Isoparametric foliation and a problem of Besse on generalizations of Einstein condition

Tang, Zizhou; Yan, Wenjiao

doi:10.1016/j.aim.2015.09.003

Cited by 26 publications

(20 citation statements)

References 25 publications

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“…Summarizing all the estimates above, and combining with the results in [TY15], [TY17], [QTY13] and [LZ16], we are able to prove the following theorem, determining all the sets C A , C P and C E . Theorem 1.1.…”

Section: Introductionmentioning

confidence: 67%

“…for any unit normal vector N . Besides, Lemma 2.1 in [TY15] provides an orthonormal basis of T x M − : for a given 1 ≤ i ≤ 8,…”

Section: Now We Would Like To List Three Observationsmentioning

confidence: 99%

“…Proof. In Proposition 4.2 of [TY15], except for the cases of (5, 2), (6, 1) and (9, 6), the proof for M − of OT-FKM type with m ≥ 2 are not Ricci parallel is actually pointwise. Now we are ready to show the following Proof.…”

Section: The Set C P -Parallel Second Fundamental Formmentioning

confidence: 99%

“…In Sections 4 and 5, we determine the sets C P and C E . There we will use the methods in [TY15], [QTY13] and [LZ16] where the parallelism and Einstein conditions were considered globally. Lastly, in Section 6, we prove Proposition 1.2, where the relations between the focal submanifolds with g = 4, (8, 7) are shown and totally isoparametric functions are constructed.…”

Section: Introductionmentioning

confidence: 99%

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Normal Scalar Curvature Inequality on the Focal Submanifolds of Isoparametric Hypersurfaces

Tang

Yan

2018

International Mathematics Research Notices

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An isoparametric hypersurface in unit spheres has two focal submanifolds. Condition A plays a crucial role in the classification theory of isoparametric hypersurfaces in [CCJ07], [Chi16] and [Miy13]. This paper determines CA, the set of points with Condition A in focal submanifolds. It turns out that the points in CA reach an upper bound of the normal scalar curvature ρ ⊥ (sharper than that in DDVV inequality [GT08], [Lu11]). We also determine the sets CP (points with parallel second fundamental form) and CE (points with Einstein condition), which achieve two lower bounds of ρ ⊥ .

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Section: Introductionmentioning

confidence: 67%

“…for any unit normal vector N . Besides, Lemma 2.1 in [TY15] provides an orthonormal basis of T x M − : for a given 1 ≤ i ≤ 8,…”

Section: Now We Would Like To List Three Observationsmentioning

confidence: 99%

Section: The Set C P -Parallel Second Fundamental Formmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Normal Scalar Curvature Inequality on the Focal Submanifolds of Isoparametric Hypersurfaces

Tang

Yan

2018

International Mathematics Research Notices

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“…This example is a S 1 -bundle over some Kähler-Einstein manifold. Recently Z. Tang and W. Yan in [12] obtained some new examples of A-manifolds as focal sets of isoparametric hypersurfaces in spheres.…”

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confidence: 99%

A-manifolds on a principal torus bundle over an almost Hodge A-manifold base

Zborowski¹

2015

Annales UMCS, Mathematica

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Abstract. An A-manifold is a manifold whose Ricci tensor is cyclic-parallel, equivalently it satisfies ∇X Ric(X, X) = 0. This condition generalizes the Einstein condition. We construct new examples of A-manifolds on r-torus bundles over a base which is a product of almost Hodge A-manifolds. Introduction.One of the most extensively studied objects in mathematics and physics are Einstein manifolds (see for example [1]), i.e. manifolds whose Ricci tensor is a constant multiple of the metric tensor. In his work [2] A. Gray defined a condition which generalizes the concept of an Einsten manifold. This condition states that the Ricci tensor Ric of the Riemannian manifold (M, g) is cyclic parallel, i.e.where ∇ denotes the Levi-Civita connection of the metric g and X, Y, Z are arbitrary vector fields on M . A Riemannian manifold satisfying this condition is called an A-manifold. It is obvious that if the Ricci tensor of (M, g) is parallel, then it satisfies the above condition. On the other hand, if Ric is cyclic-parallel, but not parallel, then we call (M, g) a strict A-manifold. A. Gray gave in [2] the first example of such strict A-manifold, which was the sphere S 3 with appropriately defined homogeneous metric. to K-contact manifolds. Namely, over every almost Hodge A-manifold with J-invariant Ricci tensor we can construct a Riemannian metric such that the total space of the bundle is an A-manifold. In the present paper we take a next step in the generalization process and we prove that there exists an A-manifold structure on every r-torus bundle over product of almost Hodge A-manifolds. Our result and that of Jelonek are based on the existence of almost Hodge A-manifolds, which was proven in [5]. Conformal Killing tensors.Let (M, g) be any Riemannian manifold. We call a symmetric tensor field of type (0, 2) on M a conformal Killing tensor field iff there exists a 1-form P such that for any X ∈ Γ(T M)where ∇ is the Levi-Civita connection of g. The above condition is clearly equivalent to the followingIt is easy to prove that the 1-form P is given bywhere X ∈ Γ(T M) and divS and tr S are the divergence and trace of the tensor field S with respect to g. If the 1-form P vanishes, then we call K a Killing tensor. Of particular interest in this work is a situation when the Ricci tensor of the metric g is a Killing tensor. We call such a manifold an A-manifold. In the more general situation, when the Ricci tensor is a conformal tensor we call (M, g) a AC ⊥ -manifold. We will use the following easy property of conformal Killing tensors.A conformal Killing form or a twistor form is a differential p-form ϕ on (M, g) satisfying the following equationA-manifolds on a principal torus bundle... 111An extensive description of conformal Killing forms can be found in a series of articles by Semmelmann and Moroianu ([11],[7]). It is known that if ϕ is a co-closed conformal Killing form (also called a Killing form), then the (0, 2)-tensor field K ϕ defined byis a Killing tensor.The following theorem generalizes the above observation....

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Differential Topology Interacts with Isoparametric Foliations

Qian

2016

Springer Proceedings in Mathematics &Amp; Statistics

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Isoparametric foliation and a problem of Besse on generalizations of Einstein condition

Cited by 26 publications

References 25 publications

Normal Scalar Curvature Inequality on the Focal Submanifolds of Isoparametric Hypersurfaces

Normal Scalar Curvature Inequality on the Focal Submanifolds of Isoparametric Hypersurfaces

A-manifolds on a principal torus bundle over an almost Hodge A-manifold base

Differential Topology Interacts with Isoparametric Foliations

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