Invariants of Quadratic Differential Forms (Dover Books on Mathematics)

Join Kobo & start eReading today

Gauge Theory and Variational Principles. Applications of Tensor Analysis. Fundamentals of Scientific Mathematics. Differential Forms with Applications to the Physical Sciences. Geometry from a Differentiable Viewpoint.

• The Other Shoe Dropped!
• The Enculturated Gene: Sickle Cell Health Politics and Biological Difference in West Africa.
• Reprogramming Microbial Metabolic Pathways: 64 (Subcellular Biochemistry);

Lectures on Quantum Mechanics. Geometry from Dynamics, Classical and Quantum. The Meaning of Relativity. Clifford Algebras and Lie Theory. Group Theory Applied to Chemistry. Structure and Geometry of Lie Groups. Ricci Flow and Geometric Applications. Mathematical Aspects of Quantum Field Theory. Cohomology and Differential Forms.

Invariants of Quadratic Differential Forms

Groups, Geometry and Applications. Tensor Algebra and Tensor Analysis for Engineers. An Invitation to Morse Theory. A Modern Approach to Classical Mechanics. Open Problems in Mathematics. The Absolute Differential Calculus: Volume 2, Superstring Theory and Beyond. Analysis and Geometry of Markov Diffusion Operators. Mathematical Adventures in Performance Analysis. Treatise on Irreversible and Statistical Thermodynamics. Spectral Geometry of the Laplacian. Optimal Control and Geometry: Geometry, Algebra and Applications: From Mechanics to Cryptography. Einstein's Theory of Unified Fields.

What is Kobo Super Points?

Application to determination of invariants of a quadratic form. The problem of the determination of differential invariants of a quadratic differential form is now seen to be that of the determination of all invariants of a certain extended group. We start with a general point transformation on the n variables: In addition there are introduced other dependent variables ar,, functions of the: The group of all such point transformations is then extended so as to include the transformation equations for the new variables introduced and their derivatives, and our problem is to determine all the invariants of this extended group.

To do this it is necessary to obtain the infinitesimal transformations of the group, and from these we obtain a complete system of linear differential equations the solutions of which are the invariants. The case of two independent variables. As a first case we consider a quadratic form in two variables te, g.

Now if z be any one of the dependent variables we must have dz - zJ: This is a complete system of six independent linear equations in eight variables, and therefore it possesses two functionally independent solutions. There are thus two invariants in these eight variables ; one is j, and the other is readily found to be Beltrami's first differential parameter There are several points to be noted in connection with this example.

In the first place, the finite equations of the group are not wanted, and therefore it is only necessary to calculate the increments of the various variables for an infinitesimal transformation. Again, there is no need to take account of more than two functions j, for any other function may be expressed in terms of these two, say j The general case of n independent variables. We shall now consider the general case of a quadratic form in n variables.

Determination of the increments. By equating to zero the coefficients of the various powers of the k's, we have all the increments of the derivatives of f. Exactly as in the particular case of two independent variables, the increments of the quantities ar, may be obtained by comparison of the quadratic form and its transformed, and then Forsyth's method will give the increments of the derivatives of the a's.

Hence m is less than or equal to! We thus have, immediately, a classificatioB of quadratic forms. Differential parameters for forms of rank zero. The problem in this particular case is therefore seen to separate into two parts: A The determination of all invariants under a general transformation on the: B 'l,he selection from these of those functions which are still invariant when a translation and an orthogonal transformation are performed on the u's. The first of these is equivalent to the determination of all the invariants of any number of functions of the set of variables rc.

Cr ' denotes the effect of the infinitesimal transformation on L 'rhe increments of the various variables in I may be obtained without. It is further assumed that there are n functions f. A complete set of solutions of this system is readily seen to be given by tPj and there remains yet one other solution, which is manifestly J, the Jacobian of the J. The second part of our problem is to determine those invariants which still remain invariant when a general translation and a general orthogonal transformation are performed on the u's.

The variables entering are 1 ttt, u.. Tbe translation may be taken account of at once. It is equivalent to the condition that the variables u do not enter explicitly into the invariants. The orthogonal transformation is a linear one on the variables u. We give the proof for two variables, though the method is quite general. Hence U'tt, "Vtt are both zero. Similarly all the other second derivatives of U and V are zero and hence U and V are linear functions of u and v.

As a translation has already been taken account of, onr transformation is homogeneous and linear on the variables u, v. Now, returning to the case of n variables, let us consider a general homogeneous linear transformation on the u's; the variables dAtu are transformed by the same linear transformation. III where the U's are a set of auxiliary variables. The transformations for these derivatives of ,; are precisely those on the coefficients of the algebraic form Am, when the U's are transformed by the contragredient transformation to the original linear transformation.

We therefore im1nediately obtain the result: The functionally independent set of invariants of orders up to and n including the pth of the quadratic for. The final result is: We proceed to calculate these forms: It therefore follows by induction that the coefficients of Ap may be expressed in terms of these magnitudes. The initial limitation that the quadratic form is of rank zero may now be removed, for it may be verified without difficulty that the invariants obtained are also invariants for a general quadratic form. When the form is of -rank zero it has no other invariants than these, which are differential parameters.

In the general case the quadratic is expressed as the sum of the square of m perfect differentials; where m is greater than n. Then it is clear that all the invariants of our quadratic must be included in m those of the form: By means of these equations the forms A for the parameters may be reduced so as to contain n instead of m variables X, and hence the covariant.

These last are not invariants of the original quadratic form since they cease to be invariants if the n dimensional manifold is deformed in the m dimensional space. Among them, however, ntust be include! An idea of the algebraic work necessary for a direct application of the Lie theory to the problem of the determination of Gaussian invariants may be gathered from a study of Forsyth's memt irs on the subject.

Explanation of the symbolic notation. Its leading principle will present no difficulty to those fatniliar with the ordinary symbolic notation of algebraic invariants. The assumption is made that the quadratic form 'la,k dtet d: Any expression involving the a's may thus be expressed in terms of t.

If, however, the expression in the a's is of order higher than the first, we must avoid ambiguity by introducing equivalent symbols. Now call the fundamental quadratic form A, and let Ft, J! F' for a new set of variables y. If it should be necessary to put in evidence the first one, two, three, etc.

For example b, c, a denotes the invariant derived from b, c, at, F is called an invalriantive constituent of A. It is to be expected that some functions of the higher derivatives off, for example, may be interpreted by means of the derivatives of magnitudes aa;, and it is easy to see that any expression built up of invariantive constituents is an invariant if it can be interpreted in terms of actual quantities, for instance, the a's and their derivatives and the arbitrary functions introduced and their derivatives. Now by actual differentiation.

By means of the relations for second derivatives we have, for example, 1 n- 1! In these examples f is used to denote a symbol by means of which the quadratic form is expressed, and u, t' are actual functions oi the variables. This notation will be generally followed, that is to say, f's distinguished by means of indices are the symbols of the quadratic form, the letters u, v, w, etc.

Exactly as in the ordinary symbolism for algebraic invariants, there are many different symbolic expressions for a given invariant, due to the fact that equivalent symbols are used. For many symbolic identities Maschke's paper quoted above should be consulted. We content ourselves with giving one or two illustrations. Again, it is easy to show that. This covariant may be written fa fz ua z. For further work we require the higher covariant derivatives of a given function of the variables. Following Maschke's notation which differs from that used in Chapter II we use: It follows by means of the expression already calculated for zfk that: The higher covariant derivatives can now be calculated as soon as we know the covariant derivative of an invariantive constituent, for we know that to differentiate a product covariantively we apply the same rule as in ordinary differentiation, e.

The expression of the quadriliuear form G":: Using this, we have n -1! Ga may now be calculated, and n-1!

These symbolic expressions for the G's put in evidence at once the fact that they are invariants. For further work on this part of' the subject Maschke's papers should be consulted. Take the equation of the surface to be then U is F. They are, however, two interpretations of one and the same general formula.

We now consider some applications of the theory of differential invariants. Suppose first that the quadratic form is interpreted as the square of the element of length in a certain n-dimensional manifold. They are differential, and involve only one set of the independent variables. They thus give the intrinsic character of the manifold at, and infinitely near, a particular point of it.

If we take account of differential parameters, they express quantities intrinsically connected with the section of the fundamentaltnanifold by another manifold. But we have a set of quantities, defined geometrically, of just this type, that is to say they depend only on the manifold at or near some particular point and are intrinsically connected with it.

Let us consider more in detail the particular case of a surface in Euclidean space of three dimensions. This is not intrinsically determinate if only one quadratic form, that for dB", is given. The catenoid the surface of revolution obtained by revolving a catenary about its directrix and the regular helicoid the surface swept out by a line which is parallel to a fixed plane, intersects a fixed line perpendicular to the plane, and rotates uniformly about the fixed line as its point of intersection moves uniformly along that line are two surfaces with the same quadratic form for dsl.

Two surfaces which have the same quadratic form for dr are said to be applicable to each other. If we regard a surface as made of some perfectly flexible inextensible material, then it is clear that the surfaces into which it may be deformed are the surfaces applicable to it. The coefficients of the second form cannot be chosen arbitrarily if the first is given, though they involve a certain amount of arbitrariness. It follows that the geometrical magnitudes at a particular point on the surface are the differential invariants of two quadratic forms.

Among all these geometrical quantities, there are some that are the same for all surfaces applicable to each other. These, it is clear, do not depend on the second quadratic form, and they are therefore invariants and parameters of a single quadratic form, that for dil'. It follows, conversely, that any invariant or parameter of the form for ds2 represents some geometrical magnitude associated 'vith the surface, and the magnitude is the same for all surfaces applicable to that given.

What is Kobo Super Points?

On this account, such iuvariants are called deformation in variants. Geometrical interpretation of invariants. In order to apply the invariant theory, we have now to interpret our invariants in terms of geometrical n1agnitudes, and also to interpret geometrical magnitudes in terms of invariants. It is perhaps worth while to point out the advantages gained by the use of invariant theory. In the first place we are able to apply in the simplest possible way the methods of analysis to geometrical problems, for we express all our data analytically, and yet avoid extraneous properties which arise through the relations of our surface to a particular coordinate frame chosen.

Again, when we express a given quantity as far as possible by means of invariants, it may happen that its expression only involves invariants, and thus its invariance becomes intuitive. It follows at ouce that 02V 02JT a: This example is a good illustration of the advantages derived from the invariant theory. V and express it as far as possible in terms of invariants. It happens that it is entirely expressed by means of invariants, and hence it is itself an invariant.

Further, when it is thus expressed in terms of invariants only, its general value may be at once written down. Another application of the differential invariant theory can be made by means of the theory of algebraic invariants. All our differential invariants and parameters are invariants of algebraic forms.

By means of the algebraic theory we can determine syzygies or algebraic relations connecting these invariants, and such syzygies, when expressed in terms of the geometrical magnitudes of the surface, lead to algebraic relations among these apparently independent quantities. A surface is intrinsically determinate if two quadratic forms are given, and hence all its geometrical magnitudes are invariants of two quadratic forms. We are thus able, by means of the invariant theory, to determine which among these magnitudes are independent, and, by means of syzygies, to determine all the relations connecting those that are not independent.

For illustrations of this part of the subject the reader should consult the latter part of Forsyth's memoir t. The case of a surface in ordinary space-the quadratic form in two variables. In this case there is only one Riemann symbol, the quantity G. The set of algebraic forms is now i the fundamental quadratic, ii cp,.

All the deformation invariants and parameters of a surface are therefore given by the algebraic invariants of these forms. We thus have the geometrical interpretation of b. An important paratneter of the second order is that known as Beltrami's second differential parameter ll. In fact it is easy to prove that if cfl and. All the parameters may be expressed in terms of three of them, and their derivatives.

We shall prove that all invariants can be obtained by algebraic combinations of these, a result due to Beltrami. Suppose tbat the quadratic form has been transformed so that cfl, tf! It follows, in this case, that all the invariants may be obtained by repeated application of the operations d, b. We know from the algebraic theory that there are three independent invariants of the second order involving one function cfl. These must be b. There are four of the third order, and these are included in b. In particular, suppose that , tf! For example, the Gaussian invariant of the third order may be written either b.

Geometrical properties expressed by the vanishing of invariants. We next consider the geometrical properties involved iu the vanishing of certain invariants. This curve of course imaginary must therefore be such that the tangent at any point meets the circle at infinity. Such curves are analogues of the straight lines '! The curves are hence such that the distance between any two points on one of them, measured along that curve, is zero.

It is proposed to express the fundamental quadratic in the form P. H au av ev' H ov au au' where f is a certain function of u and f', and the second. Thus the quadratic form becomes i. Applicability of two surfaces. We no'v consider the general problem of the applicability of two surfaces. Choose on the first surface any point P, and take as para1netric curves v the geodesics through this point, the curves 'lt being their orthogonal trajectories. Now suppose that K and Kt are not constants.

If these equations are inconsistent, the surfaces are not applicable. Again there are the three possibilities before mentioned. Suppose first that K and 4K are independent of ea. It follows at once from the general expression for the form given on p. If K and AK are dependent on one another, we take K and A 2K as the variables of the form, and the necessary and sufficient conditions become in this case tl.

The quadratic form in three variables. For this case we only note the significance of a few of the more important invariants of lowest order. The geometrical interpretation of those of orders one, two and three for a form of rank zero, the case of ordinary Euclidean space, has been considered in detail by Forsyth in his memoir, already quoted, on the differential invariants of space.

We note that there are six Riemann symbols. If they all vanish, the space is Euclidean. It follows that they are geodesics in the general space. Condition that a family of surfaces form part of a. We next consider the questions: The equation for the particular case of the form da:?. The reader will find it instructive to compare the method given here for the general case with that of Darboux for the particular case. It is in fact the algebraic invariant of four ternary forms, three quadratic and one linear, which is linear in the coefficients of each of the quadratic forms, and cubic in the coefficients of the linear form.

This result might have been obtained by contravariant differentiation, starting with the expression l ara u would have been of the same character and the final result of tbe same form, the only difference being that each contravariant derivative involved would be replaced by its reciprocal covariant derivative with reference to the' fundamental form. The quadratic form in n variables. A general method for dealiug with a manifold for which ds' is given is due to Ricci and Levi Civita. There is thus associated a direction with each point in the manifold, and, if we start from a given point and proceed always in the direction associated with the point reached, we finally obtain a curve in the manifold.

Hence the equations given above define a congruence of curves in the manifold, such that there is one and only one curve through each point of the manifold. Now take n such congruences, denoted by A. Let it be assumed that all the curves of these congruences cut each other orthogonally, then lA.

Such a set of congruences is called an ortkogonal ennuple; the notation [1], [2], Any covariant or contravariant system can be expressed in terms of invariants and the coefllcients of an orthogonal ennuple. It follows that if the members of the system X are all zero, the invariants c are all zero. Thus any absolute system of equations may be expressed by the vanishing of a set of invariants. A particular case of the theorem just proved is that the covariant derivatives of the A's may be expressed in terms of the A's themselves.

It therefore follows that y. Thus the number of independent invariants y is in" n Coefllcients of rotation of an ennuple. It is clear, from what has been said, that the geometrical properties of the ennuple are all contained in the invariants y. These invariants are called the coefficients of rotation of the ennuple. By means of this identity we can determine the condition that a congruence may be normal.

A congruence is said to be normal if its curves can be considered as the orthogonal trajectories of a family of surfaces f llh, Since a vector field on N determines, by definition, a unique tangent vector at every point of N , the pushforward of a vector field does not always exist. By contrast, it is always possible to pull back a differential form.

A differential form on N may be viewed as a linear functional on each tangent space. Precomposing this functional with the differential df: The existence of pullbacks is one of the key features of the theory of differential forms. It leads to the existence of pullback maps in other situations, such as pullback homomorphisms in de Rham cohomology.

To define the pullback, fix a point p of M and tangent vectors v 1 , There are several more abstract ways to view this definition. This is a section of the cotangent bundle of M and hence a differential 1 -form on M. Pullback respects all of the basic operations on forms. The pullback of a form can also be written in coordinates. Assume that x 1 , Each exterior derivative df i can be expanded in terms of dx 1 , The resulting k -form can be written using Jacobian matrices:.

A differential k -form can be integrated over an oriented k -dimensional manifold. There are several equivalent ways to formally define the integral of a differential form, all of which depend on reducing to the case of Euclidean space. Let U be an open subset of R n. Give R n its standard orientation and U the restriction of that orientation. Such a function has an integral in the usual Riemann or Lebesgue sense. Fixing an orientation is necessary for this to be well-defined.

Riemann and Lebesgue integrals cannot see this dependence on the ordering of the coordinates, so they leave the sign of the integral undetermined. The orientation resolves this ambiguity. First, assume that there is a parametrization of M by an open subset of Euclidean space. That is, assume that there exists a diffeomorphism. In coordinates, this has the following expression. Fix a chart on M with coordinates x 1 , In general, an n -manifold cannot be parametrized by an open subset of R n.

But such a parametrization is always possible locally, so it is possible to define integrals over arbitrary manifolds by defining them as sums of integrals over collections of local parametrizations. To make this precise, it is convenient to fix a standard domain D in R k , usually a cube or a simplex.

That is, it is a collection of smooth embeddings, each of which is assigned an integer multiplicity. Each smooth embedding determines a k -dimensional submanifold of M. If the chain is. This approach to defining integration does not assign a direct meaning to integration over the whole manifold M. However, it is still possible to assign such a meaning indirectly because every smooth manifold may be smoothly triangulated in an essentially unique way, and the integral over M may be defined to be the integral over the chain determined by a triangulation.

There is another approach, expounded in Dieudonne , which does directly assign a meaning to integration over M , but this approach requires fixing an orientation of M. On this chart, it may be pulled back to an n -form on an open subset of R n.

Invariants of quadratic differential forms - CERN Document Server

Here, the form has a well-defined Riemann or Lebesgue integral as before. It is also possible to integrate k -forms on oriented k -dimensional submanifolds using this more intrinsic approach. The form is pulled back to the submanifold, where the integral is defined using charts as before.

Fubini's theorem states that the integral over a set that is a product may be computed as an iterated integral over the two factors in the product. This suggests that the integral of a differential form over a product ought to be computable as an iterated integral as well. The geometric flexibility of differential forms ensures that this is possible not just for products, but in more general situations as well.

Under some hypotheses, it is possible to integrate along the fibers of a smooth map, and the analog of Fubini's theorem is the case where this map is the projection from a product to one of its factors. Because integrating a differential form over a submanifold requires fixing an orientation, a prerequisite to integration along fibers is the existence of a well-defined orientation on those fibers.

Let M and N be two orientable manifolds of pure dimensions m and n , respectively. Following Dieudonne , there is a unique. More precisely, define j: That is, suppose that. In particular, a choice of orientation forms on M and N defines an orientation of every fiber of f. The analog of Fubini's theorem is as follows.

As before, M and N are two orientable manifolds of pure dimensions m and n , and f: Fix orientations of M and N , and give each fiber of f the induced orientation. Moreover, there is an integrable n -form on N defined by. It is also possible to integrate forms of other degrees along the fibers of a submersion. Integration along fibers is important for the construction of Gysin maps in de Rham cohomology.

Integration along fibers satisfies the projection formula Dieudonne The fundamental relationship between the exterior derivative and integration is given by the Stokes' theorem: A key consequence of this is that "the integral of a closed form over homologous chains is equal": This case is called the gradient theorem , and generalizes the fundamental theorem of calculus. This path independence is very useful in contour integration.

This theorem also underlies the duality between de Rham cohomology and the homology of chains. On a general differentiable manifold without additional structure , differential forms cannot be integrated over subsets of the manifold; this distinction is key to the distinction between differential forms, which are integrated over chains or oriented submanifolds, and measures, which are integrated over subsets. The simplest example is attempting to integrate the 1 -form dx over the interval [0, 1]. By contrast, the integral of the measure dx on the interval is unambiguously 1 formally, the integral of the constant function 1 with respect to this measure is 1.

Similarly, under a change of coordinates a differential n -form changes by the Jacobian determinant J , while a measure changes by the absolute value of the Jacobian determinant, J , which further reflects the issue of orientation. In the presence of the additional data of an orientation , it is possible to integrate n -forms top-dimensional forms over the entire manifold or over compact subsets; integration over the entire manifold corresponds to integrating the form over the fundamental class of the manifold, [ M ].