The eta invariant and the equivariant spin bordism of spherical space form 2 groups

1996 ◽  
pp. 213-223 ◽  
Author(s):  
Peter B. Gilkey ◽  
Boris Botvinnik
Keyword(s):  
Space Form ◽  
Spherical Space ◽  
Eta Invariant ◽  
Spherical Space Form

The Eta Invariant and Equivariant SpinC Bordism for Spherical Space form Groups

1988 ◽  
Vol 40 (2) ◽  
pp. 392-428 ◽  
Author(s):  
Peter B. Gilkey
Keyword(s):  
Space Form ◽  
Spherical Space ◽  
Eta Invariant ◽  
Eta Invariants ◽  
Whitney Numbers ◽  
Free Representation ◽  
A Finite Group ◽  
Prime 2 ◽  
Coefficient Ring ◽  
Spherical Space Form

A finite group G is a spherical space form group if it admits a fixed point free representation τ:G → U(k) for some k; for the remainder of this paper, we assume G is such a group. The eta invariant of Atiyah et al [2] defines Q/Z valued invariants of equivariant bordism. In [6], we showed the eta invariant completely detects the reduced equivariant unitary bordism groups and completely detects all but the 2-primary part of the reduced equivariant SpinC bordism groups . The coefficient ring is without torsion; all the torsion in is of order 2. The prime 2 plays a distinguished role in the discussion of equivariant SpinC bordism and is quite different from at the prime 2. Let ker*(η, G) denote the kernel of all eta invariants and let ker*(SW, G) denote the kernel of the Z2-equivariant Stiefel-Whitney numbers (see Section 1 for details). Then:THEOREM 0.1. Let. If M = ker*(η, G) ∩ ker*(SW, G), M = 0.

Metrics of Positive Scalar Curvature on Spherical Space Forms

1996 ◽  
Vol 48 (1) ◽  
pp. 64-80 ◽  
Author(s):  
Boris Botvinnik ◽  
Peter B. Gilkey
Keyword(s):  
Scalar Curvature ◽  
Space Form ◽  
Positive Scalar Curvature ◽  
Spherical Space ◽  
Space Forms ◽  
Eta Invariant ◽  
Positive Scalar ◽  
Spherical Space Forms ◽  
Simply Connected ◽  
Spherical Space Form

AbstractWe use the eta invariant to show every non-simply connected spherical space form of dimension m ≥ 5 has a countable family of non bordant metrics of positive scalar curvature.

A note on a three-dimensional sphere theorem with integral curvature condition

2016 ◽  
Vol 18 (04) ◽  
pp. 1550070 ◽  
Author(s):  
Mijia Lai
Keyword(s):  
Scalar Curvature ◽  
Ricci Curvature ◽  
Space Form ◽  
Three Dimensional ◽  
Dimensional Sphere ◽  
Integral Curvature ◽  
Spherical Space ◽  
Sphere Theorem ◽  
Curvature Condition ◽  
Spherical Space Form

In this paper, we obtain a three-dimensional sphere theorem with integral curvature condition. On a closed three manifold [Formula: see text] with constant positive scalar curvature, if a certain combination of [Formula: see text] norm of the Ricci curvature and [Formula: see text] norm of the scalar curvature is positive, then [Formula: see text] is diffeomorphic to a spherical space form.

scholarly journals Bundles with even-dimensional spherical space form as fibers and fiberwise quarter pinched Riemannian metrics

2021 ◽  
Vol 149 (12) ◽  
pp. 5407-5416
Author(s):  
Diego Corro ◽  
Karla Garcia ◽  
Martin Günther ◽  
Jan-Bernhard Kordaß
Keyword(s):  
Space Form ◽  
Riemannian Metrics ◽  
Spherical Space ◽  
Spherical Space Form

scholarly journals Topological spherical space form problem. III. Dimensional bounds and smoothing

1983 ◽  
Vol 106 (1) ◽  
pp. 135-143 ◽  
Author(s):  
Ib Madsen ◽  
Charles Thomas ◽  
C. Terence C. Wall
Keyword(s):  
Space Form ◽  
Spherical Space ◽  
Form Problem ◽  
Spherical Space Form

scholarly journals The curvature and topological properties of hypersurfaces with constant scalar curvature

2004 ◽  
Vol 70 (1) ◽  
pp. 35-44
Author(s):  
Shu Shichang ◽  
Liu Sanyang
Keyword(s):  
Scalar Curvature ◽  
Space Form ◽  
Constant Scalar Curvature ◽  
Second Fundamental Form ◽  
Topological Properties ◽  
Spherical Space ◽  
Riemannian Product ◽  
3 Dimensional ◽  
The Second Fundamental Form ◽  
Spherical Space Form

In this paper, we consider n (n ≥ 3)-dimensional compact oriented connected hypersurfaces with constant scalar curvature n(n − 1)r in the unit sphere Sn+1(1). We prove that, if r ≥ (n − 2)/(n − 1) and S ≤ (n − 1)(n(r − 1) + 2)/(n − 2) + (n − 2)/(n(r − 1) + 2), then either M is diffeomorphic to a spherical space form if n = 3; or M is homeomorphic to a sphere if n ≥ 4; or M is isometric to the Riemannian product , where c2 = (n − 2)/(nr) and S is the squared norm of the second fundamental form of M.

scholarly journals On the Betti and Tachibana Numbers of Compact Einstein Manifolds

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1210 ◽  
Author(s):  
Vladimir Rovenski ◽  
Sergey Stepanov ◽  
Irina Tsyganok
Keyword(s):  
Space Form ◽  
Einstein Manifold ◽  
Einstein Manifolds ◽  
Spherical Space ◽  
And Topology ◽  
History Of ◽  
Sectional Curvatures ◽  
Einstein Constant ◽  
Spherical Space Form

Throughout the history of the study of Einstein manifolds, researchers have sought relationships between the curvature and topology of such manifolds. In this paper, first, we prove that a compact Einstein manifold ( M , g ) with an Einstein constant α > 0 is a homological sphere when the minimum of its sectional curvatures > α / ( n + 2 ) ; in particular, ( M , g ) is a spherical space form when the minimum of its sectional curvatures > α / n . Second, we prove two propositions (similar to the above ones) for Tachibana numbers of a compact Einstein manifold with α < 0 .

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