvature to positive weighted curvature. In particular, we prove generalizations of Weinstein's theorem, O'Neill's formula for submersions, Frankel's theorem, and Wilking's connectedness lemma. As applications of these results, we recover weighted versions of topological classification results of Grove-Searle and Wilking for manifolds of high symmetry rank and positive curvature.

Orientation and Symmetries of Alexandrov Spaces with Applications in Positive Curvature

We develop two new tools for use in Alexandrov geometry: a theory of ramified orientable double covers and a particularly useful version of the Slice Theorem for actions of compact Lie groups. These tools are applied to the classification of compact, positively curved Alexandrov spaces with maximal symmetry rank.

Positively curved Riemannian metrics with logarithmic symmetry rank bounds

We prove an obstruction at the level of rational cohomology to the existence of positively curved metrics with large symmetry rank. The symmetry rank bound is logarithmic in the dimension of the manifold. As one application, we provide evidence for a generalized conjecture of H. Hopf, which states that no symmetric space of rank at least two admits a metric with positive curvature. Other applications concern product manifolds, connected sums, and manifolds with nontrivial fundamental group.

Nonnegatively curved 5–manifolds with almost maximal symmetry rank

We show that a closed, simply-connected, non-negatively curved 5-manifold admitting an effective, isometric $T^2$ action is diffeomorphic to one of $S^5$, $S^3\times S^2$, $S^3\tilde{\times} S^2$ (the non-trivial $S^3$-bundle over $S^2$) or the Wu manifold $SU(3)/SO(3)$.

Topology of positively curved 8-dimensional manifolds with symmetry

In this paper we show that a simply connected 8-dimensional manifold M of positive sectional curvature and symmetry rank � 2 resembles a rank one symmetric space in several ways. For example, the Euler characteristic of M is equal to the Euler characteristic of S 8 , HP 2 or CP 4 . And if M is rationally elliptic then M is rationally isomorphic to a rank one symmetric space. For torsion-free manifolds we derive a much stronger classification. We also study the bordism type of 8-dimensional manifolds of positive sectional curvature and symmetry rank � 2. As an illustration we apply our results to various families of 8-manifolds.

Low-dimensional manifolds with non-negative curvature and maximal symmetry rank

We classify closed, simply connected n-manifolds of non-negative sectional curvature admitting an isometric torus action of maximal symmetry rank in dimensions 2 < n < 6. In dimensions 3k, k = 1,2 there is only one such manifold and it is diffeomorphic to the product of k copies of the 3-sphere.

Erratum to: The stable free rank of symmetry of products of spheres

Erratum to: The stable free rank of symmetry of products of spheres

On the classification of positive quaternionic Kähler manifolds with b 4 = 1

Let M be a positive quaternionic Kahler manifold of dimension 4m. We already showed that if the symmetry rank is greater than or equal to \( \left[ {\tfrac{m} {2}} \right] + 2 \) and the fourth Betti number b4 is equal to one, then M is isometric to ℍPm. The goal of this paper is to report that we can improve the lower bound of the symmetry rank by one for higher even-dimensional positive quaternionic Kahler manifolds. Namely, it is shown in this paper that if the symmetry rank of M with b4(M) = 1 is greater than or equal to \( \tfrac{m} {2} + 1 \) for m ≥ 10, then M is isometric to ℍPm. One of the main strategies of this paper is to apply a more delicate argument of Frankel type to positive quaternionic Kahler manifolds with certain symmetry rank.

On positive quaternionic Kähler manifolds with certain symmetry rank

Let M be a positive quaternionic Kahler manifold of real dimension 4m. In this paper we show that if the symmetry rank of M is greater than or equal to [m/2] + 3, then M is isometric to HPm or Gr2(Cm+2). This is sharp and optimal, and will complete the classification result of positive quaternionic Kahler manifolds equipped with symmetry. The main idea is to use the connectedness theorem for quaternionic Kahler manifolds with a group action and the induction arguments on the dimension of the manifold.

The stable free rank of symmetry of products of spheres

A well known conjecture in the theory of transformation groups states that if p is a prime and (ℤ/p) r acts freely on a product of k spheres, then r≤k. We prove this assertion if p is large compared to the dimension of the product of spheres. The argument builds on tame homotopy theory for non-simply connected spaces.

Positive Quaternionic Kähler manifolds and symmetry rank: II

Let $M$ be a positive quaternionic K\ahler manifold of dimension $4m$. If the isometry group $\text{Isom}(M)$ has rank at least $\frac {m}2 +3$, then $M$ is isometric to $\Bbb HP^m$ or $Gr_2(\Bbb C^{m+2})$. The lower bound for the rank is optimal if $m$ is even.

The Hopf conjecture for positively curved manifolds with discrete abelian group actions

Let M be a closed even n-manifold of positive sectional curvature. The main result asserts that the Euler characteristic of M is positive, if M admits an isometric Z p k -action with prime p ⩾ p ( n ) (a constant depending only on n) and k satisfies any one of the following conditions: (i) k ⩾ n − 4 8 and n ≠ 12 , 18 or 20; (ii) k ⩾ n − 2 10 , and n ≡ 0 mod 4 with n ≠ 12 or 20; (iii) k ⩾ n + 4 12 , and n ≡ 0 , 4 or 12 mod 20 with n ≠ 20 . This generalizes some results in [T. Püttmann, C. Searle, The Hopf conjecture for manifolds with low cohomogeneity or high symmetry rank, Proc. Amer. Math. Soc. 130 (2002) 163–166; X. Rong, Positively curved manifolds with almost maximal symmetry rank, Geom. Dedicata 59 (2002) 157–182; X. Rong, X. Su, The Hopf conjecture for positively curved manifolds with abelian group actions, Comm. Cont. Math. 7 (2005) 121–136].