ON SASAKIAN–EINSTEIN GEOMETRY

We introduce a multiplication ⋆ (we call it a join) on the space of all compact Sasakian-Einstein orbifolds [Formula: see text] and show that [Formula: see text] has the structure of a commutative associative topological monoid. The set [Formula: see text] of all compact regular Sasakian–Einstein manifolds is then a submonoid. The set of smooth manifolds in [Formula: see text] is not closed under this multiplication; however, the join [Formula: see text] of two Sasakian–Einstein manifolds is smooth under some additional conditions which we specify. We use this construction to obtain many old and new examples of Sasakain–Einstein manifolds. In particular, in every odd dimension greater that five we obtain spaces with arbitrary second Betti number.

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Sasaki-Einstein 7-Manifolds, Orlik Polynomials and Homology

Symmetry ◽

10.3390/sym11070947 ◽

2019 ◽

Vol 11 (7) ◽

pp. 947

Author(s):

Ralph R. Gomez

Keyword(s):

Betti Number ◽

Homology Group ◽

Integral Homology ◽

Einstein Manifolds ◽

The Third

In this article, we give ten examples of 2-connected seven dimensional Sasaki-Einstein manifolds for which the third homology group is completely determined. Using the Boyer-Galicki construction of links over particular Kähler-Einstein orbifolds, we apply a valid case of Orlik’s conjecture to the links so that one is able to explicitly determine the entire third integral homology group. We give ten such new examples, all of which have the third Betti number satisfy 10 ≤ b 3 ( L f ) ≤ 20 .

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Towards the theory of boundary value problems on non-smooth manifolds

10.1063/5.0040277 ◽

2021 ◽

Author(s):

Vladimir Vasilyev

Keyword(s):

Boundary Value Problems ◽

Boundary Value ◽

Smooth Manifolds

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Quaternionic contact 4n + 3-manifolds and their 4n-quotients

Annals of Global Analysis and Geometry ◽

10.1007/s10455-021-09758-5 ◽

2021 ◽

Author(s):

Yoshinobu Kamishima

Keyword(s):

Scalar Curvature ◽

Lie Group ◽

Contact Structure ◽

Einstein Manifolds ◽

Hermitian Metrics ◽

Kähler Metric ◽

Formula One ◽

Quaternion Space ◽

Kahler Metric ◽

Do So

AbstractWe study some types of qc-Einstein manifolds with zero qc-scalar curvature introduced by S. Ivanov and D. Vassilev. Secondly, we shall construct a family of quaternionic Hermitian metrics $$(g_a,\{J_\alpha \}_{\alpha =1}^3)$$ ( g a , { J α } α = 1 3 ) on the domain Y of the standard quaternion space $${\mathbb {H}}^n$$ H n one of which, say $$(g_a,J_1)$$ ( g a , J 1 ) is a Bochner flat Kähler metric. To do so, we deform conformally the standard quaternionic contact structure on the domain X of the quaternionic Heisenberg Lie group$${{\mathcal {M}}}$$ M to obtain quaternionic Hermitian metrics on the quotient Y of X by $${\mathbb {R}}^3$$ R 3 .

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Bootstrap bounds on closed Einstein manifolds

Journal of High Energy Physics ◽

10.1007/jhep10(2020)069 ◽

2020 ◽

Vol 2020 (10) ◽

Author(s):

James Bonifacio ◽

Kurt Hinterbichler

Keyword(s):

Compact Riemannian Manifold ◽

Tree Level ◽

Conformal Field Theories ◽

Einstein Manifolds ◽

Consistency Conditions ◽

Conformal Field ◽

Geometric Data ◽

Conformal Bootstrap ◽

Laplacian Operators ◽

Kaluza Klein

Abstract A compact Riemannian manifold is associated with geometric data given by the eigenvalues of various Laplacian operators on the manifold and the triple overlap integrals of the corresponding eigenmodes. This geometric data must satisfy certain consistency conditions that follow from associativity and the completeness of eigenmodes. We show that it is possible to obtain nontrivial bounds on the geometric data of closed Einstein manifolds by using semidefinite programming to study these consistency conditions, in analogy to the conformal bootstrap bounds on conformal field theories. These bootstrap bounds translate to constraints on the tree-level masses and cubic couplings of Kaluza-Klein modes in theories with compact extra dimensions. We show that in some cases the bounds are saturated by known manifolds.

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An Obata-type theorem on compact Einstein manifolds with boundary

Geometriae Dedicata ◽

10.1007/s10711-021-00598-y ◽

2021 ◽

Author(s):

Kazuo Akutagawa

Keyword(s):

Boundary Condition ◽

Smooth Boundary ◽

Type Theorem ◽

Type Inequality ◽

Einstein Metric ◽

Manifolds With Boundary ◽

Einstein Manifolds ◽

Conformal Class ◽

Totally Geodesic ◽

Yamabe Constant

AbstractWe show a kind of Obata-type theorem on a compact Einstein n-manifold $$(W, \bar{g})$$ ( W , g ¯ ) with smooth boundary $$\partial W$$ ∂ W . Assume that the boundary $$\partial W$$ ∂ W is minimal in $$(W, \bar{g})$$ ( W , g ¯ ) . If $$(\partial W, \bar{g}|_{\partial W})$$ ( ∂ W , g ¯ | ∂ W ) is not conformally diffeomorphic to $$(S^{n-1}, g_S)$$ ( S n - 1 , g S ) , then for any Einstein metric $$\check{g} \in [\bar{g}]$$ g ˇ ∈ [ g ¯ ] with the minimal boundary condition, we have that, up to rescaling, $$\check{g} = \bar{g}$$ g ˇ = g ¯ . Here, $$g_S$$ g S and $$[\bar{g}]$$ [ g ¯ ] denote respectively the standard round metric on the $$(n-1)$$ ( n - 1 ) -sphere $$S^{n-1}$$ S n - 1 and the conformal class of $$\bar{g}$$ g ¯ . Moreover, if we assume that $$\partial W \subset (W, \bar{g})$$ ∂ W ⊂ ( W , g ¯ ) is totally geodesic, we also show a Gursky-Han type inequality for the relative Yamabe constant of $$(W, \partial W, [\bar{g}])$$ ( W , ∂ W , [ g ¯ ] ) .

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On equicontinuous factors of flows on locally path-connected compact spaces

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2020.104 ◽

2020 ◽

pp. 1-18

Author(s):

NIKOLAI EDEKO

Keyword(s):

Lie Group ◽

Metric Space ◽

Betti Number ◽

Alternative Proof ◽

Simple Consequence ◽

Compact Metric Space ◽

Invariant Functions ◽

Compact Spaces ◽

Path Connected

Abstract We consider a locally path-connected compact metric space K with finite first Betti number $\textrm {b}_1(K)$ and a flow $(K, G)$ on K such that G is abelian and all G-invariant functions $f\,{\in}\, \text{\rm C}(K)$ are constant. We prove that every equicontinuous factor of the flow $(K, G)$ is isomorphic to a flow on a compact abelian Lie group of dimension less than ${\textrm {b}_1(K)}/{\textrm {b}_0(K)}$ . For this purpose, we use and provide a new proof for Theorem 2.12 of Hauser and Jäger [Monotonicity of maximal equicontinuous factors and an application to toral flows. Proc. Amer. Math. Soc.147 (2019), 4539–4554], which states that for a flow on a locally connected compact space the quotient map onto the maximal equicontinuous factor is monotone, i.e., has connected fibers. Our alternative proof is a simple consequence of a new characterization of the monotonicity of a quotient map $p\colon K\to L$ between locally connected compact spaces K and L that we obtain by characterizing the local connectedness of K in terms of the Banach lattice $\textrm {C}(K)$ .

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