Then come the theorems connecting line, surface, and volume integrals, and, later, an introductory account of linear vector functions and dyadics. This general theory occupies four chapters; the rest of the book is concerned with the applications to potential theory, the conduction of heat, hydrodynamics, rigid dynamics, elasticity and electricity, not forgetting some account of the restricted principle of relativity.
The treatment is admirably clear and interesting, and exhibits the advantages of the use of vector methods in mathematical physics, provided that they are kept in their proper place. Review by: W V D Hodge. The Mathematical Gazette 13 , One of the results of the recent work in the theory of relativity has been that a fresh impetus has been given to the study of differential geometry.
Many of the problems that have arisen are problems in differential geometry, and in the solution of these there has grown up a new calculus, the calculus of tensors, which has had a remarkable success in this connection. This has resulted in the appearance of several new volumes on differential geometry, in which the methods of the standard works on the subject have been replaced by those which have proved so successful in satisfying modern requirements.
In expressing an opinion on any of these volumes, we must consider primarily the success or failure of the use made of these new methods, for therein generally lies the reason for the existence of the book. Professor Weatherburn's volume is of a very elementary character. Great use is made of vector methods, and in this respect the author rather surprisingly claims some originality of treatment.
These methods have however been superseded by the use of tensors, which have the advantage of generality, and are not open to the criticism that they do not lead easily to new results. This criticism of vector methods is one with which Professor Weatherburn disagrees, for he adds a special chapter in order to explain results which he has obtained by these means. As a textbook for elementary students, Professor Weatherburn's volume is marred by a certain looseness of expression.
Professor Weatherburn has shown himself to be an enthusiast for vectors, and so, as one would expect, his book makes great play with vector methods and is full of rather repulsive-looking dots and crosses and Clarendon type. It is much more geometrical than Mr Campbell's [A Course of Differential geometry ], as is natural, and it is of course much more elementary, beginning at the beginning and keeping to three dimensions or less.
The subject matter is, in fact, that usual in treatises on differential geometry, of which there is a fair choice. The book will certainly be of use to those who do not read German or Italian ; moreover, there are plenty of instructive examples. But the student must not neglect Darboux, as Prof Weatherburn has done. He chooses his geometrical material well, and, in adopting Gibbs's notation for his vector treatment, he is undoubtedly in accord with the current preference.
When a specialist in vector analysis turns his attention to geometry, it is too often the result that the geometry becomes merely a foil for the aggrandizement of the vector analysis. The present writer treats geometry more kindly. He keeps it clearly in mind as the first of the two objects he set out to achieve and does exceedingly well by it.
The author's geometric insight is keen, clever, and instructive. But at times he is found offering, as rigorous proofs, intuitive geometric arguments which, though enlightening and to the point in their proper place, are lacking in substance. An able presentation of the elements of the subject by vector methods, a clearly written text with an abundance of good exercises, this book should prove a welcome addition to the literature in differential geometry. Review by: Ernest P Lane. Monthly 38 1 , The second volume of this work on metric differential geometry continues the discussion of the subject along lines which are a natural extension of those followed in the first volume, which appeared in It seems appropriate, therefore, to make a few comments on certain characteristics which the two volumes have in common, before considering the second volume specifically.
The first thing that strikes the reader on turning the pages of the two volumes is that consistent use of vector analysis is made throughout. The classical notation of Gibbs is employed, symbols for vectors being printed in Clarendon type. Certain economies are thus effected in the way of simplifying and condensing the presentation of the subject.
Those who have been in the habit of lecturing to graduate students on the subject of metric differential geometry, using the conventional methods, might do well to consider the advisability of trying out a presentation by vector methods. To anyone who ventures on this undertaking these two volumes before us will be very useful; they should be in the hands of the students as well as on the lecturer's desk. Another commendable characteristic of the entire work is that the author has not allowed himself to forget that he is, before all, writing a treatise on geometry.
The geometry is the thing that holds the centre of the stage, and the analytical machinery is relegated to a subordinate place in the background where it ought to be in a book on geometry. The author does not make the mistake of becoming so immersed in the intricacies of his machinery, or so occupied with juggling his tools, that he loses sight of his main undertaking. Moreover, it is worthy of remark in this connection that the author's geometric insight and intuition are as clear and penetrating as his geometric interest is dominant.
The treatise lacks the profundity of such monumental works as those of Bianchi and Darboux. The more elementary parts of the subject are fairly adequately treated, but the more advanced portions are touched upon rather lightly. The book is well written. The author evidently understands the fundamental principles of good mathematical exposition. The second volume of Professor Weatherburn's Differential Geometry of Three Dimensions should prove a very useful book to those familiar with the earlier volume.
The branches of the subject discussed are not usually included in an honours course in English universities, but students who have studied the subject to degree standard will find this book an excellent introduction to further work. Much of the volume is devoted to subjects to which the author has himself contributed in the last few years, particularly in the theory of families of curves and surfaces, and of small deformations. Other topics are however included, with the result that the two volumes together give an account of most of the principal branches of classical Differential Geometry.
The first volume of this work see this Bulletin, vol. The present volume, though containing certain classical material supplementing that of the first volume, is primarily devoted to a consequential exposition of the author's published contributions to the subject. The treatment in both volumes is in terms of vectors. But, whereas the first volume employs, except in the last chapter, on differential invariants, only the algebra of vectors, the second volume uses also, and to a great extent, the differential and integral calculus of vectors.
Email Address. Sign In. Access provided by: anon Sign Out. On almost orbital equivalence of nonlinear systems Abstract: This paper is concerned with equivalence of nonlinear systems from a viewpoint of geometric congruence of system orbits in the state space. The notion of orbital equivalence had been originally exploited by Sampei and Furuta in the context of the time-scale transformation approach. They gave a fundamental characterization of orbital equivalence in the form of a similarity parallelism of the system vector fields; however, it was not satisfactory in the sense that there remains significant gap between the necessary condition and the sufficient one, mainly due to its treatment of unavoidable singularity of the time-scale functions.
In such a variety the coordinates of a point are expressible in terms of a single parameter, say x l. It is therefore a curve in V ni and will be referred to as a coordinate curve of parameter x i. Thus for an infinitesimal displacement along this cu rve ,. Field of normals to a hypersurface. Holder, , 4, pp. See also Ex. They intersect orthogonally wherever the condition Vtp. They will be orthogonal at all points provided that g ij vanishes identically. This should be compared with equation 14 giving the inclination of the coordinate curves. These are referred to by some writers as differential parameters of the first order.
JV-ply orthogonal system of hyper surfaces. If in a V n there are n families of hypersurfaces such that, at every point, each hypersurface is orthogonal to the n— 1 hypersurfaces of the other families which pass through that point, they are said to form an n-ply orthogonal system of hypersurfaces. An arbitrary Riemannian V n does not admit an n-ply orthogonal system of hypersurfaces. Orthogonal ennuples. We have already mentioned a congruence of curves in a V n as a family of curves, one of which passes through each point of the V n. A congruence is determined by a vector field u, the value of u at any point being tangent to the curve of the congruence through that point.
Since there is one such curve through each point P, the vector field determines a congruence of curves, as stated above. An orthogonal ennuple in a V n consists of n mutually ortho- gonal congruences of curves. If the rela- tions 24" are regarded as referring to the rows of the deter- minant, the analogous relations hold with respect to the columns.
Principal directions for a symmetric covariant tensor of the second order. Let be the components of a symmetric covariant tensor of the second order, and k a scalar invariant.
Consequently the determinant equation 32 which is an equation of the wth degree in k , is equivalent to the equation. These quantities are determined by the equations to within a factor. That they are contra variant components of a vector follows from the fact that — K h g hj is a covariant tensor, and the zero second members of 33 are covariant components of a vector. Con- tinuing in this way we see that, if the roots of 32 are all simple, they determine uniquely n mutually orthogonal vector fields which satisfy equations of the form The directions of these vectors at any point are called the principal directions at that point determined by the symmetric tensor a tj.
Eisenhart, , 1, pp. But these are the principal directions determined by the tensor a ij7 and the theorem is proved. Also, since the funda- mental form has been assumed positive definite, it follows from 39 that k is finite for all directions. In this case we say that the space is homogeneous with respect to the tensor a ip Euclidean space of n dimensions.
If the fundamental quadratic form, g ij dx i dxf which con- stitutes the metric of the space, reduces in a particular coordinate system y l to the sum of the squares of the dif- ferentials, so that 1 , The coordinates y l will be called Euclidean coordinates. It follows that, in Euclidean coordinates, the covariant com- ponents of a vector are the same as the contravariant com- ponents; and the square of the magnitude of a vector is equal to the sum of the squares of its components.
Euclidean coordinates are a particular case of orthogonal Cartesian coordinates. We shall see later the conditions that must be satisfied by the coefficients of the fundamental form g ij dx i dx j 3 in order that the space may be Euclidean. Veblen, , 2, p. Levi-Civita, , 1, p. For example, if V n is Euclidean, n dimensions are sufficient.
In other words, if V n can be immersed in a Euclidean space of m dimensions, but not of less than m dimensions, the class of V n is m — n. The class of a Euclidean space is zero. We may remark that the inclination 6 of two vectors a, b has the same value whether these are regarded as vectors in V n , or as vectors in a Euclidean space S m in which V n is immersed.
Take a, b as unit vectors, and let s be the arc-length of a curve C passing through the point P considered in the direction of a. Conformal representation. The reader is familiar with conformal representation of one surface on another in Euclidean space of three dimensions, and is aware that the characteristic of such representation is similarity of corresponding infinitesimal portions. Also the inclination of the vectors dx l and 8x i f at a point of V n , is the same as the inclination of the corresponding vectors at the corresponding point of V n. Thus corresponding infinitesimal figures are similar.
Such correspondence is conformal , and either space is said to be conformally represented on the other. Cross product of two vectors. Let u, v be two vectors whose covariant components are and respectively. This tensor may be compounded with a vector w. Weatherbum, , 3, pp. For transformation of coordinates from x i to x l we have the relation dx 1. The Christoff el symbols. We must now introduce certain functions involving the derivatives of the components of the fundamental tensors.
From their definitions it follows that both functions are symmetric in the indices i and j. Consequently 6 Second derivatives of the x's with respect to the x's. Hence, in virtue of 3 , the symbols of the first kind do not denote components of a tensor. It may be verified that the law of transformation 8 of the Christoffel symbols possesses the group property, the symbols in any coordinate system being connected with those in any other system by equations of this form. It may also be remarked that the above formulae remain true if g tj are the components of any symmetric covariant tensor of the second order.
Co variant derivative of a co variant vector. Curl of a vector. We have seen that the partial derivatives of a scalar invariant with respect to the coordinates are the covariant components of a vector. This case is, however, unique. For a general system of coordinates, the partial derivatives of the components of a vector or a tensor are not components of a tensor. There are, however, expressions involving the first derivatives which do possess this property. Covariant differentiation is indicated as above by a subscript preceded by a comma.
But this is the only case in which it is so.
Of Congruences of Field a Parallelism Of Vectors
Thus: A necessary and sufficient condition that the first covariant derivative of a covariant vector be symmetric is that the vector be a gradient. Co variant derivative of a contravariant vector. Let u i and u l be the components of a contravariant vector in two coordinate systems x i and x l respectively. It is called the covariant derivative of the contra- variant vector vt with respect to the fundamental tensor. Derived vector in a given direction.
In extending this idea to a vector function we must employ co variant derivatives, since the ordinary partial derivatives of the components of a vector with respect to the coordinates are not components of a tensor. Let v be a vector whose covariant and contravariant com- ponents are v i and v l respectively,, and a a unit vector in any direction. Then the vector whose covariant components are a k v i k is the intrinsic derivative of v in the direction of a, or the derived vector of v in that direction.
Thus V becomes a symbol of covariant differentiation; and the position of the vector a in front of this symbol in- dicates that the contravariant index in a k is the same as the index of covariant differentiation, giving the vector a k v i k by contraction. The projection of the vector a-Vv in the direction of the unit vector a will be called the tendency of v in that direction.
Co variant differentiation of tensors. Tensors may also be obtained by covariant differentiation of tensors in the following manner. Let and A be the components of a covariant tensor of the second order in the coordinate systems x i and x l respectively. Then from the law of transformation j. From these equations it follows immediately that the covariant derivatives of the tensors g ijy g ij and 8 all vanish identically.
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Thus the above tensors may be treated as con- stants in covariant differentiation with respect to the funda- mental tensor. The process of co variant differentiation can be repeated indefinitely. The covariant derivative of the first covariant derivative is called the second covariant derivative, and so on. Covariant differentiation of sums and products. From the above formulae for covariant differentiation it is evident that the co variant derivative of the sum or difference of two tensors of the same type and order is equal to the sum or difference of their covariant derivatives.
Further, the covariant derivative of a product of tensors is given by the same rule as in ordinary differentiation. The method of proof is general, and applies to all cases of outer products of tensors. The inner product of two tensors is a tensor formed by outer multiplication and contraction.
It is therefore a sum of products, so that the same rule for differentiation applies. Consequently, covariant differentiation of sums and products of tensors obeys the same rules as ordinary differentiation. As an important illustration of covariant differentiation of a product, let us consider the gradient of the scalar product of two vectors , u and v. Covariant differentiation of this invariant gives 2 uiu i,k — Forming the scalar product of this with any unit vector a, we have. Thus: A vector of constant magnitude is orthogonal to its intrinsic derivative in any direction.
Divergence of a vector. The divergence of a contravariant vector u x may be de- fined as the contraction of its co variant derivative. Thus the theorem is proved, since the choice of the orthogonal ennuple is at our disposal.
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In the case of Euclidean space of three dimen- sions the formula 29 is well known. Laplacian of a scalar invariant. Weatherbum, , 1, p. If two unit vectors are such that, at all points of a given curve C , their intrinsic derivatives in the direction of the curve are zero, show that they are inclined at a constant angle along G. If the intrinsic derivative of a vector u along a given curve C vanishes at all points of the curve, show that the magnitude of the vector is constant along the curve.
If, at all points of C, a vector v of variable magnitude has the same direction as the above vector u, show that the intrinsic derivative of v along C has the same direction as u at each point of the curve. Let s be the arc -length of a curve 7, and t a function of s. Let derivatives with respect to t be denoted by dots over the symbols differentiated. If, at a specified point, the derivatives of the s with respect to the coordinates are all zero, the components of co variant derivatives at that point are the same as ordinary derivatives.
Some formulae involving the curl of a vector. In virtue of Chapter in, Ex. Next consider the gradient of the scalar product of the vectors u and v. An important particular case of iii is that in which the vectors u, v are equal and of constant magnitude. Principal normal.
Let C be a curve in a given V n , and let the coordinates x 1 of the current point on the curve be expressed as functions of the arc-length s. Then the unit tangent t to the curve has contravariant components Generalising the concept of the vector curvature of a curve in Euclidean space of three dimensions, we call the derived vector of t along the curve th e first curvature vector of C relative to V n.
The reader is familiar with the idea of a geodesic on a surface in Euclidean 3-space, as a curve whose tangential curvature is zero at all points; that is to say, a curve whose curvature relative to the surface is everywhere zero. We might generalise this concept by defining a geodesic in a Riemannian V n as a curve whose first curvature relative to V n is zero at all points. The differential equation of such a curve is obtained by equating to zero the second member of 4. Another method of approach is by using the property of a geodesic that it is a path of minimum or maximum length j oining two given points on it ; and we shall see that this method leads to the same differential equation for geodesics.
Let C be a curve in a V n , and A, B two fixed points on it. The coordinates x l of the current point P on G are functions of a single para- meter t. Let t 0 and t x be the values of the parameter for the points A and B respectively. Now let the curve suffer an infinitesimal deformation to G', the points A and B remaining fixed while the current point P is displaced to P' whose coordinates are ,. In other words, for deformations of a curve joining two fixed points, the length is stationary when the curve is a geodesic.
Using this property we may find the differential equations satisfied by geodesics in a V n. The first variation SI of the integral is given by the terms of the first order. Hence the necessary and sufficient conditions that Cf. Eisenhart, , 1, p. Differential equations of geodesics. In other words, a geodesic of V n is a line whose first curvature relative to V n is identically zero. The equations 11 are n differential equations of the second order. Their complete integral involves 2 n arbitrary constants. These may be determined by the n coordinates of a point P on the curve, and the n components of the unit vector in the direction of the curve at P.
Thus, in general, one and only one geodesic passes through a given point in a given direction. Or the 2 n arbitrary constants are determined by the coordinates of two points on the curve; so that, in general, one and only one geodesic passes through two given points. If G is a geodesic, the coefficient of -s 2 is equal to —. Geodesic coordinates. We have already remarked that a Cartesian coordinate system is one relative to which the coefficients of the funda- mental form are constants. Such a system of coordinates is said to be geodesic with pole at P 0.
It is usual to choose geodesic co- ordinates so that they all vanish at the pole, which is then also the origin for the coordinate system. From the definition of co variant differentiation it is clear that: At the pole of a geodesic coordinate system , the components of first covariant derivatives are ordinary derivatives.
It is this property which accounts for the simplification in proof often achieved by the use of geodesic coordinates. The conditions that a system of coordinates be geodesic, with pole at P 0 , may be expressed in another form. For a fixed value of d the function x d is a scalar invariant, and the second member of 15 is its second covariant derivative x l t j with respect to the metric of V n. The existence of a geodesic coordinate system for any V ni with an arbitrary pole P 0 , is easily proved. Then at the point P 0 we have dxn. They also vanish at this point, which is thus the origin for the system.
Fermi f has extended this result by proving that, for an arbitrary curve C in a V n , it is possible to choose coordinates which are geodesic at every point of O, that is to say, which are such that every point of C is a pole at which the conditions 14 are satisfied. Duschek-Mayer, , 2, Vol. Riemannian coordinates. We shall consider briefly a particular type of geodesic coordinates introduced by Riemann, and known as Riemannian coordinates. As there is one geodesic from P 0 to any point of V n , each point of the space has definite coordinates y i assigned to it.
These are the Riemannian coordinates referred to. It will be shown that they are geodesic coordinates with pole at P 0. They also vanish at P 0 , since s is zero for that point. These are of the same form as the equations of straight lines through the origin in Euclidean geometry. Geodesic form of the linear element. From the differential equations satisfied by geodesics, viz. We have thus proved the theorem: If a hypersurface S be taken in a V n , and along the geodesics orthogonal to 8 the same length be measured from 8, the locus of the points so found is a hypersurface orthogonal to the geodesics.
The hypersurfaces so constructed, one corresponding to each length of arc measured from 8 , are said to be geodesic- ally parallel to 8.
Congruences of Parallelism of a Field of Vectors.
Thus a system of parallel hypersurfaces are orthogonal to a family of geodesics; and the distance along one of these geodesics between two of the hypersurfaces is the same for all the geodesics. Straight lines. Consider a Euclidean space S n of n dimensions. These are a generalisation of the straight lines of ordinary space.
We speak of this length as the distance between the two points. Since these are the same for all points of S n , the coordinate curves are straight lines. Parallelism of Vectors Parallel displacement of a vector of constant magnitude. Consider a vector field whose direction at any point is that of the unit vector t. In ordinary space the field is said to be parallel if the derivative of t vanishes for all directions and at every point. Similarly in a Riemannian V n the field is said to be parallel if the derived vector of t vanishes at each point for every direction at that point. It can be shown, however, that with a general Riemannian metric this is not possible.
It is easy to show that they are inclined at a constant angle. It is evident that the argument applies when the constant magnitudes of the vectors are different from unity; so that: If two vectors , of constant magnitudes , undergo parallel displacements along a given curve , they are inclined at a constant angle. Comparing 11 and 32' we see that the unit tangent to a geodesic of V n suffers a parallel displacement along the geodesic.
This is sometimes expressed by saying that geodesics are auto-parallel curves. From the above theorem it then follows that: Any vector which undergoes a parallel displacement along a geodesic is inclined at a constant angle to the curve. They may also be expressed ds ikj ds.
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Hence the result. Parallelism for a vector of variable magnitude. We can extend the above definition of parallelism with respect to a given curve, so as to apply to vectors which are not of constant magnitude. Two vectors at a point are said to be parallel, or to have the same direction, if their corresponding components are proportional.
If u is of constant magnitude, it satisfies These show that the derived vector of v along G has the same direction as v at each point of the curve. Conversely, if a vector v satisfies 37 , it is parallel to itself along C. It is then evident from 38 that v is parallel to itself along C. Subspaces of a Riemannian manifold.
Before extending the above ideas of parallelism to subspaces of a Riemannian V m. But the components of a vector of magnitude l are l times the corresponding components of the unit vector in the same direction. Consequently relations of the form 42 hold for a vector of any magnitude.
Corresponding to the n independent variables x 1 there are n such independent vector fields in V n , in terms of which any vector field in V n is linearly expressible. Parallelism in a subspace. Then by 42 rpa. In order that the vector field t may be parallel with respect to V n along C, its derived vector p in the direction of G, with respect to the metric of V n , must vanish at all points of C.
Similarly, in order that the vector field may be parallel with respect to V m along C , its derived vector q in the direction of C, with respect to the metric of V m , must vanish at all points of C. In general the derived vectors p and q are not the same. We propose to find a relation between their components, which will show how parallelism in V m is connected with parallelism in the subspace V n. In virtue of 45 and 46 this may be written dt l dy a ,. Hence the theorem: If a curve C lies in a subspace V n of V m , and a vector field in V n is parallel along C with respect to V m , it is also parallel with respect to V n.
Then the parallelism of the vectors along C with respect to either manifold means that the curve is a geodesic in that manifold. The theorem thus becomes: If a curve is a geodesic in a Riemannian space, it is also a geodesic in any subspace in which it lies. But these are the conditions that the vector q be normal to V n. Hence: A necessary and sufficient condition that a vector of constant magnitude be parallel with respect to V n along a curve in that subspace , is that its derived vector relative to V m for the direction of the curve be normal to V tl.
As a particular case let the vector be the unit tangent to the curve. The theorem then becomes: A necessary and sufficient condition that a curve be a geodesic in V n is that its principal normal , relative to the enveloping space V m , , be normal to V n at all points of the curve. If two subspaces of V m , of the same number of dimensions, are such that, along a curve C in both, every normal to either subspace is also normal to the other, the subspaces are said to touch each other along C. From the above theorem it then follows that: If two subspaces , of the same number of dimensions , touch each other along a curve C, any vector which undergoes a parallel displacement along C with respect to one of the subspaces suffers a parallel displacement with respect to the other also.
Tendency and divergence of vectors with respect to subspace or enveloping space. Thus the relations 48 hold for the derived vectors of any vector t in V n. Similarly, p i a i is the tendency of t with respect to V n for the same direction. Con- sequently: For any vector field in a subspace V n the tendency , in any direction in that subspace , has the same value with respect to V n as to the enveloping space V m.
If a vector field v lies in V m , but not in V n , its tendency must be calculated with respect to V m. This will be denoted by div m v. Since these m directions may be chosen arbitrarily, we may take n of them in V n , and the remaining m — n normal to V n. The choice of the n orthogonal directions in V n is arbitrary, and is independent of the choice of the m — n normal directions ; for every direction in V n is orthogonal to these normal directions.
Weatherburn, , 1, p. It may be expressed where k 1 is its magnitude and n x the unit principal normal. Resolve it into a component perpendicular to t and n 1? The latter com- ponent has the direction of t, since the derived vector of ri x is per- pendicular to n x. The latter component has the direction of n x. Blaschke, , 1; also Levy, , 7. We may call K r the rth curvature of the curve. As an illustration of the use of geodesic coordinates let us con- sider the rate of divergence of a given curve G from the geodesic which touches it at a given point P 0.
Take geodesic coordinates x i with origin at P 0. If k is zero, the curve and its tangent geodesic have contact of the second or higher order. Subtract this equation from the sum of two others obtained from it by interchanging i and k, and j and k respectively, and write the result in the form , ,. If u, v are unit vectors, the first member is the projection on v of the derived vector of u in its own direction; the second member is minus the tendency of v in the direction of u. It will, of course, be observed that the indices l, h , k in these symbols are not indices of covariance.
They simply indicate the congruence whose unit tangent is considered, and the two directions for differentiation and projection. The second index Cf. Levy, , 3. Geodesic congruences. The first curvature vector p z[ of a curve of the congruence, whose unit tangent is e z! The above theorem may therefore be stated: A necessary and sufficient condition that a congruence C of an orthogonal ennuple be a geodesic congruence is that the ten- dencies of all the other congruences of the ennuple in the direction of C vanish identically.
Commutation formula for the second derivatives along the arcs of the ennuple. Let s be the arc-length of one of the curves of the ennuple, say the curve through the point P considered in the direction e Ai. Then, along this curve, the coordinates x l are functions of s. Let C m be a definite curve of the congruence whose unit tangent is e m!
Let u be a unit vector, which coincides with the vector e, at P 0 , and undergoes a parallel displacement along the curve C m. Conditions that a congruence be normal. A normal congruence is one which intersects orthogonally a family of hypersurfaces. We require the conditions that must be satisfied by a congruence in order that it may possess this property.
Let t be the unit tangent to the congruence considered. Suppose now that the congruence is one of an orthogonal ennuple. Let it be taken as that whose unit tangent is e wl. The conditions that the congruence be normal are, of course, obtained from 15 replacing t by e n ;. Then, since e p!
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Hence: Necessary and sufficient conditions that the congruence e ni of an orthogonal ennuple be normal are expressed by If all the congruences of an orthogonal ennuple are normal, all the coefficients of rotation with three distinct indices must be zero. To an orthogonal ennuple of normal congruences corre- sponds an w-ply orthogonal system of hypersurfaces. Curl of a congruence. We proceed to examine the curl of the unit tangent to a con- gruence of curves, which may be called briefly the curl of the congruence. If it vanishes identically, the congruence will be described as irrotational.
Consider an orthogonal ennuple of which the given con- gruence is the nth, so that its unit tangent is e n! In the first let h and k take the values 1, 2,. If the conditions of this theorem are satisfied, the congruence is orthogonal to a family of hypersurfaces, and its unit tangent is the unit normal to the hypersurfaces. Consequently the theorem shows that: A necessary and sufficient condition that a family of hyper- surfaces be parallel is that the curl of the unit normal should vanish identically. Congruences canonical with respect to a given congruence.
By a suitable choice of these other congruences the analysis may often be simplified. They form with the given congruence an orthogonal ennuple. If, now, in the first of equations 23 , we insert the value of given by 20 and 18 , we obtain 1 , Then, if k is not equal to h or n, the last two terms disappear; and, in the sum with respect to Z, all the terms disappear except that in which l has the value k, We thus obtain Ynhk Ynkh Conversely, if these relations hold for the values 1, Consider the family of hypersurfaces orthogonal to the normal congruence e nJ.
If are the unit tangents to the canonical system, equations 27 hold; and these show that, for a given value of k , the derived vector of e n! But this derived vector is also orthogonal to e n! Consequently it must have the direction of e kl. Now if the derived vector of the unit normal to a hyper- surface, for any direction in the hypersurface, has this same direction, this is called a principal direction for the hyper- surface at the point considered.
And a curve in the hyper- surface, whose direction at each point is a principal direction at that point, is called a line of curvature of the hypersurface. These will be considered more fully in Chapter vm. In the meantime we have shown that: The congruences canonical with respect to a normal congruence are the lines of curvature of the hypersurfaces orthogonal to the congruence. If the congruences of an orthogonal ennuple are all normal, the equations 17 are satisfied. Consequently any n— 1 of the congruences are canonical with respect to the other congruence, and we have the theorem: When a V n admits an n-ply orthogonal system of hyper- surfaces, any hypersurface of the system is cut by the hypersurfaces of the other families in the lines of curvature of the former.
Curvature tensor and Ricci tensor. We have already seen that the order of covariant differen- tiations is not in general commutative. From its definition 2 the tensor is clearly skew-symmetric in j and k. The steps in the argument are exactly similar to those given above. The curvature tensor may be contracted in two different ways. One of these leads to a zero tensor. Covariant curvature tensor. The identity of Bianchi. Let us choose geodesic coordinates x i with pole at the point P. Then on differentiating 2 covariantly with respect to xf, remembering that all the Christoffel symbols vanish at the pole P, we see that, at that point, R a.
It is called the Bianchi identity in honour of its discoverer. As coordinates in V n let us choose Riemannian coordinates with origin at P. Thus, with the present notation, [r. Let R' a fi yS denote the Riemann symbols of the first kind for the surface S and the coordinates u a. Then, since the indices Cf. Weatherburn, , 3, pp. For a change of coordinates to u a let the corre- sponding Riemann symbols be R[ And this invariant is equal to the Gaussian curvature of the surface S.
Weatherburn, , 3, p. Formula for Riemannian curvature. An explicit expression for the Riemannian curvature, in terms of the covariant curvature tensor of V n , may be deduced from And, since this expression is an invariant, the equation holds for all coordinate systems. In that case the Riemannian curvature is identically zero, and the space is said to be flat. A flat space of n dimensions is denoted by S n.
If, for example, the coeffi- cients of the fundamental form are constants, the com- ponents of the curvature tensors are all zero, and the space is flat. Conversely it can be shown that if Ri! If the fundamental form is positive definite, the space is then Euclidean. Theorem of Schur. Since n may be assumed greater than 2, we may take three distinct values for j, k , l. Such a space is said to be of constant Rieman- nian curvature. The equations 24 , with K constant, are necessary and sufficient conditions that the space be one of constant curvature. Mean curvature of a space for a given direction.
The first member of this equation is the sum of the Riemannian curvatures of V n for the orientations determined by and each of the n — 1 directions of the ennuple which are orthogonal to e A! We shall denote it by M h. These are called the Ricci principal directions of the space. Consequently an Einstein space is characterised by the property n. Eisenhart, , 3. If in Ex. Prove that an Einstein space V 3 has constant curvature.
Show that, at corresponding points of two conformal spaces cf. Ill, 1 , the quantities have the same values. Unit normal. Generalised co variant differentiation.
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Let C be any curve in V n , and s its arc-length. Then, along this curve, the and the y ' s may be expressed as functions of s only. Duschek-Mayer, , 2, vol. In this section we follow McConnell closely. Its derivative with respect to s is also a scalar invariant. If C is any curve in V n , and the functions are defined throughout the subspace, we may write the above intrinsic derivative Now dx j jds is an arbitrary unit vector in V n9 because the direc- tion of C is arbitrary; and therefore, by the quotient law, its coefficient in square brackets is a tensor.
The reader may easily verify that tensor differentiation of sums and products obeys the ordinary rules. A few other important points in connection with tensor differentiation may be mentioned explicitly. In particular the tensor derivative of a scalar invariant is its gradient with respect to V n. For, if u a are the contravariant components of this vector in the yf s, the intrinsic derivative with respect to s along any curve C as stated. These may therefore be treated as constants in tensor differentiation. Second fundamental form. In the geometry of a surface in Euclidean space of three dimensions, the second derivatives of the Cartesian co- ordinates of the current point, with respect to the parameters on the surface, play an important part.
Normal curvature. Let u be a vector field in the hypersurface. The two derived vectors thus differ by a vector normal to the hypersurface. From this follow the theorems connecting parallelism in a hypersurface with parallelism in the enveloping space, given for any subspace in the section referred to.