55 research outputs found
Conformal Killing forms on Riemannian manifolds
Conformal Killing forms are a natural generalization of conformal vector
fields on Riemannian manifolds. They are defined as sections in the kernel of a
conformally invariant first order differential operator. We show the existence
of conformal Killing forms on nearly Kaehler and weak G_2-manifolds. Moreover,
we give a complete description of special conformal Killing forms. A further
result is a sharp upper bound on the dimension of the space of conformal
Killing forms.Comment: 24 page
Scalar curvature estimates for compact symmetric spaces
We establish extremality of Riemannian metrics g with non-negative curvature
operator on symmetric spaces M=G/K of compact type with rk(G)-rk(K)\le 1. Let
g' be another metric with scalar curvature k', such that g'\ge g on 2-vectors.
We show that k'\ge k everywhere on M implies k'=k. Under an additional
condition on the Ricci curvature of g, k'\ge k even implies g'=g. We also study
area-non-increasing spin maps onto such Riemannian manifolds.Comment: 13 pages, LaTeX, uses amsar
The First Eigenvalue of the Dirac Operator on Quaternionic Kaehler Manifolds
In a previous paper we proved a lower bound for the spectrum of the Dirac
operator on quaternionic Kaehler manifolds. In the present article we show that
the only manifolds in the limit case, i.e. the only manifolds where the lower
bound is attained as an eigenvalue, are the quaternionic projective spaces. We
use the equivalent formulation in terms of the quaternionic Killing equation
and show that a nontrivial solution defines a parallel spinor on the associated
hyperkaehler manifold.Comment: 19 pages, LaTeX2e, fullpage styl
Killing spinors are Killing vector fields in Riemannian Supergeometry
A supermanifold M is canonically associated to any pseudo Riemannian spin
manifold (M_0,g_0). Extending the metric g_0 to a field g of bilinear forms
g(p) on T_p M, p\in M_0, the pseudo Riemannian supergeometry of (M,g) is
formulated as G-structure on M, where G is a supergroup with even part G_0\cong
Spin(k,l); (k,l) the signature of (M_0,g_0). Killing vector fields on (M,g)
are, by definition, infinitesimal automorphisms of this G-structure. For every
spinor field s there exists a corresponding odd vector field X_s on M. Our main
result is that X_s is a Killing vector field on (M,g) if and only if s is a
twistor spinor. In particular, any Killing spinor s defines a Killing vector
field X_s.Comment: 14 pages, latex, one typo correcte
A Reilly formula and eigenvalue estimates for differential forms
We derive a Reilly-type formula for differential p-forms on a compact
manifold with boundary and apply it to give a sharp lower bound of the spectrum
of the Hodge Laplacian acting on differential forms of an embedded hypersurface
of a Riemannian manifold. The equality case of our inequality gives rise to a
number of rigidity results, when the geometry of the boundary has special
properties and the domain is non-negatively curved. Finally we also obtain, as
a by-product of our calculations, an upper bound of the first eigenvalue of the
Hodge Laplacian when the ambient manifold supports non-trivial parallel forms.Comment: 22 page
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