Search

(the tangent space at p) such that expU has no points of M onthe \inside" of L. By the inside of Lwe mean the component of a tubularneighbor hood with Ldeleted which corresponds to the negative multiplesof the orienting normal vector eld under exp. The exponential maps arethose of M.It is an immediate consequence of one of the standard interpretations ofthe second fundamental form, as the Hessian of a normal coordinate func-tion restricted to L[BC, p. 198], that local convexity implies in nitesimalconvexity. When Mhas constant curvature, the converse has been provedby doCarmo and Warner [CW]. It was doCarmo who brought my attentionto the fact that the general case is not as obvious as it sounds.The di erentiability requirements for our proof of the converse seem tobe C

The general theory of rods which has recently been developed by Green and Laws is specialized to the case of an elastic rod with constant curvature. It is assumed that the displacement from the initial configuration is small and the equations which are obtained are compared with those of Love and Morley. The present theory predicts the existence of three waves not predicted by other theories. Some numerical results are presented in graphical form.

1. Introduction. The optimum design of plane reinforced concrete slabs frequently requires consideration of deflected shapes for the slab such that through every point there is at least one principal curvature (κ1) direction on the deflected surface which is a constant curvature direction. The most interesting cases are those where only one principal direction has this property.

Let ( M , g ) (M,g) and ( M ′ , g ′ ) (M’,g’) be compact orientable Riemannian manifolds with the same spectrum of the Laplacian for 1 1 -forms. We prove that, for dim ⁡ M = 2 , 3 , 16 , 17 , ⋯ , 93 , ( M , g ) \dim M = 2,3,16,17, \cdots ,93,(M,g) is of constant curvature if and only if ( M ′ , g ′ ) (M’,g’) is so.

and has the constant curvature -4. 2. Consider now an analytic function o =f(z) from the circle I zx < 1 to a Riemann surface W. The analyticity is expressed by the fact that every local parameter w is an analytic function of z. To a differential element dz corresponds an element dw whose length does not depend on the dlirection of dz. The corresponding value of ds =X j dw|N dz I is therefore uniquely de* Presented to the Society, September 8, 1937; received by the editors April 1, 1937. t For the definition of a Riemann surface see T. Rad6, Uber den Begriff der Riemannschen Fldche, Acta Szeged, vol. 2 (1925). 359

This paper reports some relations between the curvature due to deflection and the frequency or mode of the vibration in long thin cantilevers, as the helicopter blades. For the lateral-torsional vibration coupled through a constant curvature, which is introduced in order to simplify the problem, the frequency equation and coupled mode are obtained. In the case where the torsional vibration is predominant, the free vibrations are studied under the curvature varying along the cantilever, their modes and frequencies are obtained. It is assumed through the calculation that a term involving the curvature in the equation of motion is smaller than others.The consideration for such the uniform cantilever with the deflection that the torsional vibration is predominant, has revealed the following: the curvature makes the torsional frequency increase the lower the natural frequency is, the greaten the effect of the curvature is on the frequency and mode, and moreover the effect can be approximately concentrated in a nondimensional parameter.

It is shown that, for certain second-order linear homogeneous partial differential equations, there exists a set of equivalence transformations to the form Δ2u+c′(x)u = 0 in the metric of spaces conformal to the space Vn related to the equation. An element of this space is a transformation of the equation to the canonical form $$\Delta _2 u + \frac{{n - 2}}{{4(n - 1)}}Ru = \pm u$$ in the metric of a space V′n, where R is the scalar curvature of V′n. Examples are examined of equations reducible to canonical form in spaces of constant curvature and in Kagan's subprojective space. The bibliography contains five references.

The following questions are presented in this paper: A geometric method for obtaining two-sided estimates for general quasilinear elliptic equations and its applications to problems of the calculus of variations and the problem of recovering a hypersurface from its mean curvature in spaces of constant curvature. Estimates of the modulus of the gradient for a hypersurface with boundary in a Riemannian space by means of its mean curvature and the metric tensor of the space. Estimates of the modulus of the gradient of a hypersurface depending on the distance of a point from the boundary and its mean curvature in Euclidean space. Estimates of these three types are of independent interest and play a fundamental role in the proofs of existence theorems for a hypersurface with prescribed mean curvature in Riemannian spaces. Bibliography: 3 items.

A method is reported for determining mathematically the combined disturbance field, and in certain cases the minimum drag, of wings at supersonic speeds. The simplest analytic example is provided by the wing of elliptic planform, which achieves its minimum drag when the lift is distributed uniformly over the surface. With a symmetrical distribution of thickness, the requirement of minimum drag for a given total volume is found to lead to profiles of constant curvature.

The five Platonic solids can be drawn in elliptic or hyperbolic space just as well as in Euclidean space. Their numerical properties are, of course, the same in all three. So are the various angles subtended at the centre. But the face-angles and dihedral angles are greater in elliptic space, smaller in hyperbolic. It is a special feature of the non-Euclidean spaces that we cannot change the size of a solid without changing its shape.

In order to numerically clarify the effect of small curvature on vibration prevention of panels for random loading, an analytical study is carried out by the linear theory for a shallow shell having a constant curvature in x and y directions, respectively, and a rectangular plane figure. The boundary condition is "simply supported" for bending. And two cases, i.e. free and constraint, are considered for in-surface displacements. The mean-square of the amplitude of each component mode is calculated. As a result, it is clarified that the lowest mode is predominant and the curvature is effective to diminish the mean-square of the amplitude. The difference of the natural frequency due to the in-surface boundary condition becomes small according as the depth-thickness ratio of the shallow shell becomes large. But the "constraint" in-surface displacement condition more diminishes the mean-square of the amplitude.

The transfer matrix method is extended to the analysis of non-circular cylindrical panels. The exact solution for the transfer matrix of a panel with exponential curvature is obtained by solving exactly the variable coefficient differential equations of motion of the shell using a Laplace transform—difference equation technique. The results are compared with respect to accuracy and computer time with various approximate methods of computing the transfer matrix for the same panel. Natural frequencies and mode shapes for typical non-circular panels are computed and compared with a constant curvature panel to show the effects of variable curvature.

The idea has recently been proposed of embedding the isometric groups of motion which apply to elementary particles in curved Riemannian manifolds (~). The use of the manifold V 4 with ~inkowskian signature already imposes very sever6 restrictions on the possible isometry groups (2). The simplest possibility is a space-time of constant curvature, and this corresponds to the two de Sitter groups S04.1 and S03.~. Ro~faN and AG~tSSl (~) have investigated and compared the energy spectra of the de Sitter world theories, i.e. regarded as global symmetry groups. On the other hand ~IICKnL and Li3vY (~) have stressed that the only extension of the Poincar4Lie algebra compatible with relativistic invariance is to the de Sitter algebras. Another interpretation recently proposed (z) showed that if the Wigner-In6nii contraction process is applied simultaneously to S04,1 and SOa,2, unified within the conformal group, then the algebra of quantum mechanics is recovered. The present method adopts a slightly different interpretation. S04.~, called D +, is adopted as the external event space and S03.2, called D-, as the internal structure space, both embedded in V4( + ~--~---). I t will be seen that all points of view are closely related, but this subject will not be investigated here. The use of Das internal structure space is reminiscent of Dirac's original extended particle model (G) for the theory ef the radiating electron. As in his theory the speed of propagation of a light signal inside the particle is greater than c. I t should be noted aIso that the speed of propagation in the external space is less than e, although the difference is locally undetectable. The maximal compact subgroup of S03. ~ is SO a x SO 2 ~ S U 2 • U1, and can therefore be used to label the isospin, I, and hypercharge, Y, quantum numbers. The

The coefficients of the complete set of n fundamental forms of a hypersuface Vn−1 imbedded in an n-dimensional Riemannian space Vn, as recently introduced[(5)], are used to construct certain tensor fields over Vn−1 which display some remarkable features. In particular, the divergences of these tensor fields can be expressed very simply in terms of polynomials involving the curvature tensor of Vn, the coefficients of the n fundamental forms, and the rth curvatures of Vn−1. As the result of an application of the generalized divergence theorem of Gauss to these relations a set of integral formulae on Vn−1 is obtained. The integrands of these integral formulae can be expressed very simply in terms of the n fundamental forms of Vn−1. By successive specialization it is indicated how known integral theorems([2], [3], [6], [7], [8]) can be derived as particular cases, which is possible partly as a result of the fact that the polynomial referred to above vanishes identically whenever Vn is a space of constant curvature.

The pattern of light signals, which was proposed before for the measurement of the curvature, is investigated in a two‐dimensional manifold of constant curvature (deSitter space). The pattern consists of light signals between two freely moving bodies, each signal being emitted when the signal from the other body arrives. It is shown that the coordinates of the arrivals (or emissions) of the light signals can be obtained from the coordinates of the emission of the first signal by means of projective transformations [see Eq. (18)] which are iterates of a single such transformation. The same applies to the proper times at which these signals are received.

Extension of theorems ofF. Riesz on subharmonic functions to spaces of constant curvature by the use of hyperbolic partial differential equations.

A generalization of Whitehead's law of gravitation for a discrete set of particles having arbitrary world-lines in a flat background space-time is formulated for a space-time of constant curvature. It is then extended by a method of Synge (Rayner 1954) to include systems having finite and continuous proper-density and velocity distributions. The new law is particularly adapted to a cosmological theory like that previously considered by the author in which every gravitating particle recedes with uniform velocity from a common spatio-temporal origin.

We consider a countable number of independent random uniform lines in the hyperbolic plane (in the sense of the theory of geometrical probability) which divide the plane into an infinite number of convex polygonal regions. The main purpose of the paper is to compute the mean number of sides, the mean perimeter, the mean area and the second order moments of these quantities of such polygonal regions. For the Euclidean plane the problem has been considered by several authors, mainly Miles [4]-[9] who has taken it as the starting point of a series of papers which are the basis of the so-called stochastic geometry. GEOMETRICAL PROBABILITY; RANDOM LINES; RANDOM POLYGONS; PLANE OF CONSTANT CURVATURE; HYPERBOLIC PLANE; CONVEX DOMAINS; POISSON LINE PROCESS

Publisher Summary This chapter focuses on n-dimensional riemannian spaces admitting a group of motions of order n(n-1)/2+ 1. If an n-dimensional Rieinannian space admits a group of motions of the maximum order n(n+l)/2, then, the space is of constant curvature. An n-dimensional finslerian space for n>2, n ≠4, admits a group of motions of order greater than n(n-l)/2+ 1, then the space is Riemannian and of constant curvature(2). The largest group of motions in a space of constant curvature being of order n(n+1)/2, if it denote it by G, then Gr must be a subgroup of G. In an n-dimensional Euclidean space for n ≠4, n ≠8, any subgroup of the rotation group of order (n - 1) (n - 2)/2 fixes one and only one direction. If an n-dimensional Riemannian space for n ≠4 admits a group of motions of order n(n - 1)/2+ 1, then the group is transitive.

We report a beginning in a project to determine the consequences of the following assumption, "that a physical theory in flat space is obtainable as the limit of a physical theory in a curved space." Because of the absence of groups of motion in general curved spaces, we discuss only the case of constant curvature. Then operators of angular and linear momentum exist, and we show that the interesting irreducible unitary representations of the group of motions reduce very simply to those of the inhomogeneous Lorentz group in the limit of zero curvature.

We prove: Let S be a closed n-dimensional surface in an(n+1)-space of constant curvature (n ≥ 2); k1 ≥ ... ≥ kn denote its principle curvatures. Let φ(ξ1, ..., ξn) be such that $${}_{\partial \xi _i }^{\partial \varphi } > 0$$ . Then if φ(k1, ..., kn)=const on S and S is subject to some additional general conditions (those(II 0) or(II) no 1), S is a sphere.

The displacement of lines in the spectrum of a distant star, on the assumption that space-time is of constant curvature (the “de Sitter world”) has been extensively studied. The question, however, is not yet settled, different investigators reaching conclusions which are not apparently concordant. The origin of some of the discordances may be traced to the ambiguities which are involved in the use of the terms “ time,” “spatial distance” and “velocity,” when applied by an observer to an object which is remote from him in curved space-time. The “interval” which is defined by ds 2 = ∑ p, q g pq dx p dx q involves space and time blended together; and although any particular observer at any instant perceives in his immediate neighbourhood an “instantaneous three-dimensional space,” consisting of world-points which he regards as simultaneous, and within which the formulæ of the “restricted relativity theory” are valid, yet this space cannot be defined beyond his immediate neighbourhood; for with a general Riemannian metric, it is not possible to define simultaneity (with respect to a particular observer) over any finite extent of space-time. The concept of “spatial distance between two material particles” is, however, not really dependent on the concept of “simultaneity.” When the astronomer asserts that “the distance of the Andromeda nebula is a million light-years,” he is stating a relation between the world-point occupied by ourselves at the present instant and the world-point occupied by the Andromeda nebula at the instant when the light left it which arrives here now; that is, he is asserting a relation between two world-points such that a light-pulse, emitted by one, arrives at the other; or in geometrical language, between two world-points which lie on the same null geodesic. The spatial distance of two material particles in a general Riemannian space-time may, then, be thought of as a relation between two world-points which are on the same null geodesic . It is obviously right that “spatial distance” should exist only between two world-points which are on the same null geodesic; for it is only then that the particles at these points are in direct physical relation with each other. This statement brings out into sharp relief the contrast between “spatial distance” and the “interval” defined by ds 2 = ∑ p, q g pq dx p dx q ; for between two points on the same null geodesic the “interval” is always zero. Thus “ spatial distance ” exists when, and only when, the “interval” is zero.

1. The elastic stability of a long flat plate when acted on by a shearing force along its edges has been discussed by Southwell and Skan*, and Leggett† has solved the shearing problem when the plate has a small constant curvature. The present paper deals with the elastic stability of a long corrugated plate which is acted on by a shearing force along its generators, the work being applicable to plates with any given number of bays forming the corrugations. It is supposed that the plate is thin and that the depthdof a bay is a small multiple of the semithicknessh;little progress has been made without some such assumption as the second, which enables us to reduce the fundamental equations to a soluble form.

This paper gives a sufficient condition for a submanifold of a Riemannian manifold of nonnegative constant curvature to be totally umbilical. The condition will be given by an inequality which is established between the length of the second fundamental tensor and the mean curvature. Introduction. In a previous paper [4] the present author proved the following theorem. Theorem A. Let M be an n-dimensional, connected, compact hypersurface with constant mean curvature immersed in an (n + l)-dimensional Riemannian manifold of nonnegative constant curvature. If the second fundamental tensor H satisfies (n 1) trace H2 < (trace H)2, then M is a totally umbilical hypersurface and consequently a sphere. In this theorem, if the length of the second fundamental tensor is constant, then we can replace the condition that M is compact by requiring that M is complete. In this case, the estimation in Theorem A is the best possible since Sn~ x E1 in F" + I satisfies (22 l) trace H2 = (trace H)2. In this paper, the author tries to generalize this theorem to a submanifold of any codimension which is immersed in a Riemannian manifold of nonnegative constant curvature. 1. Preliminaries. Let M be an 72-dimensional submanifold of an (22 +p)-dimensional Riemannian manifold M of constant curvature c. The Riemannian connections of M and M ate denoted by V and V respectively, whereas the connection in the normal bundle of M in M is denoted by D. Let N., • ■ • , N be mutually orthogonal unit normal vectors at a point P £ M and extend them to vector fields in a neighborhood of P. We define H . X (A = 1, 2, • • • , p) to be the tangential component of ^x^ A ^or ^ £ "^f/^ an

The theory of transformations Bk of deforms by flexure of quadricst may be regarded as a later development of the concepts introduced by Sophus Lie in his researches on the transformations of surfaces of constant curvature. The attempt to make a possible extension of the same principles to researches of another class of applicable surfaces leads us naturally to propose a problem of arranging plane elements, orfacettes, in space-which for the sake of clearness we will explain in detail.t A facette consists of a plane and a point of it called the center. We think of a surface S as the totality of its ooI2 facettes f, the planes of the facettes being tangent to S at. their respective centers. We consider associated with each facette f a simple infinity of facettes f', in accordance with any continuous law whatever. We imagine also that in every deformation of S, each facette f and the oo associated facettes f' are carried along as an invariable system. In each configuration of S the associated facettes f' form a triply infinite system, and in general they cannot be arranged into a series of oo1 surfaces S', each consisting of o2 facettesf'. The problem in view consists precisely in determining all the cases for which the above circumstance is true in all deformations of S. Of the general problem thus stated it is easy to indicate an infinity of particular solutions amongst which are immediately evident those in which each of the ool surfaces S' remains constituted always of the same oo2 facettes f'. But, in view of the eventual applications to problems of deformation, it is opportune to limit the problem much more, and to suppose that everv facette f and each of its associated facettes f' has the center of one in the plane of the other. Thus the surface S and each of its transforms S' are always the focal surfaces of the rectilinear congruence formed by the joins of corresponding points.

Let N N be a complete simply connected Riemannian manifold of constant sectional curvature ≠ 0 \ne 0 . Let M M be an immersed Riemannian hypersurface of N N . The Gauss map on M M based at a point p p in N N is defined. Suppose a Gauss map on M M has constant rank less than the dimension of M M ; then M M is generated by Riemannian submanifolds with constant sectional curvature. The sectional curvature of each of these generating submanifolds of M M has the same sign as the sectional curvature of N N .

Continued analysis was performed for the lateral vibration of an axially moving linear material. The boundary conditions are assumed constant curvature at two points supported by pullies. An algebraic method of analysis gives a simplified relation of parameters concerning the mode of vibration.

obtain the variational equation the commutation relations for oqi and oqi are settled so as to be consistent with the Euler-Lagrange equation. Under the premise of oqi= (djdt)oqi, it is shown that consistent Lagrangian and Hamiltonian formalisms exist when a space defined by metrics gij is of constant curvature. All consequences of a previous paper are derived and the dynamical model reduces to the case of flat space, if the commutability between the Euler equation and oqi is supposed to hold. As a non-trivial example of the irreducible dynamical models, the case of the three-dimensional space with constant curvature is investigated. It is shown that for this model the explicit form of [qi, oqj] and [qi, oqj] are all determined and the consistent quantization is carried out.