scholarly journals A Kastler–Kalau–Walze type theorem for 7-dimensional spin manifolds with boundary about Dirac operators with torsion

2016 ◽  
Vol 110 ◽  
pp. 213-232
Author(s):  
Kai Hua Bao ◽  
Ai Hui Sun ◽  
Jian Wang
Keyword(s):  
Type Theorem ◽  
Dirac Operators ◽  
Manifolds With Boundary ◽  
Spin Manifolds

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 Invertible Dirac operators and handle attachments on manifolds with boundary

2014 ◽  
Vol 06 (03) ◽  
pp. 339-382
Author(s):  
Mattias Dahl ◽  
Nadine Grosse
Keyword(s):  
Dirac Operator ◽  
Riemannian Metrics ◽  
Dirac Operators ◽  
Classification Theorem ◽  
Manifolds With Boundary ◽  
Spin Manifolds

For spin manifolds with boundary we consider Riemannian metrics which are product near the boundary and are such that the corresponding Dirac operator is invertible when half-infinite cylinders are attached at the boundary. The main result of this paper is that these properties of a metric can be preserved when the metric is extended over a handle of codimension at least two attached at the boundary. Applications of this result include the construction of non-isotopic metrics with invertible Dirac operator, and a concordance existence and classification theorem.

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 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 ¯ ] ) .

Twisted dirac operators and Kastler-Kalau-Walze theorems for six-dimensional manifolds with boundary

2020 ◽  
Vol 17 (14) ◽  
pp. 2050211
Author(s):  
Sining Wei ◽  
Yong Wang
Keyword(s):  
Vector Bundle ◽  
Dirac Operators ◽  
Manifolds With Boundary ◽  
Unitary Connection ◽  
Twisted Dirac Operators

In this paper, we establish two kinds of Kastler-Kalau-Walze type theorems for Dirac operators and signature operators twisted by a vector bundle with a non-unitary connection on six-dimensional manifolds with boundary.

scholarly journals A long neck principle for Riemannian spin manifolds with positive scalar curvature

2020 ◽  
Vol 30 (5) ◽  
pp. 1183-1223
Author(s):  
Simone Cecchini
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Pairwise Disjoint ◽  
Index Theory ◽  
Spin Manifold ◽  
Manifolds With Boundary ◽  
Topological Information ◽  
Manifold With Boundary ◽  
Spin Manifolds ◽  
Nonzero Degree

AbstractWe develop index theory on compact Riemannian spin manifolds with boundary in the case when the topological information is encoded by bundles which are supported away from the boundary. As a first application, we establish a “long neck principle” for a compact Riemannian spin n-manifold with boundary X, stating that if $${{\,\mathrm{scal}\,}}(X)\ge n(n-1)$$ scal ( X ) ≥ n ( n - 1 ) and there is a nonzero degree map into the sphere $$f:X\rightarrow S^n$$ f : X → S n which is strictly area decreasing, then the distance between the support of $$\text {d}f$$ d f and the boundary of X is at most $$\pi /n$$ π / n . This answers, in the spin setting and for strictly area decreasing maps, a question recently asked by Gromov. As a second application, we consider a Riemannian manifold X obtained by removing k pairwise disjoint embedded n-balls from a closed spin n-manifold Y. We show that if $${{\,\mathrm{scal}\,}}(X)>\sigma >0$$ scal ( X ) > σ > 0 and Y satisfies a certain condition expressed in terms of higher index theory, then the radius of a geodesic collar neighborhood of $$\partial X$$ ∂ X is at most $$\pi \sqrt{(n-1)/(n\sigma )}$$ π ( n - 1 ) / ( n σ ) . Finally, we consider the case of a Riemannian n-manifold V diffeomorphic to $$N\times [-1,1]$$ N × [ - 1 , 1 ] , with N a closed spin manifold with nonvanishing Rosenebrg index. In this case, we show that if $${{\,\mathrm{scal}\,}}(V)\ge \sigma >0$$ scal ( V ) ≥ σ > 0 , then the distance between the boundary components of V is at most $$2\pi \sqrt{(n-1)/(n\sigma )}$$ 2 π ( n - 1 ) / ( n σ ) . This last constant is sharp by an argument due to Gromov.

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.

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