Search

A procedure based on the envelope concept of differential geometry is described that permits the reconstruction of the contour of a smooth, moving, conducting target, satisfying the geometrical optics approximation. The target reflections are assumed to be specular in nature with either one reflection point or multiple resolvable reflection points. The time variation of the range to the reflection point of the target (assumed derivable from a high-resolution radar) and the general motion of the target (assumed derivable from tracking or trajectory information) are employed to reconstruct the contour of that portion of the assumed target surface that is illuminated by the radar. The reconstruction is accomplished by the simultaneous solution of two nonlinear differential equations which are derived using the envelope concept of differential geometry. Several reconstruction examples based on computer analysis are presented which indicate the results obtainable using this method.

Carathéodory’s formulation of classical thermodynamics is cast into a mathematically rigorous form, in which such well-known results as the existence of entropy and of (positive) absolute temperature, the principle of increase of entropy and the positivity of isometric specific heats are shown to follow logically from a set of postulates closely related to the traditional laws of thermodynamics and more economical than those of the original formulation. The basic tools are differential geometry and topology, the latter being needed in order to eliminate those flaws in the old arguments that arise from the erroneous application of local results in a global context.

The present part of the 'Theory of Bessel Potentials' contains Chapter 4 dealing with potentials on regular Riemannian manifolds. (Author)

The primary purpose of this paper is to expose, in as simple and clear a form as is possible, the fundamentals of the geometric structure of a Riemannian space. It is a general truth that the methods which pierce most deeply into the heart of a geometric theory are invariant methods, that is, methods which are independent of the choice of the coordinates in terms of which the theory is expressed analytically. In the case of Riemannian geometry, these are the methods of tensor analysis. As important, perhaps, as the use of invariant methods is the expression of the analytic theory, so far as possible, in terms of invariant quantities alone. For it is in this form that the theory becomes most illuminating and suggestive. But, in ordinary tensor analysis, the components of a tensor are not invariants. A first step toward our goal will be, then, to introduce for Riemannian geometry an intrinsic tensor analysis, that is, a form of tensor analysis in which the components of all tensors are invariants. Any theory of the geometry of a Riemannian space presupposes that the space is referred to a certain ennuple of congruences of curves. In the ordinary theory, this ennuple consists of the parametric curves. In the intrinsic theory, it is an ennuple E, whose choice, as will presently be evident, is entirely arbitrary. The ordinary components of a tensor, that is, the components in the ordinary theory, are referred to the differentials of the coordinates xi pertaining to the ennuple of the parametric curves. The intrinsic components are referred to the differentials of arc, dsi, of the curves of the ennuple E. There is no need in the intrinsic theory of actual coordinates pertaining to the ennuple E; the differentials of arc dsi suffice. Accordingly, E can be chosen arbitrarily; it does not have to be a parametric ennuple, that is, an ennuple with which coordinates can be associated. It may be, and we shall ordinarily take it to be, an ennuple of general type, and hence, of course, not necessarily orthogonal. Intrinsic components, referred to E, of the covariant derivative of a tensor are made possible by the introduction of invariant Christoffel symbols in

n/a

In the theory of curves, one often makes differentiability assumptions in order that analytic methods can be used. Then one tries to weaken these assumptions as much as possible. The theory of curves which is presented here uses geometric methods, such as central projection, rather than analysis. In this way, no analytic assumptions are needed and a purely geometric theory results. Since this theory is not so well known as the analytic one, I have tried to make the treatment as self-contained as possible. It is hoped that this paper will form a quick introduction for a reader who has had no previous acquaintance with the subject.We assume that our curves satisfy a condition, which we call direct differentiability. Roughly this condition is that, at each point of the curve, all the osculating spaces exist.

Various questions of classical differential geometry lead to the problem of determining all those hypersurfaces of euclidean space for which the covariant derivative of the second fundamental form vanishes identically. In the first part of this note, we investigate these hypersurfaces; the result is summarized in the theorem of the opening paragraph, whose proof is based in the decomposition theorem of De Rham (see e.g. [6], pp. 180–186). Applying this result in the second part of the note, we give local characterizations of the Euclidean sphere.

Publisher Summary This chapter discusses a set of axioms for differential geometry. The chapter presents the axioms that are intended to describe the class of manifolds of n dimensions to which the theories, grouped together under the heading of differential geometry, are applicable. The manifolds are classes of elements called points, having a structure, which is characterized by means of coordinate systems. A coordinate system is a (1-1) correspondence, P → x , between a set of points, [ P ], of the manifold, and a set, [ x ], of ordered sets of n real numbers, x= (x 1 ,…, x n ). Each point P, which corresponds in a coordinate system, P → x , to an arithmetic point x, is said to be represented by x. The set, [ P ], of all points represented in a given coordinate system is called the domain of the coordinate system, and the set, [ x ], of the arithmetic points that represents them is called its arithmetic domain. If P → x and P → y are two coordinate systems having the same domain, the transformation x y of the arithmetic space, which is the resultant of x → P followed by P → y, is called the transformation of coordinates from P → x to P → y, or the transformation between P → x and P → y . The general scheme of the axioms is to characterize a class of allowable coordinate systems by means of the analytic properties of the transformations between them. The chapter explains a few arithmetic theorems and definitions to state these properties clearly.

The main purpose of this paper is the representation by algebraic varieties in projective space, of certain aggregates, in which each element is a sequence P 0 ... P n of n +1 points consecutive on an algebraic (or algebroid) branch. Two types of aggregate are to be considered: W r , n , of all sequences of n + 1 points with a given origin P 0 in S r ; and W r ,* n , of all such sequences with origin anywhere in S r . The problem has been studied by various writers from 1901 onwards, and a complete solution for n = 1,2 only was given in 1955. In this period no progress at all had been made for higher values of n , and its intractability led to conjectures that the aggregates in question were not irreducible, or had some other defects inhibiting the type of representation sought. In the present paper a complete method is given of parametrizating both W r , n and W r * n for all values of r , n , the co-ordinates of a point of the model being expressed systematically in terms of certain invariants of an arbitrary branch through the sequence represented. These models are irreducible and non-singular, and their points are in one-one correspondence without exception with the sequences in the appropriate aggregates. In addition, the geometrical properties of the models are studied in some detail for r = 2 (the clarity of the picture obtained naturally fading off as n increases) and (JV2,3 having been dealt with in the author’s short contribution to a symposium in honour of Bompiani) a complete geometrical description is given of W 2 , 4 , W 3 , 3 , and W 2 * 3 , including the base and intersection theory on these. The main importance claimed for the work is twofold: (i) it offers in theory (i.e. by a well defined method, which however involves sharply increasing labour with increase of n and of r ) a complete solution of a problem which has proved intractable for 60 years; (ii) it gives detailed descriptions of some, and less full description of some others, of a family of varieties which have the same type of importance for local algebraic geometry that for instance Grassmannians have for line geometry, etc. It is possible also that the very ample system of invariants defined in the course of the parametrization of the models may prove to be of use in differential geometry (the formal power series from which they are obtained being interpreted as Taylor series) but this is a pure surmise, being rather outside the author’s sphere of interest.

n/a

n/a

The differential geometry of circle systems has received a large amount of attention of late. The subject has been approached from two quite distinct sides. On the one hand we have Koenigs, Cosserat, Moore, Bompiani and others who fix their attention on what we may call the descriptive differential properties of circles. The totality of circles in three-dimensional space may be represented by an S6, Iying in an Ss, and this variety may be studied by the now familiar methods of projective diBerential geometry. The theorems so reached are invariant under the twenty-four parameter group of sphere transformations. The other class of writers, wherein we may include Bianchi, Tzitzeica, Guichard and Eisenhart, have confined themselves largely to congruences (two-parameter systems) of circles, frequently to normal congruences. The methods employed have been the general ones of differential geometry and the center of interest has been rather more in the axes of the circles than the circles themselves. It has seemed for some time to the present writer that the last word on these subjects had not by any means been written, and that by a different method of approach not only might we obtain simpler proofs of known theorems but discover a number of new theorems as well. The most interesting properties of circles are those which are invariant under inversion, or under the tenparameter group of conformal transformations of space; the best approach to this group is through the use of pentaspherical coordinates. It is true that some of the writers mentioned above have made use of these coordinates, and still more has been done by Darboux in his Theorie des Surfaces, yet the possibilities of these coordinates have been by no means exhausted, and in the present paper they are more systematically applied to problems in differential circle geometry than has been the case in the past. A circle may be regarded in two different aspects, either as a locus of points, or an envelop of spheres. The two points of view are, of course, closely related, but the change of emphasis leads naturally to rather different sets of theorems. The first section of the present article is devoted to preliminary formulse for points and spheres in pentaspherical coordinates, and certain

Introduction. One of the main problems of local differential geometry is to determine when a given differentiable structure (for example, a Riemannian structure, a projective structure or an almost complex structure) is integrable. There are really two problems involved in this: (1) the problem of finding a consistent system of differential equations whose unknowns yield a solution to the problem; and (2) the problem of solving these equations. It goes without saying that the second problem is more or less trivial if we are considering analytic structures. On the other hand for C*-structures it is extremely difficult, since the equations that occur in (1) are generally not elliptic. In the paper we will only deal with the first problem which we will refer to as the problem of formal flatness or formal integrability. Also, for simplicity, we will restrict ourselves to studying G-structures, though most of the results of this paper can be extended to more general kinds of pseudogroup structures (such as projective structures). We will also formulate the problem in slightly more intuitive terms: In ?2 we will define what we mean by a mapping of one G-structure into another which is structure preserving to kth order at some point, a notion which is almost self-evident. We will try to find conditions on a given G-structure, E -4 M, such that every point of M admits a mapping into a neighborhood of an integrable G-structure which is structure preserving to order k at that point, for arbitrarily large k. The obstructions to constructing such mappings turn out to be tensors of type Hk2(g), defined on E, where the Hki(g) are the bigraded homology groups of a certain chain complex due to Spencer (cf. ?1). Therefore, the character of the integrability problem will be determined by the cohomology sequence Hk( g). We will also show in this paper that for G-structures of finite type formal integrability in the sense above implies integrability. For analytic Gstructures we will give a direct proof of the equivalence of these two notions (not using the Cartan-Kaehler theorem) in a paper to follow this one. Finally we mention that the following are known for CG-structures

Immersions or maps of closed manifolds in Euclidean space, of minimal absolute total curvature are called tight in this paper. (They were called convex in [25].) After the definition in Chapter 1, many examples in Chapter 2, and some special topics in Chapter 3, we prove in Chapter 4 that topological tight immersions ofn-spheres are only of the expected type, namely embeddings onto the boundary of a convexn+1-dimensional body. This generalises a theorem of Chern and Lashof in the smooth case. In Chapter 5 we show that many manifolds exist that have no tight smooth immersion in any Euclidean space.

Let X1 X2, ... be independent and identically distributed observations with probability densities p(x; θ) with respect to some measure µ0. Extending some work of Barankin and Gurland [2] we shall in this paper first give a proof of the existence of consistent, asymptotically normal distributed (c.a.n.) estimators under relative mild restrictions on the family P = p(x; θ) of probability densities. Instead of assuming, as did Barankin and Gurland, for the existence proof that socalled separators exist, we shall be able to rely on some results of differential geometry, viz. the existence of tubular neighborhoods, in carrying out the proof. In actually constructing c.a.n. estimators separators are of great use, but they are irrelevant for the general existence theorem.

Periodic solutions of the generalized Lienard equation are investigated in the phase and Lienard planes. Certain comparison equations are obtained by modifying the functions f(x), g(x), F(x) = integral evaluated from 0 to x f(u)du, so that the resulting equations may be integrated to within an error of order mu to the - 2.

The purpose of this paper is to classify holomorphic principal fibre bundles which admit a smooth section (i.e. are real product bundles). This is accomplished if the structure group is solvable of type (E). In the general case, a sufficient condition is obtained for a real product bundle to be equivalent to the complex product bundle. A necessary and sufficient condition for the existence of a holomorphic connection on a real product bundle is also obtained. Using this criterion in the case where the structure group is abelian, a generalization of a theorem due to Atiyah (in the case the structure group is C ∗ {C^ \ast } ) is obtained.

To the pure mathematician of the present day the tensor calculus is a notation of differential geometry, of special utility in connection with multi-dimensional spaces; to the applied mathematician it is the backbone of the general theory of relativity. But when it is recognised that every problem in applied mathematics may be regarded as a geometrical problem (in the widest sense) and that the geometrical forms which many of these problems take are such that the tensor calculus can be directly applied, it is realised that the possibilities of this calculus in the field of applied mathematics can hardly be overestimated. It has a dual importance: first, by its help, known results may be exhibited in the most compact form; secondly, it enables the mathematician to exercise his most potent instrument of discovery, geometrical intuition. In the present paper we are concerned with the development of general dynamical theory with the aid of the tensor calculus. In view of the present close association of the tensor theory with the theory of relativity, it should be clearly understood that this paper only attempts to deal with the classical or Newtonian dynamics of a system of particles or of rigid bodies. The subject is presented in a semi-geometrical aspect, and the reader should visualise the results in order to realise the close analogy between general dynamical theory and the dynamics of a particle. Mathematicians display a strange reluctance in summoning to their assistance the power of visualisation in multidimensional space. They forget that they have studied the geometry of three dimensions largely through the medium of a schematic representation on a two dimensional sheet of paper. The same method is available in the case of any number of dimensions.