Differential Geometric Viewpoint Research Articles

PL DENSITY INVARIANT FOR TYPE II DEGENERATING K3 SURFACES, MODULI COMPACTIFICATION AND HYPER-KÄHLER METRIC

AbstractA protagonist here is a new-type invariant for type II degenerations of K3 surfaces, which is explicit piecewise linear convex function from the interval with at most $18$ nonlinear points. Forgetting its actual function behavior, it also classifies the type II degenerations into several combinatorial types, depending on the type of root lattices as appeared in classical examples.From differential geometric viewpoint, the function is obtained as the density function of the limit measure on the collapsing hyper-Kähler metrics to conjectural segments, as in [HSZ19]. On the way, we also reconstruct a moduli compactification of elliptic K3 surfaces by [AB19], [ABE20], [Brun15] in a more elementary manner, and analyze the cusps more explicitly.We also interpret the glued hyper-Kähler fibration of [HSVZ18] as a special case from our viewpoint, and discuss other cases, and possible relations with Landau–Ginzburg models in the mirror symmetry context.

Black holes, marginally trapped surfaces and quasi-minimal surfaces

The concept of trapped surfaces introduced by Sir Roger Penrose in [Phys. Rev. Lett. 14 (1965), 57-59] plays an extremely important role in cosmology and general relativity. A black hole is a trapped region in a space-time enclosed by a marginally trapped surface. In term of mean curvature vector, a space-like surface in a space-time is marginally trapped if its mean curvature vector field is light-like at each point. In this article, we survey recent classification results on marginally trapped surfaces from differential geometric viewpoint. Also, we survey recent results on a closely related subject; namely, quasi-minimal surfaces in pseudo-Riemannian manifolds.

A differential geometric viewpoint on local identifiability and identification part II: application

The robustness of parameter identification for nonlinear dynamical systems is investigated by interpreting the parameter estimation problem from a differential geometric point of view. Results on robust identification are proved in a deterministic setting and an algorithm is developed to derive a robust estimator for the parameters of nonlinear dynamical systems. The proposed algorithm is tested on the offline identification of the rotor resistance of an electric machine for stochastic and deterministic disturbances. It could be shown that the robustness is significantly improved compared to conventional algorithms.

A differential geometric viewpoint on local identifiability and identification part I: theory

The questions of local identifiability and identification of nonlinear systems are treated from a differential geometric point of view. It is shown how identifiability can be interpreted in the framework of Lie groups, which provides convenient tools for a unifying concept of identifiability. Three different notions of identifiability are interpreted in this framework. Furthermore parameter identification itself is also treated. It is shown how a parameter estimator can be formulated in a coordinate-free setting. In this setting it can be clearly determined to what extent the underlying geometric structure and the coordinate-specific parameterization of the system influence the properties of the estimator.

IGB-offset for plane curves—loop removal by scanning of interval sequences

The generation of NC-tool paths for 2 1 2 - axes machining is based on offset curves to arbitrary simple piecewise smooth G 0 generator curves. Global and local properties that are of interest for the offset generation from the NC and the differential geometric viewpoint are discussed and described in this paper. These properties lead to an approach for generating global offset curves in a more direct and technological way. The key point of this approach is the restriction of the offset image to nonsingular parts of the generator curve with respect to the given offset distance. In general, this restricted offset image is discontinuous, but the restriction guarantees a minimal number of self-intersection points. The correspondence of self-intersections to parameter intervals is used to describe uniquely a set of relevant trim intervals. The result after the trimming of these intervals is a continuous offset curve that is, up to some technological constraints, equal to the offset curve of the right- or left-sided rolling ball blend of the generator curve. This offset carries the Gauss-Bonnet values of the generator as invariants and is, therefore, called IGB-offset. Global loops of the IGB-offset are classified by the superposition of remaining self-intersection intervals and are described by the union of elements with even index of finite interval sequences in the parameter space of the IGB-offset.

Nonparametric Estimation of a Probability Density on a Riemannian Manifold Using Fourier Expansions

Supposing a given collection $y_1, \cdots, y_N$ of i.i.d. random points on a Riemannian manifold, we discuss how to estimate the underlying distribution from a differential geometric viewpoint. The main hypothesis is that the manifold is closed and that the distribution is (sufficiently) smooth. Under such a hypothesis a convergence arbitrarily close to the $N^{-1/2}$ rate is possible, both in the $L_2$ and the $L_\infty$ senses.

Representations of loop groups, Dirac operators on loop space, and modular forms

FOR M a compact Spin manifold of dimension d= 2k, the unicersal elliptic genus p(M), in the sense of D. V. Chudnovsky, G. V. Chudnovsky, P. Landweber, S. Ochanine, and R. E. Stong, is a modular form of weight k for a level 2 congruence subgroup of SL(2, Z) [SJ, [17], [19], [26], and [28]. E. Witten showed that p(M) could be interpreted as the index of a sort of twisted Dirac operator on the loop manifold LM [26] and [27]. He actually considered several sorts of twistings of this Dirac operator; all of them give rise to modular forms for level 2 congruence subgroups. Similar results were obtained by Alvarez, Killingback, Mangano, and Windey [l] and by Schellekens and Warner [23]. In this article we consider, in a purely formal way, the index of the Dirac operator on LM, with coefficients in the vector bundle Reassociated to a “positive energy” representation of the universal central extension e Spin(d) of the loop group LSpin(d). We do not claim to understand anything about this operator, except that its index may be formally written down. The main result (Theorem 2.2) is that this index is a modular form of weight k for a congruence subgroup of SL(2, Z), which depends on the level of the representation of zSpin(d). The proof uses the results of V. KaE and D. Peterson [ll] and [12] on the characters of such representations, in particular the fact they are Jacobi modular forms in several variables. We present a conjecture relating the representations of tSpin(d) with elliptic cohomology, the theory of Landweber, Ravenel, and Stong [17], [19]. More precisely, we extend elliptic cohomology to higher levels (i.e. for each integer N, we construct a homology theory Ellz such that Ellt(pt) is the ring of modular forms for the principal congruence subgroup F(N), whose Fourier expansions at every cusp have coefficients in Z [ e2ni’N]). The point of this construction is that Ellz(pt) is a faithfully flat module over Ell,(pt); so we do not claim that Ellt(pt) is a really new homology theory, from the topological veiwpoint. From the differential-geometric viewpoint, we show that the vector bundle E, mentioned above, is equivariant under the Virasoro algebra, which acts on LM through its quotient vecr(S’). We think it might be useful to view the index of the Dirac operator on LM as a virtual representation of the Virasoro algebra, rather than merely the group of rotations of S’.

On kinematical invariances of the equations of motion

Invariance properties of the equations of motion are considered from the differential geometric viewpoint, on making use of vector fields and differential forms. It will be shown that the kinematical symmetries generated from first integrals and the invariance such as the dilation can be treated on an equal footing.