Harmonic sections of vector bundles with spherically symmetric metrics

We equip an arbitrary vector bundle over a Riemannian manifold, endowed with a fiber metric and a compatible connection, with a spherically symmetric metric (cf. [4]), and westudy harmonicity of its sections firstly as smooth maps and then as critical points of the energy functional with variations through smooth sections.We also characterize vertically harmonic sections. Finally, we give some examples of special vector bundles, recovering in some situations some classical harmonicity results.

Local Well-Posedness of Skew Mean Curvature Flow for Small Data in $$d\ge 4$$ Dimensions

The skew mean curvature flow is an evolution equation for d dimensional manifolds embedded in $${{\mathbb {R}}}^{d+2}$$ R d + 2 (or more generally, in a Riemannian manifold). It can be viewed as a Schrödinger analogue of the mean curvature flow, or alternatively as a quasilinear version of the Schrödinger Map equation. In this article, we prove small data local well-posedness in low-regularity Sobolev spaces for the skew mean curvature flow in dimension $$d\ge 4$$ d ≥ 4 .

Stitching Data: Recovering a Manifold’s Geometry from Geodesic Intersections

Let (M, g) be a Riemannian manifold with boundary. We show that knowledge of the length of each geodesic, and where pairwise intersections occur along the corresponding geodesics allows for recovery of the geometry of (M, g) (assuming (M, g) admits a Riemannian collar of a uniform radius). We call this knowledge the ‘stitching data’. We then pose a boundary measurement problem called the ‘delayed collision data problem’ and apply our result about the stitching data to recover the geometry from the collision data (with some reasonable geometric restrictions on the manifold).

Foliation of an Asymptotically Flat End by Critical Capacitors

We construct a foliation of an asymptotically flat end of a Riemannian manifold by hypersurfaces which are critical points of a natural functional arising in potential theory. These hypersurfaces are perturbations of large coordinate spheres, and they admit solutions of a certain over-determined boundary value problem involving the Laplace–Beltrami operator. In a key step we must invert the Dirichlet-to-Neumann operator, highlighting the nonlocal nature of our problem.

Multiple graph fusion based on Riemannian geometry for motor imagery classification

In motor imagery-based brain-computer interfaces (BCIs), the spatial covariance features of electroencephalography (EEG) signals that lie on Riemannian manifolds are used to enhance the classification performance of motor imagery BCIs. However, the problem of subject-specific bandpass frequency selection frequently arises in Riemannian manifold-based methods. In this study, we propose a multiple Riemannian graph fusion (MRGF) model to optimize the subject-specific frequency band for a Riemannian manifold. After constructing multiple Riemannian graphs corresponding to multiple bandpass frequency bands, graph embedding based on bilinear mapping and graph fusion based on mutual information were applied to simultaneously extract the spatial and spectral features of the EEG signals from Riemannian graphs. Furthermore, with a support vector machine (SVM) classifier performed on learned features, we obtained an efficient algorithm, which achieves higher classification performance on various datasets, such as BCI competition IIa and in-house BCI datasets. The proposed methods can also be used in other classification problems with sample data in the form of covariance matrices.

Twistor Bundle of a Neutral Kähler Surface

In this paper, we characterize neutral Kähler surfaces in terms of their positive twistor bundle. We prove that an O+,+(2,2)-oriented four-dimensional neutral semi-Riemannian manifold (M,g) admits a complex structure J with ΩJ∈⋀−M, such that (M,g,J) is a neutral-Kähler manifold if and only if the twistor bundle (Z1(M),gc) admits a vertical Killing vector field.