Weingarten Map Research Articles

Estimates for the curvature of a submanifold which lies inside a tube

We give some estimates for the supremun of the sectional curvature of a submanifold N of a Riemannian or Kaehlerian manifoldM such that N is contained in a tube about a submanifoldP ofM. These estimates depend on the sectional curvature ofM, the Weingarten map ofP and the radius of the tube. Then we apply them to get theorems of non immersibility.

Linearization of response mappings in constrained elasticity

The linearization of response mappings for elastic materials with internal constraints is discussed with a geometric perspective. It is shown that, when the active residual stress is not zero, the Weingarten map of the constraint manifold plays an important role in the deduction of linearized constitutive equations. As an example, the Weingarten map is computed for a material which is incompressible and inextensible in a given direction.

GEODESIC TUBES ON LOCALLY SYMMETRIC SPACES


 
 
 In this paper we state and prove a characteristic relation which exists, between the eigenspaces of the Ricci transformation $R(N, - )N$ acting on the orthocomplement space of $N$ in $T_mM$ where $m \in M$, $M$ being a locally symmetric space, and the Weingarten map $S_N$ of small enough geodesic tubes of $M$. 
 
 

Path Integral Formulation of Curved Low Dimensional Space

When a low dimensional space has a curvature, there is an effective potential in the Schrodinger equation as a geometrical correction. In this paper, we have shown that a confined space in three dimensional space can be regarded as a curved low (one or two) dimensional space when the thickness of the space multiplied by the Weingarten map of each space is sufficiently smaller than unity. Under the condition, we have also evaluated the effective potential using the path integral method.

Representation and recognition of surface shapes in range images: A differential geometry approach

Theory and matching algorithms are developed for accurate orientation determination and recognition of 3D surface shapes in range images. Two corollaries to the fundamental theory of surface theory are proved. The first corollary proves the invariance of the fundamental coefficients when lines of curvature are used as the intrinsic parameter curves. The second corollary proves that a diffeomorphism which preserves the intrinsic distance along the principal directions, in addition to preserving the eigenvectors and eigenvalues of the shape operator (Weingarten map), is necessarily an isometry. Based on these two corollaries, a set of geometric descriptors which satisfy the uniqueness and invariance requirements are theoretically identified for all classes of surfaces, namely, hyperbolic, elliptic, and developable surfaces. The unit normal and shape descriptors list array (UNSDLA) representation and the corresponding matching algorithm are developed. The UNSDLA is a generalization of the extended Gaussian image (EGI). The EGI has a fundamental limitation; that is, it can only uniquely represent convex shapes. The new representation overcomes this limitation of the EGI and extends the scope of unique representation to all classes of surfaces. Moreover, it still has all the advantages of the EGI. This is achieved by preserving the connectivity of the original data. Connectivity here should include not only the adjacency relation of points or patches on a surface, but also the direction and order in which the points or patches are traversed in a connected path. The importance of the direction and order of connectivity is emphasized. Surface matching can be performed more accurately using the UNSDLA than the EGI. Based on the UNSDLA representations, surfaces can be matched via the Gaussian map by optimization over all possible rotations of a surface shape. The representation and matching algorithm can deal with hyperbolic and elliptic surfaces whose Gaussian maps are not one-to-one. Developable surfaces whose Gaussian maps of lines of curvature with nonzero principal curvature are not one-to-one can also be accommodated. Two theorems on developable surfaces are proved.

A characterization of the symmetric space SU( n)⧸SO( n) by geodesic spheres

We characterize the symmetric space AI, i.e. SU(n) SO(n) , and its noncompact dual by means of the Weingarten map on geodesic spheres and two particular tensor fields T and S of type (1,3) and (1,2) respectively, satisfying certain algebraic conditions.

The motion of membranes in space-time

The equation of a hypersurface interacting with external fields in space-time is established. A class of models is based on the properties of a symmetric divergence-free distribution-valued stress tensor for the interacting system. The elastodynamic properties are encoded into the induced metric and shape tensor of the immersion together with invariants constructed from the Weingarten map. Two particular examples are discussed that are relevant to recent developments in the theories of relativistic extended particles.

A characterization of the Grassmann manifold (Gp,2(IR)). another review

The purpose of the present study is to characterise the Grassmann manifold G (R) and its non compact dual G* (R) by means of a particular parallel tensor p,2 p,2 field T of type (1,3) and the Weingarten map on geodesic spheres.

Dupin's cyclide and the cyclide patch

To retrieve models from a database for recognizing objects in stereo, a new formulation of Dupin's cyclide patches provides a succinct representation of surface shape. The parameters can be extracted from the Weingarten map and its derivatives at a point where a contour meets an extremal boundary.