In this paper, we investigate the geometry of asymptotically flat manifolds with controlled holonomy. We show that any end such manifold admits a torus fibration over an ALE end. addition, prove Hitchin-Thorpe inequality for oriented Ricci-flat $4$-manifolds curvature decay and As application, complete metric on $4$-manifold which is homeomorphic to $\mathbb R^4$ must be isometric Euclidean or ...

1. Write the formula for the Liouville measure Λ on the unit tangent bundle T 1 H n. Prove that Λ is invariant under the actions by Isom(H n) and the geodesic flow g t on T 1 H n. 2. Let δ be a geodesic in the hyperbolic space H n and let π : H n → δ denote the nearest point projection map. Suppose that for some x, y ∈ H n and some r > 0 we have d(x, δ), d(y, δ) ≥ r. Prove that d(π(x), π(y)) ≤ ...

We introduce a notion of rectifiability modeled on Carnot groups. Precisely, we say that a subset E of a Carnot group M and N is a subgroup of M , we say E is Nrectifiable if it is the Lipschitz image of a positive measure subset of N . First, we discuss the implications of N-rectifiability, where N is a Carnot group (not merely a subgroup of a Carnot group), which include N-approximability and...

The goal is to perform geometric studies on cone-beam CT scanning along a three-dimensional (3D) spiral of variable radius. First, the background for variable radius spiral cone-beam scanning is given in the context of electron-beam CT/micro-CT. Then, necessary and sufficient conditions are proved for existence and uniqueness of PI lines inside the variable radius 3D spiral. These results are n...

In Reference [1] the authors have shown that numerical differentiation is a competitive alternative to analytical derivatives for the computation of consistent tangent matrices. Relatively simple models were treated in that reference. The approach is extended here to a complex model: the MRS-Lade model [2,3]. This plastic model has a cone-cap yield surface and exhibits strong coupling between t...

Totally geodesic property of the Hopf vector field. Abstract We prove that the Hopf vector field is a unique one among geodesic unit vector fields on spheres such that the submanifold generated by the field is totally geodesic in the unit tangent bundle with Sasaki metric. As application, we give a new proof of stability (instability) of the Hopf vector field with respect to volume variation us...

Generating-line method, which is based on the generating process of spherical involute curve, is a new processing theory of cutting ideal spherical involute gears. Based on the principle of the new method, this paper expands the traditional geometrical relationship into the cutting geometry of generating-line method, and proposes the base-cone parameters with their formulas to provide necessary...

We study the formation of singularities for the mean curvature flow of monotone Lagrangians in C. More precisely, we show that if singularities happen before a critical time then the tangent flow can be decomposed into a finite union of area-minimizing Lagrangian cones (Slag cones). When n = 2, we can improve this result by showing that connected components of the rescaled flow converge to an a...