Killing spinors on Lorentzian manifolds

The aim of this paper is to describe some results concerning the geometry of Lorentzian manifolds admitting Killing spinors. We prove that there are imaginary Killing spinors on simply connected Lorentzian Einstein–Sasaki manifolds. In the Riemannian case, an odd-dimensional complete simply connected manifold (of dimension n≠7) is Einstein–Sasaki if and only if it admits a non-trivial Killing spinor to . The analogous result does not hold in the Lorentzian case. We give an example of a non-Einstein Lorentzian manifold admitting an imaginary Killing spinor. A Lorentzian manifold admitting a real Killing spinor is at least locally a codimension one warped product with a special warping function. The fiber of the warped product is either a Riemannian manifold with a real or imaginary Killing spinor or with a parallel spinor, or it again is a Lorentzian manifold with a real Killing spinor. Conversely, all warped products of that form admit real Killing spinors.     

Lower Bounds for Nodal Sets of Dirichlet and Neumann Eigenfunctions

Let ${\phi}$ be a Dirichlet or Neumann eigenfunction of the Laplace-Beltrami operator on a compact Riemannian manifold with boundary. We prove lower bounds for the size of the nodal set ${\{\phi = 0\}}$ .     

Cohomological tautness for Riemannian foliations

In this paper, we present new results on the tautness of Riemannian foliations in their historical context. The first part of the paper gives a short history of the problem. For a closed manifold, the tautness of a Riemannian foliation can be characterized cohomologically. We extend this cohomological characterization to a class of foliations which includes the foliated strata of any singular Riemannian foliation of a closed manifold.     

Minimal orbits of metrics

The group of diffeomorphisms of a compact manifold acts isometrically on the space of Riemannian metrics with its L² metric. Following Arnaudon and Paycha (1995) and Maeda, Rosenberg and Tondeur (1993), we define minimal orbits for this action by a zeta function regularization. We show that odd dimensional isotropy irreducible homogeneous spaces give rise to minimal orbits, the first known examples of minimal submanifolds of infinite dimension and codimension. We also find a flat 2-torus giving a stable minimal orbit. We prove that isolated orbits are minimal, as in finite dimensions.     

Gravitational Descendants in Symplectic Field Theory

It was pointed out by Y. Eliashberg in his ICM 2006 plenary talk that the rich algebraic formalism of symplectic field theory leads to a natural appearance of quantum and classical integrable systems, at least in the case when the contact manifold is the prequantization space of a symplectic manifold. In this paper we generalize the definition of gravitational descendants in SFT from circle bundles in the Morse-Bott case to general contact manifolds. After we have shown using the ideas in Okounkov and Pandharipande (Ann Math 163(2):517–560, 2006) that for the basic examples of holomorphic curves in SFT, that is, branched covers of cylinders over closed Reeb orbits, the gravitational descendants have a geometric interpretation in terms of branching conditions, we follow the ideas in Cieliebak and Latschev ( [math.s6], 2007) to compute the corresponding sequence of Poisson-commuting functions when the contact manifold is the unit cotangent bundle of a Riemannian manifold.     

Spectral geometry of eta-Einstein Sasakian manifolds

We extend a result of Patodi for closed Riemannian manifolds to the context of closed contact manifolds by showing the condition that a manifold is an η$\eta$-Einstein Sasakian manifold is spectrally determined. We also prove that the condition that a Sasakian space form has constant ?$?$-sectional curvature c$c$ is spectrally determined.     

Closed star products and cyclic cohomology

We define the notion of a closed star product. A (generalized) star product (deformation of the associative product of functions on a symplectic manifold W) is closed iff integration over W is a trace on the deformed algebra. We show that for these products the cyclic cohomology replaces the Hochschild cohomology in usual star products. We then define the character of a closed star product as the cohomology class (in the cyclic bicomplex) of a well-defined cocycle, and show that, in the case of pseudodifferential operators (standard ordering on the cotangent bundle to a compact Riemannian manifold), the character is defined and given by the Todd class, while in general it fails to satisfy the integrality condition.     

The geometry of sectional curvatures

A large class of questions in differential geometry involves the relationship between the geometry and the topology of a Riemannian (= positive-definite) manifold. We briefly review the status of the following question from this class: given that a compact, even-dimensional manifold admits a Riemannian metric of positive sectional curvatures, what can one say about its topology? Very few manifolds are known to admit such metrics. For example, is it not known whether or not the product of then-sphere with itself (n ≥ 2) does. One answer to the question above is provided by Synge's theorem: if the manifold is orientable, then it is simply connected. Another possible answer is given by the Hopf conjecture: such a manifold necessarily has positive Euler number. The Hopf conjecture is known to be true for homogeneous manifolds, and for arbitrary manifolds in dimensions two and four. This last result has two, apparently entirely different, proofs, one using Synge's theorem and the other the Gauss-Bonnet formula. Neither, it is shown, can be generalized directly to dimensions six or greater. The Hopf conjecture in these higher dimensions remains open.     

Nuclear and string fields on a Kähler manifold

An arbitrary oriented Riemannian manifold of real dimension two is a complex manifold that is also the world sheet of an oriented closed string. Another example is the complex Grassmann manifold ofp-planes inC _2p, which is shown to carry the most symmetric state of⁹Li. In both cases we are concerned with a chiral spinor field on a curved surface that gives rise to anyons in the nuclear case. Specifically we find a distorted torus on a Kähler manifold which is also a Calabi-Yau space.     

On the Dirac spectrum of Riemannian foliations admitting a basic parallel 1-form

In this paper, in the special setting of a Riemannian foliation with basic, non-necessarily harmonic mean curvature we introduce a Weitzenböck-Lichnerowicz type formula which allows us to apply the classical Bochner-Lichnerowicz technique. We show that the lower bound for the eigenvalues of the basic Dirac operator can be calculated using only classical techniques. As another application, for general Riemannian foliations we calculate the above eigenvalue bound in the presence of a basic parallel 1-form, as an extension of a known result on a closed Riemannian manifold. Some results concerning the limiting case are obtained in the final part of the paper.     

Scalar Curvature Deformation and a Gluing Construction for the Einstein Constraint Equations

On a compact manifold, the scalar curvature map at generic metrics is a local surjection [F-M]. We show that this result may be localized to compact subdomains in an arbitrary Riemannian manifold. The method is extended to establish the existence of asymptotically flat, scalar-flat metrics on ℝ ⁿ (n≥ 3) which are spherically symmetric, hence Schwarzschild, at infinity, i.e. outside a compact set. Such metrics provide Cauchy data for the Einstein vacuum equations which evolve into nontrivial vacuum spacetimes which are identically Schwarzschild near spatial infinity. Received: 8 November 1999 / Accepted: 27 March 2000     

The existence of homogeneous geodesics in homogeneous pseudo-Riemannian and affine manifolds

It is well known that, in any homogeneous Riemannian manifold, there is at least one homogeneous geodesic through each point. For the pseudo-Riemannian case, even if we assume reductivity, this existence problem is still open. The standard way to deal with homogeneous geodesics in the pseudo-Riemannian case is to use the so-called “Geodesic Lemma”, which is a formula involving the inner product. We shall use a different approach: namely, we imbed the class of all homogeneous pseudo-Riemannian manifolds into the broader class of all homogeneous affine manifolds (possibly with torsion) and we apply a new, purely affine method to the existence problem. In dimension 2, it was solved positively in a previous article by three authors. Our main result says that any homogeneous affine manifold admits at least one homogeneous geodesic through each point. As an immediate corollary, we prove the same result for the subclass of all homogeneous pseudo-Riemannian manifolds.     

Rigidity of pseudo-isotropic immersions

Several notions of isotropy of a (pseudo-)Riemannian manifold have been introduced in the literature, in particular, the concept of pseudo-isotropic immersion. The aim of this paper is to look more closely at this notion of pseudo-isotropy and then to study the rigidity of this class of immersion into the pseudo-Euclidean space. It is worth pointing out that we first obtain a characterization of the pseudo-isotropy condition by using tangent vectors of any causal character. Then, rigidity theorems for pseudo-isotropic immersions are proved, and in particular, some well known results for the Riemannian case arise. Later, we bring together the notions of pseudo-isotropy, intrinsically and extrinsically isotropic manifolds, and prove interesting relations among them. Finally, we pay special attention to the case of codimension two Lorentz surfaces.     

Dimensionality Reduction of SPD Data Based on Riemannian Manifold Tangent Spaces and Isometry

Symmetric positive definite (SPD) data have become a hot topic in machine learning. Instead of a linear Euclidean space, SPD data generally lie on a nonlinear Riemannian manifold. To get over the problems caused by the high data dimensionality, dimensionality reduction (DR) is a key subject for SPD data, where bilinear transformation plays a vital role. Because linear operations are not supported in nonlinear spaces such as Riemannian manifolds, directly performing Euclidean DR methods on SPD matrices is inadequate and difficult in complex models and optimization. An SPD data DR method based on Riemannian manifold tangent spaces and global isometry (RMTSISOM-SPDDR) is proposed in this research. The main contributions are listed: (1) Any Riemannian manifold tangent space is a Hilbert space isomorphic to a Euclidean space. Particularly for SPD manifolds, tangent spaces consist of symmetric matrices, which can greatly preserve the form and attributes of original SPD data. For this reason, RMTSISOM-SPDDR transfers the bilinear transformation from manifolds to tangent spaces. (2) By log transformation, original SPD data are mapped to the tangent space at the identity matrix under the affine invariant Riemannian metric (AIRM). In this way, the geodesic distance between original data and the identity matrix is equal to the Euclidean distance between corresponding tangent vector and the origin. (3) The bilinear transformation is further determined by the isometric criterion guaranteeing the geodesic distance on high-dimensional SPD manifold as close as possible to the Euclidean distance in the tangent space of low-dimensional SPD manifold. Then, we use it for the DR of original SPD data. Experiments on five commonly used datasets show that RMTSISOM-SPDDR is superior to five advanced SPD data DR algorithms.     

Self-Dual Conformal Gravity

We find necessary and sufficient conditions for a Riemannian four-dimensional manifold (M, g) with anti-self-dual Weyl tensor to be locally conformal to a Ricci-flat manifold. These conditions are expressed as the vanishing of scalar and tensor conformal invariants. The invariants obstruct the existence of parallel sections of a certain connection on a complex rank-four vector bundle over M. They provide a natural generalisation of the Bach tensor which vanishes identically for anti-self-dual conformal structures. We use the obstructions to demonstrate that LeBrun’s anti-self-dual metrics on connected sums of \({\mathbb{CP}^2}\) s are not conformally Ricci-flat on any open set. We analyze both Riemannian and neutral signature metrics. In the latter case we find all anti-self-dual metrics with a parallel real spinor which are locally conformal to Einstein metrics with non-zero cosmological constant. These metrics admit a hyper-surface orthogonal null Killing vector and thus give rise to projective structures on the space of β-surfaces.     

Eigenvalue Boundary Problems for the Dirac Operator

 On a compact Riemannian spin manifold with mean-convex boundary, we analyse the ellipticity and the symmetry of four boundary conditions for the fundamental Dirac operator including the (global) APS condition and a Riemannian version of the (local) MIT bag condition. We show that Friedrich's inequality for the eigenvalues of the Dirac operator on closed spin manifolds holds for the corresponding four eigenvalue boundary problems. More precisely, we prove that, for both the APS and the MIT conditions, the equality cannot be achieved, and for the other two conditions, the equality characterizes respectively half-spheres and domains bounded by minimal hypersurfaces in manifolds carrying non-trivial real Killing spinors. Received: 12 November 2001 / Accepted: 25 June 2002 Published online: 21 October 2002 RID="*" ID="*" Research of S. Montiel is partially supported by a Spanish MCyT grant No. BFM2001-2967 and by European Union FEDER funds     

The Effective Dynamics of the Volume Preserving Mean Curvature Flow

We consider the dynamics of small closed submanifolds (‘bubbles’) under the volume preserving mean curvature flow. We construct a map from (\(\text {n}+1\))-dimensional Euclidean space into a given (\(\text {n}+1\))-dimensional Riemannian manifold which characterizes the existence, stability and dynamics of constant mean curvature submanifolds. This is done in terms of a reduced area function on the Euclidean space, which is given constructively and can be computed perturbatively. This allows us to derive adiabatic and effective dynamics of the bubbles. The results can be mapped by rescaling to the dynamics of fixed size bubbles in almost Euclidean Riemannian manifolds.     

Scalar Curvature and Asymptotic Symmetric Spaces

A well known fact is that a complete Riemannian (spin) manifold which is strongly asymptotically flat and has nonnegative scalar curvature must be isometric to the Euclidean space. Using Witten’s positive mass argument this paper proves the analogous rigidity result for certain asymptotic symmetric spaces. Supported by the German Science Foundation.     

On nodal solutions of the Yamabe equation on products

Let (M,g), (N,h) be closed Riemannian manifolds of constant scalar curvature. We prove the existence of nodal solutions of the Yamabe equation on the Riemannian product which depend on only one of the factors. We do this by studying the second Yamabe invariant introduced by Ammann and Humbert. We work out the case when M=S¹ explicitly showing the existence of an infinite number of solutions.     

A note on the p-elastica in a constant sectional curvature manifold

In this paper, we study the p-elastica, the critical point of the total polynomial curvature functional on those immersed curves satisfying suitable boundary conditions in a Riemannian manifold with constant sectional curvature. We express the torsion of the p-elastica in terms of its curvature in a closed form and completely solve the Euler–Lagrange equation by quadratures. We study the Frenet equation of the p-elastica by using the Killing field.