Gary R. Jensen

Higher Order Contact of Submanifolds of Homogeneous Spaces

The general theory.- Surfaces in R 3 under the Euclidean group of proper motions.- Curves in real Grassmannians.- Holomorphic curves in complex projective space.- Holomorphic curves in complex Grassmannians.- Special affine surface theory.

Jump–diffusion Markov processes on orthogonal groups for object pose estimation

In the problem of recognizing targets from their observed images, the estimation of target orientations, as elements of the rotation group SO(3), plays an important role. For k-objects the unknown parameter is an element of SO(3) k . Since k may be unknown a priori, the parameter space is extended to X = � ∞=0 SO(3) k . In this representation, both the target orientations and their numbers have to be estimated simultaneously. We present a Bayesian approach that builds

Projective deformation and biholomorphic equivalence of real hypersurfaces

The concepts of first order projective deformation, biholomorphic equivalence, and equivalence of induced Cauchy-Riemann structure are all equivalent for real analytic hypersurfaces in complex projective space. Studying the first concept leads to a realization of the Cauchy-Riemann structure bundle as a submanifold of the projective group. The Chern-Moser connection on this bundle can then be given in terms of the Maurer-Cartan form of the projective group, and equations analogous to the Gauss equations of Euclidean geometry give the Chern-Moser invariants.

The geometric Cauchy problem for the membrane shape equation

We address the geometric Cauchy problem for surfaces associated to the membrane shape equation describing equilibrium configurations of vesicles formed by lipid bilayers. This is the Euler-Lagrange equation of the Canham-Helfrich-Evans elastic curvature energy subject to constraints on the enclosed volume and the surface area. Our approach uses the method of moving frames and techniques from the theory of exterior differential systems.

Isoparametric functions and flat minimal tori in

It is proved that all flat minimal tori in CP2 are unitarily congruent to the Clifford torus by studying a certain associated isoparametric function.

HOLOMORPHIC CURVES IN THE COMPLEX QUADRIC

Holomorphic curves in a complex quadric arise naturally as thecomplex conjugate of the Gauss map of a minimal surface in Euclideanspace. In addition, such holomorphic curves play a central role ingenerating a special class of harmonic maps of surfaces into spheres,complex projective space, and the complex Grassmannians. (See Eells-Wood[6], Bryan [2]t , Chern-Wolfson [51, Ramanathan L141, and many referencescited in these papers.)Received 11 March 1986.Copyright Clearance Centre, Inc. Serial-fee code: 0004-9729/87

Harmonically Immersed Surfaces of R n

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Compact Surfaces with No Bonnet Mate

Some generalizations of classical results in the theory of minimal surfaces f: M Rn are shown to hold in the more general case of harmonically immersed surfaces. Introduction. Let (M, g) be a connected Riemann surface with a prescribed metric g in its conformal class and let f: M > Rn be an immersion. It is well known that f realizes M, with the induced metric from Rn, as a minimal surface if and only if f is a conformal (with respect to g) harmonic map (cf., for example, [3 or 8]). That is, the theory of minimal surfaces is substantially the theory of conformally immersed harmonic surfaces. Our purpose is to analyze the case when f is simply a harmonic immersion, to introduce an appropriate Gauss map and, as the main achievement, to establish in this new setting the analogue of three fundamental results (cf. Theorems 1.1 and 2.1, and §3) in the theory of minimal surfaces: the harmonicity of the Gauss map (Ruh and Vilms [13]), equidistribution properties of the Gauss map in cPn (Chern and Osserman [1 and 11]), and the Enneper-Weierstrass representation formulas recently due, for arbitrary codimension, to Hoffman and Osserman [7]. The last two topics (the third for n = 3) have alrealy been treated by T. K. Milnor (see for instance her survey article [9]), but as we remark in §2, our results complement hers in an interesting way. The method of the moving frame as well as the Einstein summation convention are used throughout this paper. 1. The Gauss map and first properties. Let (M, g) be as in the Introduction. We fix the index ranges 1 Rn be an immersion. A Darboux frame al g f is a map E: U c M E(n), U ope in M, such that Received by the editors April 6, 1987. Selected results of this paper were presented by the first named author November 1, 1987 at the 837th meeting of the American Mathematical Society held at Lincoln, Nebraska. 1980 Mathematics Subject Classification (1985 Revision). Primary 53A07, 53A10.

Lie Sphere Geometry

This note gives sufficient conditions (isothermic or totally nonisothermic) for an immersion of a compact surface to have no Bonnet mate.

Complex Structure and Möbius Geometry

This chapter presents the method of moving frames in Lie sphere geometry. This involves a number of new ideas, beginning with the fact that some Lie sphere transformations are not diffeomorphisms of space S3, but rather of the unit tangent bundle of S3. This we identify with the set of pencils of oriented spheres in S3, which is identified with the set \( \varLambda \) of all lines in the quadric hypersurface \( Q \subset \mathbf{P}(\mathbf{R}^{4,2}) \). The set \( \varLambda \) is a five-dimensional subspace of the Grassmannian G(2, 6). The Lie sphere transformations are the projective transformations of P(R4, 2) that send Q to Q. This is a Lie group acting transitively on \( \varLambda \). The Lie sphere transformations taking points of S3 to points of S3 are exactly the Mobius transformations, which form a proper subgroup of the Lie sphere group. In particular, the isometry groups of the space forms are natural subgroups of the Lie sphere group. There is a contact structure on \( \varLambda \) invariant under the Lie sphere group. A surface immersed in a space form with a unit normal vector field has an equivariant Legendre lift into \( \varLambda \). A surface conformally immersed into Mobius space with an oriented tangent sphere map has an equivariant Legendre lift into \( \varLambda \). This chapter studies Legendre immersions of surfaces into this homogeneous space \( \varLambda \) under the action of the Lie sphere group. A major application is a proof that all Dupin immersions of surfaces in a space form are Lie sphere congruent to each other.