scholarly journals A Kastler-Kalau-Walze Type Theorem for 7-Dimensional Manifolds with Boundary

2014 ◽  
Vol 2014 ◽  
pp. 1-18 ◽  
Author(s):  
Jian Wang ◽  
Yong Wang
Keyword(s):  
Type Theorem ◽  
Brute Force ◽  
Manifolds With Boundary

We give a brute-force proof of the Kastler-Kalau-Walze type theorem for 7-dimensional manifolds with boundary.

scholarly journals An Obata-type theorem on compact Einstein manifolds with boundary

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 ¯ ] ) .

A general Kastler–Kalau–Walze type theorem for manifolds with boundary

2016 ◽  
Vol 13 (01) ◽  
pp. 1650003 ◽  
Author(s):  
Jian Wang ◽  
Yong Wang
Keyword(s):  
General Expression ◽  
Type Theorem ◽  
Manifolds With Boundary ◽  
Theor Phys ◽  
Geometric Data ◽  
Lower Dimensional

In this paper, we establish a general Kastler–Kalau–Walze type theorem for any dimensional manifolds with boundary which generalizes the results in [Y. Wang, Lower-dimensional volumes and Kastler–Kalau–Walze type theorem for manifolds with boundary, Commun. Theor. Phys. 54 (2010) 38–42]. This solves a problem of the referee of [J. Wang and Y. Wang, A Kastler–Kalau–Walze type theorem for five-dimensional manifolds with boundary, Int. J. Geom. Meth. Mod. Phys. 12(5) (2015), Article ID: 1550064, 34 pp.], which is a general expression of the lower dimensional volumes in terms of the geometric data on the manifold.

scholarly journals A Kastler-Kalau-Walze Type Theorem and the Spectral Action for Perturbations of Dirac Operators on Manifolds with Boundary

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Yong Wang
Keyword(s):  
Type Theorem ◽  
Dirac Operators ◽  
Compact Manifolds ◽  
Manifolds With Boundary ◽  
Spectral Action ◽  
Gravitational Action

We prove a Kastler-Kalau-Walze type theorem for perturbations of Dirac operators on compact manifolds with or without boundary. As a corollary, we give two kinds of operator-theoretic explanations of the gravitational action on boundary. We also compute the spectral action for Dirac operators with two-form perturbations on 4-dimensional compact manifolds.

scholarly journals A Kastler–Kalau–Walze type theorem for five-dimensional manifolds with boundary

2015 ◽  
Vol 12 (05) ◽  
pp. 1550064 ◽  
Author(s):  
Jian Wang ◽  
Yong Wang
Keyword(s):  
General Relativity ◽  
Dirac Operator ◽  
Type Theorem ◽  
Manifolds With Boundary ◽  
Wodzicki Residue ◽  
Hilbert Action

The Kastler–Kalau–Walze theorem, announced by A. Connes, shows that the Wodzicki residue of the inverse square of the Dirac operator is proportional to the Einstein–Hilbert action of general relativity. In this paper, we prove a Kastler–Kalau–Walze type theorem for five-dimensional manifolds with boundary.

scholarly journals General Kastler–Kalau–Walze type theorems for manifolds with boundary II

2019 ◽  
Vol 16 (02) ◽  
pp. 1950028
Author(s):  
Yong Wang
Keyword(s):  
Type Theorem ◽  
Manifolds With Boundary ◽  
Math Phys

In this paper, we establish some general Kastler–Kalau–Walze type theorems for any dimensional manifolds with boundary which generalize the results in [J. Wang and Y. Wang, The Kastler–Kalau–Walze type theorem for six-dimensional manifolds with boundary, J. Math. Phys. 56 (2015), Article ID: 052501, 14 pp.].

scholarly journals An equivariant Kastler–Kalau–Walze type theorem

2017 ◽  
Vol 14 (04) ◽  
pp. 1750056
Author(s):  
Yong Wang
Keyword(s):  
General Formula ◽  
Type Theorem ◽  
Dimensional Manifold ◽  
Manifolds With Boundary ◽  
Spin Manifolds

In this paper, we prove an equivariant Kastler–Kalau–Walze type theorem for spin manifolds without boundary. For [Formula: see text]-dimensional spin manifolds with boundary, we also give an equivariant Kastler–Kalau–Walze type theorem. Then we generalize this theorem to the general [Formula: see text]-dimensional manifold. An equivariant Kastler–Kalau–Walze type theorem with torsion is also proved.

scholarly journals Dirac–Witten Operators and the Kastler–Kalau–Walze Type Theorem for Manifolds with Boundary

2021 ◽  
Author(s):  
Tong Wu ◽  
Jian Wang ◽  
Yong Wang
Keyword(s):  
Type Theorem ◽  
Compact Manifolds ◽  
Manifolds With Boundary

AbstractIn this paper, we obtain two Lichnerowicz type formulas for the Dirac–Witten operators. And we give the proof of Kastler–Kalau–Walze type theorems for the Dirac–Witten operators on 4-dimensional and 6-dimensional compact manifolds with (resp. without) boundary.

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