Ilka Agricola

3-Sasakian manifolds in dimension seven, their spinors and G2-structures

Abstract It is well known that 7-dimensional 3-Sasakian manifolds carry a one-parametric family of compatible G 2 -structures and that they do not admit a characteristic connection. In this note, we show that there is nevertheless a distinguished cocalibrated G 2 -structure in this family whose characteristic connection ∇ c along with its parallel spinor field Ψ 0 can be used for a thorough investigation of the geometric properties of 7-dimensional 3-Sasakian manifolds. Many known and some new properties can be easily derived from the properties of ∇ c and of Ψ 0 , yielding thus an appropriate substitute for the missing characteristic connection.

Spinorial description of SU(3)- and G2-manifolds

Abstract We present a uniform description of SU ( 3 ) -structures in dimension 6 as well as G 2 -structures in dimension 7 in terms of a characterising spinor and the spinorial field equations it satisfies. We apply the results to hypersurface theory to obtain new embedding theorems, and give a general recipe for building conical manifolds. The approach also enables one to subsume all variations of the notion of a Killing spinor.

On the Ricci tensor in the common sector of type II string theory

Let ∇ be a metric connection with totally skew-symmetric torsion T on a Riemannian manifold. Given a spinor field Ψ and a dilaton function Φ, the basic equations in type II B string theory are ∇Ψ = 0 , δ(T) = a · dΦ T , T · Ψ = b · dΦ · Ψ + µ · Ψ. We derive some relations between the length ||T|| 2 of the torsion form, the scalar curvature of ∇, the dilaton function Φ and the parameters a, b, µ. The main results deal with the divergence of the Ricci tensor Ric ∇ of the connection. In particular, if the supersymmetry Ψ is non-trivial and if the conditions (dΦ T) T = 0 , δ ∇ (dT) · Ψ = 0 hold, then the energy-momentum tensor is divergence-free. We show that the latter condition is satisfied in many examples constructed out of special geometries. A special case is a = b. Then the divergence of the energy-momentum tensor vanishes if and only if one condition δ ∇ (dT) · Ψ = 0 holds. Strong models (dT = 0) have this property, but there are examples with δ ∇ (dT) = 0 and δ ∇ (dT) · Ψ = 0. 1. Type II B string theory with constant dilaton The mathematical model discussed in type II B string theory consists of a Riemannian manifold (M n , g), a metric connection ∇ with totally skew-symmetric torsion T and a non-trivial spinor field Ψ. Putting the full Ricci tensor aside for starters, there are three equations relating these objects: (*) ∇Ψ = 0 , δ(T) = 0 , T · Ψ = µ · Ψ. The spinor field describes the supersymmetry of the model. The first equation means that the spinor field Ψ is parallel with respect to the metric connection ∇. The second equation is a conservation law for the 3-form T. Since ∇ is a metric connection with totally skew-symmetric torsion, the divergences δ ∇ (T) = δ g (T) of the torsion form coincide (see [2], [8]). We denote this unique 2-form simply by δ(T). The third equation is an algebraic link between the torsion form T and the spinor field Ψ. Indeed, the 3-form T acts as an endomorphism in the spinor bundle and the last equation requires that Ψ is an eigenspinor for this endomorphism. There are models with µ …Let ∇ be a metric connection with totally skew-symmetric torsion T on a Riemannian manifold. Given a spinor field Ψ and a dilaton function Φ, the basic equations in the common sector of type II string theory are for some auxiliary parameters a, b, μ. We derive some relations between the length ||T||2 of the torsion form, the scalar curvature of ∇, the dilaton function Φ and the parameters a, b, μ. We show that for constant dilaton and μ = 0 (the physically relevant case), there cannot be even local solutions to this system of equations with vanishing scalar curvature. The main results deal with the divergence of the Ricci tensor Ric∇ of the connection. In particular, if the supersymmetry Ψ is non-trivial and if the conditions hold, then the energy–momentum tensor is divergence free. We show that the latter condition is satisfied in many examples constructed out of special geometries. A special case is a = b. Then the divergence of the energy–momentum tensor vanishes if and only if one condition δ∇(dT) ⋅ Ψ = 0 holds. Strong models (dT = 0) have this property, but there are examples with δ∇(dT) ≠ 0 and δ∇(dT) ⋅ Ψ = 0.

Upper bounds for the first eigenvalue of the Dirac operator on surfaces

Abstract In this paper we will prove new extrinsic upper bounds for the eigenvalues of the Dirac operator on an isometrically immersed surface M 2 ↪ R 3 as well as intrinsic bounds for two-dimensional compact manifolds of genus zero and genus one. Moreover, we compare the different estimates of the eigenvalue of the Dirac operator for special families of metrics.

The Casimir operator of a metric connection with skew-symmetric torsion

For any triple (Mn,g,∇) consisting of a Riemannian manifold and a metric connection with skew-symmetric torsion we introduce an elliptic, second-order operator Ω acting on spinor fields. In case of a naturally reductive space and its canonical connection, our construction yields the Casimir operator of the isometry group. Several non-homogeneous geometries (Sasakian, nearly Kahler, cocalibrated G2-structures) admit unique connections with skew-symmetric torsion. We study the corresponding Casimir operator and compare its kernel with the space of ∇-parallel spinors.

Geometric structures of vectorial type

Abstract We study geometric structures of W 4 -type in the sense of A. Gray on a Riemannian manifold. If the structure group G ⊂ SO ( n ) preserves a spinor or a non-degenerate differential form, its intrinsic torsion Γ is a closed 1-form (Proposition 2.1 and Theorem 2.1). Using a G -invariant spinor we prove a splitting theorem (Proposition 2.2). The latter result generalizes and unifies a recent result obtained in [S. Ivanov, M. Parton, P. Piccinni, Locally conformal parallel G 2 - and Spin ( 7 ) -structures, mathdg/0509038 ], where this splitting has been proved in dimensions n = 7 , 8 only. Finally we investigate geometric structures of vectorial type and admitting a characteristic connection ∇ c . An interesting class of geometric structures generalizing Hopf structures are those with a ∇ c -parallel intrinsic torsion Γ . In this case, Γ induces a Killing vector field (Proposition 4.1) and for some special structure groups it is even parallel.

Killing spinors in supergravity with 4-fluxes

We study the spinorial Killing equation of supergravity involving a torsion 3-form T as well as a flux 4-form F. In dimension seven, we construct explicit families of compact solutions out of 3-Sasakian geometries, nearly parallel G2-geometries and on the homogeneous Aloff–Wallach space. The constraint F Ψ = 0 defines a non-empty subfamily of solutions. We investigate the constraint T Ψ = 0, too, and show that it singles out a very special choice of numerical parameters in the Killing equation, which can also be justified geometrically.

Sp(3) structures on 14-dimensional manifolds

Abstract The present article investigates Sp ( 3 ) structures on 14 -dimensional Riemannian manifolds, a continuation of the recent study of manifolds modeled on rank two symmetric spaces (here: SU ( 6 ) / Sp ( 3 ) ). We derive topological criteria for the existence of such a structure and construct large families of homogeneous examples. As a by-product, we prove a general uniqueness criterion for characteristic connections of G structures and that the notions of biinvariant, canonical, and characteristic connections coincide on Lie groups with biinvariant metric.

Quaternionic Heisenberg groups as naturally reductive homogeneous spaces

In this note, we describe the geometry of the quaternionic Heisenberg groups from a Riemannian viewpoint. We show, in all dimensions, that they carry an almost

Tangent Lie Groups are Riemannian Naturally Reductive Spaces

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