Theory of projective connections on n-dimensional spaces.

Transcription

1 Theorie des projektiven Zusammenhangs n-dimensionaler Räume, Math. Ann. 6 (932), Theory of projetive onnetions on n-dimensional spaes. By D. van Dantzig in Delft. Introdution.. The relations between the various projetive differential geometries of Cartan [- 3], Shouten [-3], J.M. Thomas [], T. Y. Thomas [-6], Veblen [-7], and Weyl [-2] have reently been larified in two works of Shouten and Golab ). It has thus been demonstrated that these various geometries an be regarded as speial ases of a more general theory of projetive onnetions under a ertain assumption (f. 4, 49). Nonetheless, the theory an hardly be regarded as omplete; for one thing, it has many unsatisfying aspets from a purely formal standpoint. First, the asymmetry of the indies for projetive quantities is an impediment, i.e., the speial role of the index as opposed to the indies,, n. Seond, it is disturbing that in general the ovariant derivative is not assoiated with a ovariant differential. Indeed, the impossibility of defining suh a well-defined affine displaement without restritions has already been proved by Shouten and Golab 2 ), but the deeper basis for that fat is still not entirely obvious. Third, it is not entirely lear whether one is neessarily led to the assoiated admission of densities. Fourth, the frequent appearane n+ of the fator is partiularly astounding 3 ). One does not ompletely see whether the exponent must be preisely, muh less whether is a determinant of degree n and n + not (n+). Fifth, up till now there is no theory for the indution of a projetive onnetion in an embedded spae. 2. The goal of the present work is to seek to eliminate these beauty marks from the theory by way of a different type of representation. The key to making this possible is given by the first of the aforementioned remarks: the speial role of the index. This Translated by D.H. Delphenih. J. A. Shouten and St. Golab, Über projetive Überträgungen und Ableitungen, I. Math. Zeitsr. 32 (93), pp , II Annali di Mat. (4) 8 (93), pp (The first of these artiles will be denoted by Sh and G.) 2 Lo. it. ), pp Cf., e.g., Sh. and G. 2; Veblen [4] (u M ); Veblen and J.M. Thomas (θ); T.Y. Thomas [], log n x σ et. +

2 Projetive onnetions in n-dimensional spaes 2 very same diffiulty has been enountered previously in ordinary projetive geometry; there, one sueeded in removing it by the introdution of homogeneous oordinates. To be sure, homogeneous oordinates were also introdued into projetive differential geometry by Cartan, but always only in linear spaes that were assoiated with eah point of spae, never in the spae itself 4 ). (By ontrast, Weyl has even reently replaed the homogeneous oordinates in linear spaes with inhomogeneous ones.) However, it will now be shown that the introdution of homogeneous ur-variables asts a thoroughly illuminating light on some of the phenomena that were just mentioned. For example, one immediately sees that a type of density an appear. If v is a point of the assoiated linear spae then the numbers that determine it are generally only defined up to a ommon fator, i.e., they are homogeneous funtions (e.g., of degree r) of the homogeneous ur-variables x (f. 3, 7) If one then replaes the x with ρx then v takes on the fator ρ r. If one regards the replaement x ρx as a oordinate transformation (whih is usually not reommended), and is the determinant of the transformation then one has ρ n+, and v r is therefore a density 5 ) of weight r n +. The densities of weight are therefore, to a ertain extent, quantities of first n + degree (on this, f., however, 4). Furthermore, the lak of a ovariant differential in the general ase beomes selfexplanatory. There also exists no ordinary differential: the dx are not homogeneous funtions of the x (f. 3, 9). The basis for the lak of a ovariant differential is then the same as the basis for the lak of a point differene in ordinary projetive geometry. However, this shows that one an ahieve the existene of a ovariant differential by a speialization of the displaement ( 7, 23) and that this an even ome about by means of a path-preserving hange of the displaement ( 2, 38, 4). The theory of geodeti lines ( ), of path-preserving transformations ( 2), and of m+ P in n+ P is then effortlessly arried out. Then one sees that a projetive onnetion is uniquely determined for a given urve by inter alia the requirement that the quantities Q µ that are defined in 6, 7 must vanish ( 2, 43). However, the vanishing of the salar Q µ x µ + is ruial for the existene of the ovariant differential ( 7). This existene is then inseparable from the unique determination of the projetive onnetion, as Shouten and Golab 6 ) have already proved. 3. A ruial element of this entire style of representation was the wish that the essential features of ordinary projetive geometry should be preserved to the greatest extent possible, and that projetive differential geometry should not degenerate into a thinly-veiled affine geometry. The relationships with inhomogeneous oordinates ( 3, 4 Exept for Eulidian spaes and as loal oordinates at a single point of a general spae. In both ases, one needs to onsider only the ordinary projetive group L n+, not, however, our general homogeneous group H n+ ( 2, 6). 5 For the analysis of densities, f. J.A. Shouten and V. Hlavatý, Zur Theorie der allgemeinen linearen Überträgung, Math. Zeitshr. 3 (929), pp Cf., also footnote? 6 Cf. lo. it.

3 Projetive onnetions in n-dimensional spaes 3 4) are also established only by means of more reassuring links with the older theory; as usual, they are quite onsiderable. The geometry of Shouten and Golab is briefly skethed in 4 on the same basis; we show that it follows from Veblen s theory (and thus all of the older theories), as well as our own. Furthermore, a standpoint will be prinipally taken that is as general as possible. Thus, we sueed in establishing not just suffiient, but also neessary onditions for, e.g., the existene of ovariant differentials, the map to an infinitesimally lose n+ E, and geodeti lines. An overview of the omplete set of onditions that were introdued, along with the most important simplifiations in the general formulas, will be given in. The relationships that are employed draw upon Shouten s Rii Calulus 7 ) in an essential way, in whih most of the hanges that have, in the ourse of time, proved to be useful will be inorporated. Some inessential deviations (f. footnotes?,?,?) are, however, introdued, largely for the purpose of avoiding the vast array of indiial notations, whih ompletely obsure any intuition about the nature of the formulas. As for the form of the representation that is desribed here, exept for the Shouten Rii alulus, the only older works that had an essential influene on it were those of Cartan and Veblen; indeed, the method of Veblen is losest to the one that is presented here. 4. By the introdution of the homogeneous group H n+, whih is atually new to the theory, projetive differential geometry takes on the harater of a generalization of ordinary projetive geometry to a far greater degree that has been true up till now, in partiular, when one does without the existene of a ovariant differential and sets Q µ. Differential geometry may then be immediately disussed in the n+ E by means of the Ansatz Π ; the system of geodeti lines will then be given by the system of straight lines in n+ E. Therefore, the most important thing is then that one an immediately disuss the differential geometry of an embedded manifold m+ H, assuming that this is desired. In the event that m+ H is given by equations in the x, instead of a parametri representation, the theory is easily altered orrespondingly in 9. Moreover, the onnetion with the A 8 n+ ) yields the possibility of a generalization of the theory. Just as in ordinary projetive geometry, the n+ E are easily seen to be the geometry of the lines in an E n+ that inludes a fixed point O relative to the subgroup of the affine group of E n+ that leaves this system of lines invariant, and thus one an also regard the n+ P as an L n+ in whih a system of n lines (i.e., X will be indued under Eulidian translation) through a fixed point O is defined; the group of arbitrary oordinate transformations of the L n+ will be orrespondingly replaed by the only group that takes the system of lines to itself and indues a linear group (with O as fixed point) on any line. The aforementioned generalization now onsists of replaing the sphere L n+ with a ruled surfae L n+, or even L n+m, i.e., with a spae in whih a system of n pairwise distint lines, E n, resp., hene, Eulidian subspaes, that an be regarded as points of new n-dimensional spae. The group of L m+n will be orrespondingly redued to those transformations that, first of all, leave the system of n E m invariant, and 7 J.A. Shouten, Der Rii-Kalkül, Berlin; Julius Springer 924. Denoted by R.K. 8 Cf. T.Y. Thomas [4], Sh. and G., 6, Veblen and Hoffmann, pp. 8, and the works ited there by H. Mandel and J. H. C. Whitehead.

4 Projetive onnetions in n-dimensional spaes 4 seondly, indue a linear transformation in eah E m. A possible generalization of another sort will be briefly indiated in the onluding setion. Finally, we must remark that Veblen s projetive differential geometry gives us a quadrati differential form (Veblen [6]) quite easily by means of the method followed here. One will probably obtain a new onnetion and a new glimpse into onformal differential geometry, whih I hope to show on a later oasion. 2. The group. 5. In elementary projetive geometry, one ordinarily introdues the (n+)-ary ( n- dimensional) projetive spae n+ E 9 ) in the following way: In an ordinary n-dimensional spae E n, whih is given by Cartesian oordinates ξ i (h,, l,, n), one sets: () x x, i ξ x i (α,, ω,,, n), in whih x is an arbitrary variable that is. Any n+ numbers x that do not all vanish and satisfy the relation () will be regarded as the homogeneous oordinates of the point ξ i. One then ompletes E n by means of an imaginary hyperplane x and seeks invariants under the group L n+ of homogeneous linear oordinate transformations x x ), in whih: 9 The left-hand upper index indiates the point value of the spae (f., P.H. Shoute, Mehrdimensionale Geometrie, Sammlung Shubert, I, pp. 2), i.e., the number of dimensions plus one. We will all a spae with a point value of n+ (n+)-ary (binary, tertiary, et.). In the older theory, oordinate transformations were mostly indiated by hanging the kernel symbol (x y ) or by attahing a prime to the kernel symbol ( x x ). However, beause we (with J. A. Shouten) will view the quantities as geometrial strutures throughout, independently of any oordinate system, it is desirable to always indiate geometrial struture by the same kernel symbol and avoid the rather inessential hange in the oordinates (numbers that determine them, resp.) defined by hanging the indies. Eah oordinate system will then be assoiated, firstly, with a series of n+ fixed symbols, and seondly, with a type of notation in whih an index of the type in question is regarded as a variable that ranges through the assoiated series of symbols. The fat that the series of symbols must also be different for different oordinate systems thus implies, e.g., that the null omponent of a vetor must be distint in two different oordinate systems. Consequently, in R.K., the newer method in the later works of J.A. Shouten (sine, perhaps, 928) is still not ompletely followed (different series of symbols, but only one type of notation and primes on the kernel symbol). In the long run, however, this leads to a muh too large number of notations, as in Sh. and G, I: ι,, ω a,, a n ; Ι,, Ω A,, A n ; h,, m,, n; H,, M, ɺ, nɺ ; a,, g,,, n; A,, G θ,, ɺ, nɺ, all of whih are still in the loal E n. To simplify, we would thus like to indiate different oordinate systems in the same spae by the same notation (for the index) and distinguish them from eah other by attahing a prime to the index, or a point, or underlining the index, et. (e.g., x ɺ, x, x, x, x, x, et.).

5 Projetive onnetions in n-dimensional spaes 5 (2) x A x µ µ ( α,, ω,,, n ) and the A µ are any (n+) 2 onstants with non-vanishing determinant. One an also proeed in a somewhat different way as follows: One starts with an E n+, whih is given by the Cartesian oordinates x, and subjets them to homogeneous linear oordinate transformations of the given group L n+. Let E + be the spae that results from omitting the origin O from E n+. Two points x, x of E n + are ollinear with O when and only when: (3) x ρx, ρ. This relation between x and x is reflexive, symmetri, and transitive, and is invariant under the group L n+ ; it may then be regarded as an equivalene relation, whih we will also refer to as oinidene. The sets of oinident points, hene, the lines through O, may be regarded as points of a new n-dimensional spae that we denote by n+ E. In order to avoid onfusion, we briefly refer to a point of n+ E as a (ontravariant) position, whereas we reserve the expression (ontravariant) point for the points of E n +. There exist the following relations between the three notions of point, position, and system of n+ numbers : A system of n+ numbers that do not all vanish uniquely determines a point, as well as a position, as long as one says whih oordinate system they belong to. A given point will thus be represented by different systems of numbers that determine it in different oordinate systems, whih relate to eah other as in (2); in a single oordinate system, however, different systems of numbers determine different points. However, in a single oordinate system two different systems of numbers an very well determine the same position, namely, from (2), when they differ only by the same fator ρ. In pure projetive geometry, only the onept of position has any meaning. Points, along with systems of numbers, serve only to failitate omputation; they may not, however, enter into projetive-geometrial theorems. Finally, one may onsider funtions f(x ) that depend upon only the ratios of the x up to a fator ϕ(ρ) that depends only upon ρ: n The assoiated series of symbols is a series of number with the same alteration (e.g.,,,,, ɺ, ). We thus use: α,, ω,,, n, α,, ω,,, n, in n+ H, α,, ω,,, n, et. h,, l,, n, in X n, h,, l,, n, et. a,, g,, m, in an embedded n+ H. a,, g,, m, et.

6 Projetive onnetions in n-dimensional spaes 6 (4) f f(ρx ) ϕ(ρ) f(x ). One easily proves that ϕ(ρ) always has the form ρ r, in whih r is any onstant number (the degree of f(x)): The funtions are homogeneous: (5) f ρ r f. Equation (5) is equivalent to the Euler homogeneity ondition: (6) x µ µ f rf, µ x µ or also to: x µ µ log f r. If r, hene, f f is homogeneous of degree zero, then we all f a funtion of position. If f is homogeneous of r th degree then the partial derivatives µ f are homogeneous of (r ) th degree: f ( ρ x ) (7) µ f ρ r- µ µ f, ρ x (8) x f (r ) µ f;. µ. 6. We would also like to introdue homogeneous oordinates when the n- dimensional spae is subjeted to the general group G n of all uniquely ontinuously invertible and suffiiently many times ontinuously differentiable transformations (in this ase, the spae will be denoted by X n ). If X n is given by the ur-variables ξ i then we further take an arbitrary variable x ) and introdue the x preisely as before by way of (). Now, if the ξ i are subjeted to an arbitrary transformation ξ i ξ i from G n, then the ratios of the new homogeneous oordinates x are funtions of the ratios of the x, i.e., the x are themselves homogeneous funtions of null degree of the x, up to a ommon fator. We further assume that the funtion is homogeneous of first degree 2 ) 3 ) in the x (f., however, 5). It follows that the x are also homogeneous of first degree, and, moreover, that they are unique and suffiiently many times differentiable in the neighborhood in question, whereas the funtional determinant itself is nowhere vanishing. This x x orresponds to the e of Veblen [5]. [6], [7]. 2 Suh a thing is ompletely different from a linear funtion. The latter has the form a x, in whih the a are onstants; the former an also be given this form (but not uniquely); however, the a are then arbitrary funtions of the ratios of the x. 3 The ondition that the degree shall equal one an also be omitted, provided that it is? (otherwise, the x would be dependent). Cf., also footnotes?) and?), along with 5.

7 Projetive onnetions in n-dimensional spaes 7 The situation beomes learer when we hoose the seond path. The total system of n+ numbers x define an (n+)-dimensional spae X n+. If it is subjeted to the group of all uniquely invertible and ontinuous and suffiiently many times differentiable transformations then neither the notion of line through the origin nor the notion of origin itself is invariant. We thus redue from the group G n to the subgroup H n+, whih onsists of all transformations x x degree 4 ) in the x : (9) x µ x ( ρ x ) ρ x µ ρ x ( x ). in G n+ for whih the x are homogeneous of first We denote an X n+ with this sort of redued transformation group by H n+. The (now invariant) lines through the (now invariant) origin of this H n+ an, moreover, be regarded as points of a new spae that we denote by n+ H. We one again all them (ontravariant) positions, and one again reserve the term (ontravariant) position for the points of H n +, i.e., H n+ without the origin. In order to give a position, one must give, first, one oordinate system, and seond, a system of n+ numbers that do not all vanish. The group H n+ inludes a subgroup H that onsists of the oordinate transformations in H n+ for whih x x, and whih is (einstufig) isomorphi with the group G n of transformations of the ξ i ; the group H n+ is then no less enveloping than the group G n. The fat that H n+ an be ompleted by an imaginary manifold x is inessential sine one must usually restrit oneself to a neighborhood (that is hosen to be arbitrary small) in differential geometry, in suh a way that the imaginary manifold an be ignored. One must observe that the map x x, although it is a hange in the numbers that determine the same point, is not to be viewed as a oordinate transformation. Namely, from the degree ondition (9), this is possible when and only when the proportionality fator ρ is a funtion of position, hene, homogeneous of null degree in the x. The invariane requirement, upon whih the definition of projetors ( 3) rests, thus relates only to the group H n+, not, however, to the hange of fator (3) when ρ is not homogeneous of null degree. We now all a differential geometry in n+ H purely projetive when its theorems involve only the notion of position, but not the notion of point. The differential geometry that is introdued and developed in the following setions is not purely projetive, but projetive in a broader sense, sine we are indeed working with homogeneous funtions exlusively, but the degree of these funtions will also be onsidered, whih introdues a non-projetive element. 4 Cf., the previous footnote.

8 Projetive onnetions in n-dimensional spaes 8 3. The quantities. 7. The group H n+ indues a linear group L n+ L n+ (x ) at every point x. It is omposed of matries with onstant oeffiients, whih ome about when one substitutes the oordinates for the point in question into the funtions: () A µ x µ, µ. x µ The inverse matrix to () is: () A µ x µ, µ x µ. For the sake of later use, we remark that the A µ (and likewise the A µ ) are homogeneous of null degree, hene, they are pure funtions of position: (2) A µ A µ (ρx ) A µ. Due to the Euler homogeneity ondition this is equivalent to: ρ (3) x A. ρ µ We now assoiate eah point x with an E n+, in suh a way that eah oordinate transformation of H n+ from H n+ indues a oordinate transformation of E n+ from L n+, namely, the one that is given by the assoiated matrix (). In order to give a point of E n+ one must first give a oordinate system (e.g., x ) in H n+, and seondly a system of n+ numbers. The oordinate system in H n+ is, in fat, uniquely assoiated with a oordinate system in E n+ ; the given numbers are the oordinates of the points of E n+ relative to this assoiated oordinate system. If v are the oordinates of the same point relative to the oordinate system in E n+ that is assoiated with the oordinate system x then the v must go over to the v by means of a transformation of L n+ : (4) v A µ v µ ; v A v µ µ. A point field in H n+ is omposed of a system of n+ (single-valued, ontinuous, suffiiently many times differentiable) funtions of the oordinates that one of the oordinate systems in H n+ is assoiated with, in suh a way that the funtions that are assoiated with two different oordinate systems go to eah other by means of assoiated oordinate transformations of L n+ aording to (4). Geometrially, a point field simply means that eah point of H n+ is assoiated with a point of the assoiated ( loal ) E n+.

9 Projetive onnetions in n-dimensional spaes 9 Now, if x, x are any two oinident points of H n+ then we identify the two assoiated E n+ with eah other in suh a way that we an identify eah point of one E n+ with that point of the other E n+ that has the same oordinates. This ondition is satisfied for any oordinate system whenever it is satisfied for one oordinate system, sine the A µ have the same values at x and x, due to (2). Thus, the loal n+ E (whih we onstrut out of the E n+ as in 2, 5) to whih the oinident points of H n+ belong will be equivalent and an be identified with eah other. In general, a point field assoiates a one-dimensional family of points with eah position in n+ H, hene, a urve in the loal E n+. In the event that this urve is a straight line through the origin of E n+, or also ontrats to this origin, we all the point field a position field. Suh a onstrution thus assoiates eah position in n+ H with a unique position in the assoiated loal n+ E or with the origin. Thus, in order for a point field to be a position field it is neessary and suffiient that the v v (ρ x ) be proportional to the v v (x µ ). We would like to further assume something rather far-reahing, that (i.e., we restrit ourselves to suh positions fields for whih) the proportionality fator is a power of ρ whose exponent r (the degree 5 ) of v ) is onstant over the n+ H, i.e., that the system of numbers are all homogeneous funtions of the same (r th ) degree in the ur-variables: (5) v ρ r v. 6 ) 5 Our notion of degree has nothing to do with the notion of the degree of an affinor (R.K., pp. 23), i.e., it is atually the degree (number of indies) of the assoiated form, whih is often also alled the degree. It appears here for the first in the ordinary elementary sense (the degree of the system of numbers when onsidered to be funtions of the ur-variables) of differential geometry, in whih, up till now, one does not ordinarily bother with the type of funtional dependeny of the quantities upon the ur-variables (exept for differentiability, resp., analytiity, requirements). 6 One an also regard the assoiation x ρx as a oordinate transformation (whih is not possible for a general ρ in our theory; f. 2, 6, onlusion) and introdue densities of index (weight) r n + instead of quantities of r th degree. Suh a transition must then transform under the unimodular group ( Det (v x ) ) aording to (4), although like a point density aording to: (4a) r n+ V A µ µ V. However, it seems to me that the notion of volume (and thus, the notion of density) possesses a typially non-projetive (affine) harater to a far greater degree than the notion of degree, in suh a way that statement of the theory that is given in the text has merit. One will first arrive at a purely projetive theory (f. 2, 6, onlusion) when one ombines the Ansatz (4a) with the footnote?) that was mentioned, and defines a position by way of: (4b) v r A µ v µ with a ompletely arbitrary fator r; the geometry that is thus defined will show a ertain agreement with the theory of pseudo-quantities of Shouten and Hlavatý (f.?). On this, f. 5.

10 Projetive onnetions in n-dimensional spaes Two position fields u r, v r are oinident when and only when they differ by a salar fator (see below): u p v. In order for this to be true, it is neessary and suffiient that: u [ v µ]. 8. Analogously, we define a ovariant position field w µ of degree r by means of: (6) w µ A w, w µ A w (7) w µ ρ r w µ. µ µ, Geometrially, w µ desribes a hyperplane ( n E) in the loal n+ E. ontravariant position v when and only when: It inludes the (8) v ρ w ρ. We will define general quantities that we would like to all projetors, orresponding to the affinors of affine geometry 7 ), as a system of (n+) r+s homogeneous funtions of r th degree that transform like a produt of r ovariant and s ontravariant points: s ρ ρr v vs i i σ σ s (9) X A X 8 ) (2) r r σ σ s ρ ρr s X ρ r s X r. r Geometrially, a projetor desribes some algebrai relationship in the loal n+ E. As usual, one an also refer a quantity to two or more different oordinate systems. Thus, e.g., in X µ, both of the ovariant indies are referred to the system of x and the ontravariant index, to the system of x : (2) X ρσ A X A X µ µ ρσ ρ µ. Finally, a salar is a homogeneous funtion of r th degree that assumes values that are independent of oordinate system at eah point. A salar is a position funtion when and 7 We use the expression affine geometry, as opposed to projetive (differential) geometry, for the general linear displaement IIIA α in the lassifiation sheme of R.K., pp. 75, hene, without regard for the symmetry ondition. 8 For the sake of simpliity, we shall not reall the notation for unit-projetors and differential symbols: A σ A A ρ σ, ρ Bσ B B σ, 2 ρ ρ µ, µ. x x µ

11 Projetive onnetions in n-dimensional spaes only when its degree is zero. Two quantities that differ only by a salar fator have the same meaning geometrially. A salar of r th degree may always be brought into the form p r q (e.g., (x ) r q), in whih p is an arbitrarily hosen, but fixed, non-vanishing salar of first degree and q is a salar of null degree. This was done by Veblen (f., footnote?) and?)). 9. Examples. a) The ontat position x. By means of the Euler homogeneity ondition: (22) x ρ A ρ ρ x ρ x x, whih is equivalent to (9), v x satisfies equation (4), i.e., the n+ numbers determine a point of the first degree in E n+ that is assoiated with the point x in H n+ ; we an then think of the point x in H n+ as being identified with the point x in the assoiated E n+, just like the assoiated position in n+ H and n+ E. In affine geometry (f., last footnote) the x v transform nonlinearly; there, the ontat position orresponds to the null point (null-vetor) in E n+. b) The unit projetor A. If both of its indies refer to the same oordinate system then its values are equal to (, resp.) whenever both of the indies are equal (different, resp.). However, if one refers the unit projetor to two different oordinate systems e.g., x, x then it represents the funtional matrix x ( x, resp.), whih we have denoted by the same kernel symbol A all along (f., footnote?)). The degree of A is equal to zero (f. (3)). Geometrially, a quantity of the type X always means a singlevalued (but not neessarily one-to-one) projetive map (ollineation) of the loal n+ E onto itself: an arbitrary position v in n+ E will be mapped to the position X v ; A is the identity map. ) The differentials of the x do not determine any position field. Indeed, if they transform aording to (4) (23) dx A µ dx µ ; however, they are not homogeneous: (24) dx dx ρ (dx + x d log ρ). They do define a point field (as long as eah point in H n+ is uniquely assoiated with a line element dx, hene, a point in the loal E n+ ), but the point of E n+ that is assoiated with the oinident points of the position x does not define a line through the origin, but a ompletely undetermined urve in the plane in E n+ that is spanned by origin and both

12 Projetive onnetions in n-dimensional spaes 2 of the points x and dx. The basis for the non-existene of a ovariant differential that was mentioned in the introdution resides in equation (24). d) On the other hand, any infinitesimally neighboring position y x + dx in n+ H uniquely determines, up to quantities of seond order, a position in n+ E that has the same oordinates relative to any oordinate system in n+ E, up to quantities of seond order, as the position y in n+ H relative to the assoiated oordinate system in n+ H. One then has (up to quantities of seond order!): (25) y µ µ x + dx A x + A dx A y µ µ µ µ and: (26) y x + dx ρx + ρ dx + x dρ (ρ + dρ) y. The position y in n+ H an therefore be identified with the position y in the loal n+ H (that is assoiated with the position x in n+ H), i.e., an infinitesimal neighborhood of the ontat point x in n+ H will be uniquely embedded in the loal n+ E, up to quantities of seond order. Sine y is then embedded in the n+ E belongs to x, along with the n+ E that belongs to y, there thus exists a link (of first order) between the various loal n+ E. We thus point out that the differentials dx are not determined by the being given the positions x and y ; this is the ase only when eah point x (y, resp.) is distinguished. Nonetheless, the position y is determined by being given the position x and the differentials dx ; on the ontrary, any position on the line through x and y may be obtained by a partiular hoie of the point that represents x. e) On the other hand, this onneting line is indeed determined, and therefore the bi-position J x [ yµ] x [ dxµ] : (27) J µ µ A J (28) J ρ (ρ + dρ) J. ρσ ρσ, Therefore, there are no line elements in projetive differential geometry, but only diretions, i.e., lines through the ontat point and an infinitesimally neighboring point in the loal n+ E. J n f) A (ontravariant) (n+)-point is a projetor whose values are +J, J,, whenever the indies v, v,, v n represent an even (odd, resp.) permutation of the numbers,,, n (ontain two equal numbers, resp.). Therefore, J is any funtion ( density ) that takes on the fator under a oordinate transformation. One obtains an (n+)-point as an alternating produt of any n+ linearly independent points. If J is a homogeneous funtion then the (n+)-point also determines an (n+)-position. If J is (in n a ertain oordinate system) equal to one then (for this oordinate system) is the J

13 Projetive onnetions in n-dimensional spaes 3 unit (n+)-point (-position, resp.) 9 ). If one were to take J to be equal to one in any n oordinate system then J would no longer be a projetor, but a density; however, the latter (whih we will not usually use) is not uniquely determined, without further assumptions. g) A ovariant (n+)-point (-position, resp.) is defined analogously. One an hoose ±, for its values, in whih J has the meaning desribed in f). If the o- and J ontravariant (n+)-points are assoiated with eah other in that way then the omputations in 3,, and 3 yield the frequently-used relations: + (29) J σ σ r r n and, in partiular: (3) J ρ (3) J J n ρρ (32) J ρ σ σ (n r)! (r + )! r r+ n J n ρ J n (n r)! ρ J n ρ n! A, ρ ρ n ρ n A (n )! 4. [ r ], [ r ] The m+ H in n+ H. A [ r ], [ r ]. Let an H m+ (m < n) be given in H n+ in suh a way that the x are given as homogeneous funtions of the first degree 2 ) of the m+ homogeneous parameters x a (a,, g,,, m); let these parameters be subjet to the transformation of the group H m+ that orresponds to the group H n+ that was introdued above. At the same time, one is then given a m+ H in n+ H. Furthermore, let the loal spaes E m+ and m+ E be introdued, as above. As usual, there exist the quantities: (33) B a a x, a a x, whih assoiate eah (ontravariant) point (position, resp.) v a in the loal m+ E with a unique point (position, resp.): (34) v B a v a in the loal n+ E, whih we an think of being identified with v a, and for this reason we shall denote them with the kernel symbol v. Thus, the loal m+ E also appears to be a 9 In R.K., pp. 42, the non-vanishing values of the unit n-vetor were hosen to be equal to, instead of n!. We have hanged the fator, sine it is more ustomary to hoose the unit volume of a parallelotope to be a simplex with side. However, formulas (29) to (32) will then beome a little less simple. 2 The ondition that the degree an also be omitted here. Cf., footnote?).

14 Projetive onnetions in n-dimensional spaes 4 manifold that is embedded in n+ E. In partiular, by means of the Euler homogeneity ondition (6), one has: (35) x a B a x a a x x, suh that, in partiular, the point x a in m+ E will be identified with the point x in n+ E. Thus, there are four points (positions, resp.) x that are identified with eah other and lie in n+ H, m+ H, n+ E, and m+ E, respetively. Correspondingly, we have immediately denoted the parameter in m+ H by x a (instead of, e.g., u a ). Eah hyperplane (hene, eah n E) w in the loal n+ E is assoiated with a hyperplane (hene, an m E): (36) w a B a w in the loal m+ E. Geometrially, w a is the intersetion of w with m+ E. Objets on m+ E that are indued from objets in n+ E (or onversely) by intersetion or projetion, whih might not be uniquely determined, will be denoted by attahing a prime to the kernel symbol. For the sake of later use, we remark that B is homogeneous of null degree (in the x a ): (37) x b b B a x b ab x. m. If one introdues a quantity J in m+ H that is analogous to the one in 9. f) then the ontat m+ E in m+ H will also be represented by the (m+)-point: m (38) J m B J or by the ovariant (n m)-point: (39) Thus, one has: (4) J m B t [ ] ρ a m (m + )! B m [ m, m] J J ( m + )! m m+ n. n p, tm + n a (m + p n). Geometrially, (4) represents the ondition for the points of m+ E to be inident with the m+ E that is represented in n E-oordinates. In general, one an represent the tangential m+ E in mixed oordinates by way of: r m (4) t r! B r m r J B n r J n ( r m + ). 2. The ase m is trivial. Therefore, sine the x must be homogeneous funtions of first degree in the single oordinate x, the equations for H ultimately read like: (42) x a x,

15 Projetive onnetions in n-dimensional spaes 5 in whih the a are onstants; H is therefore a single position. For m, 2 H is a binary manifold, hene, a urve. Correspondingly, the tangent through the bi-point will be represented by: (43) J ab B J 2B []. ab For m n the tangential hyperplanes n E or hypersurfaes n H will be given, analogously to (39), (4), by: (44) t n a an J Ba a n n J n! [ n n] (n + ) B[ n ], or by: (45) t A J B J ( n )! n 2 a an 2 a 2 a n n 2 [ 2 ] n(n + ) B[ 2] A B et. The inidene ondition (4) reads like: (46) t B a, or: (47) t B, et. ab n n n n [ µ ] n, 3. m+ H is taut if an n-m E is given at every position in the loal n+ E that has no points in ommon with the tangential m+ E. Let this n-m E be represented by the (n m)- n m point n ; let it be normalized suh that: n m (48) t n (n m)! 2 ). n m If one then sets: µ µ n m (49) B t n µ n m ( n m)! + J J B n m A n m!( n m)! (m + )(n m) then, due to (3), (3), (39), (48), one has: m m m m n B m [ m m+ n] [ m-m] A n, 2 Cf., the first normalization ondition (93) R.K., pp. 57; onsistent with footnote?), the fator in R.K is hosen differently.

16 Projetive onnetions in n-dimensional spaes 6 (5) B B B a a a B a x, a,, a. As in affine geometry, one an also assoiate any ontravariant position v in n+ E with its projetion onto n-m E : (5) v B v, and any ovariant position (i.e., m E) w a in m+ E an be assoiated with its ontat hyperplane (i.e., n E) with n-m E : (52) w B a w a, whih is not uniquely possible without tautness. 5. Projetive onnetions. 4. A projetive onnetion in n+ H an be given in four ways: A. By defining a ovariant derivative. B. By defining a ovariant differential. C. By defining a map from any n+ E to the infinitesimally lose ones ( displaement ). D. By defining a system of urves (geodeti lines; paths ). The four orresponding types of definition of an affine onnetion are equivalent to eah other and will all be governed by a system of n 3 funtions Γ with well-known transformation laws. We will see that here in projetive geometry, as well, the four types of definition will be governed by a orresponding system of funtions, together with a ovariant point, but they are by no means equivalent to eah other: Namely, the orresponding funtions Π annot be hosen arbitrarily, but they must satisfy ertain onditions that are different for A, B, and D. 5. We would now like introdue a orresponding system of (n + ) 3 funtions Π in n+ H, with a orresponding law of transformation: k ij (53) ρ Π A Π + A A, µ µ ρ µ without onern for whih method of definition of a projetive onnetion we have hosen. We would further like to assume that the Π are funtions of position, hene, homogeneous funtions, e.g., of k th degree.

17 Projetive onnetions in n-dimensional spaes 7 6. As in affine geometry there exist: the torsion quantities: (54) S Π [ ] and the urvature quantities: (55) N ωµ 2 [ ] 2 ρ ω µ ρ ω µ ] Π Π Π, whih are homogeneous of k th ((k + ) th, resp.) degree. In ontrast to affine geometry, however, there exist two quantities 22 ) of null order (and (k + ) th degree): (56) P Π x µ, (57) Q µ Π x Pµ + 2S x, as writing out the transformation formulas, by the use of (3), yields immediately. 6. Covariant derivatives. 7. As usual, we would like to establish the ovariant derivative of a quantity by the following three onditions: A. The ovariant derivative of a quantity is itself a quantity (in partiular, it is therefore homogeneous). B. The differene between the ovariant derivative and the ordinary (partial) derivative of a quantity is a homogeneous linear funtion of the values of the quantity. C. The ovariant derivative of produts and ontrations of arbitrary quantities satisfies the Leibniz rule for differentiation. From the seond ondition, it follows that the most general form for a ovariant derivative of a salar of r th degree must be: (58) µ q µ q + Q µ q. r r Thus, Q µ an ertainly be of degree r, but it an no longer be dependent upon q itself. If one defines µ q r aording to (58) then ondition C yields: µ q r rq r µ q; hene, when q is a salar of null degree: Q µ. Furthermore, if the degree r of q is arbitrary and p is a salar of first degree then C, when applied to µ q p r, and together with, yields: 22 These two quantities roughly orrespond to the A ( a A, resp.) in Sh. and G., et. Cf., 4. a Q µ

18 Projetive onnetions in n-dimensional spaes 8 r Q µ r Q µ, in whih we have written Q µ, in stead of Q µ. Hene, (58) takes the form: (59) µ µ q + r Qµ q. Sine µ q is homogeneous of (r ) th degree and µ q must be homogeneous, Q µ must be homogeneous of degree. Sine µ q, like µ q, transforms like a ovariant point, Q µ must also be a ovariant point. 8. Condition B then gives the most general form for the ovariant derivative of a point of degree r: (6) µ v µ v + Π r v. If one substitutes v p r u in this, in whih p is an arbitrary salar of first degree, then u is a point of null degree, and one finds: (6) Π r Π + r A Q µ, when we write Π takes the form:, instead of Π. If one substitutes (6) in (6) then this equation (62) v µ v µ + Π v + r Q v.. µ µ µ The ondition that µ v must be a quantity yields the well-known transformation laws (53) for the Π. Sine the µ v are homogeneous of (r ) th degree, the Π (like the Q µ ) are homogeneous of ( ) th degree: I 23 ) x ρ ρ Π Π ; x Q µ Q µ The quantities P and Q that were introdued in 5, 6 thus have degree. The torsion quantities and urvature quantities have degrees and 2, resp. Thus, we have: 23 ) The numbering by Roman numerals relates to.

19 Projetive onnetions in n-dimensional spaes 9 Theorem. In order for a system of funtions Π with the transformation law (53), together with a ovariant point Q µ, to define a ovariant derivative, it is neessary and suffiient that the Π and the Q µ are homogeneous of degree 24 ). 9. For a ovariant point w of r th degree, appliation of the Leibniz rule to µ w ρ v ρ yields: (63) µ w µ w Π r w, (64) w w Π w + r Q w µ µ µ For a general quantity of degree r one finds, e.g.: (65) s s s i i ρi+ i r r r i X X + Π X µ µ ρµ r ρ Π X + r Q X i µ ρ µ s s i i i+ r r. We will denote a spae in whih a ovariant derivative is defined by means of (59), (62), (64), (65) and satisfies ondition I by n+ P. We should point out that Q µ itself transforms like a ovariant point, although this is not the ase for Π. Unlike in the analysis of densities one therefore does not set Q µ ρ ρµ ρ Π 25 ρµ ). On the other hand, the equation: IV Q µ is ompletely invariant, i.e., one an define a ovariant derivative for whih the independent of r. E.g., the operator: (66) µ r Q µ, µ Π r are whih agrees with the operator µ in affine geometry, beause of (59), (62), (64), (65), is suh a differential operator that is independent of r. However, we would like to not make 24 ) Whereas, for us, the degree of a quantity is redued by one under ovariant differentiation, for Veblen the weight of a quantity is invariant under ovariant derivative. On this, f., 4, ) Exept when one restrits oneself to unimodular (or at least onstant modular) transformations and, as in footnote?), introdues densities, whih is what was done in most of the older presentations.

20 Projetive onnetions in n-dimensional spaes 2 the assumption IV, sine we will see in 7 that it exludes the existene of a ovariant differential 26 ). 2. Covariant differentiation of (59) and alternation yields: ρ (67) [ωµ] q T q + r U ωµ q, in whih we have set: (68) T ωµ ωµ ρ Sωµ A[ ω Qµ ] +, (69) U ωµ [ω Q µ] T ρ Q. ωµ ρ Covariant differentiation of (62), (64), and alternation yields: (7) [ωµ] v 2 N v T + v r ρ ωµ ωµ ρ 2 N v T + v + r U v ρ ωµ ωµ ρ ωµ (7) [ωµ] w 2 N w T ρ + + w r ωµ ωµ ρ 2 N w T ρ + + w + r U w. ωµ ωµ ρ ωµ Formulas (67), (7), (7) differ from the orresponding formulas of affine geometry 27 ) solely by the fats that, firstly, S ωµ is replaed byt ωµ, and seondly, that the term in U ω appears, whih has degree r and is proportional to differentiated quantities, as the following relation, whih is a result of (66), yields: (72) [ ωµ ] S ρ ωµ ρ [ωµ] T ρ r U ωµ. ωµ ρ 2. In addition to the first identity 28 ), whih is trivial, the urvature quantities satisfy the seond identity 29 ): (73) N ωµ ρ 2 [ ω Sµ ] 4S[ ωµ Sω ] ρ 2Q[ ω Sµ ] + + ρ 2 [ ω Tµ ] 4T[ ωµ Tω ] ρ 2U [ ωµ A ] + +, 26 ) For this reason, assumption IV, in its essential features, may be said to belong to a true projetive differential operator. Without it, in any position-like n+ E a ovariant point Q µ, hene, an n E, would be distinguished, whih one an regard as at an infinite distane in n+ E; geometry would then take on a ertain affine harater again in the small. 27 ) Cf., R.K., pp ) Cf., R.K., pp ) Cf., R.K., pp. 88.

21 Projetive onnetions in n-dimensional spaes 2 whih takes on the onventional form N ωµ for the symmetri ase, as well as for the quasi-symmetri ase (f., below 22) when U ω, moreover. For the Bianhi identity 3 ), one finds: (74) [ N ] 2T ρ N ]. κ ωµ κω µ ρ Furthermore, there exist the following two identities, whih do not our in affine geometry: (75) N ωµ x ρ 2 [ ω Pµ ] + 4Pρ Tωµ + 2x [ ω Sµ ] (76) N ωµ x ω P. µ Finally, we note the following relations: (77) x µ A + Q + Q x QA + P 2T x, κ (78) x v + P v + r Q v, (79) x w µ P ρ w + r Q w µ, in whih we have set: (8) Q + x ρ Q ρ. ρ Equation (77) follows from (62) by means of the identity x A by applying (57) and (68); (78) and (79) follow from (62) ((64), resp.) by means of (56) and the Euler homogeneity ondition. 22. We all the projetive onnetion quasi-symmetri when we have: Vβ T. ωµ The ondition of quasi-symmetry says that the urvature [ωµ] q, i.e., the rotation of the gradient of a salar q of null degree, vanishes. Quasi-symmetry expresses an essential property of projetive onnetions by the symmetry ondition: Vα S. ωµ The additional term A [ Q in (68) originates in the fat that the gradient ω µ ] µ q of a salar of null degree is not of null degree, but has degree. Theorem 2. The ondition: VI U µ 3 ) Cf., R.K., pp. 9.

22 Projetive onnetions in n-dimensional spaes 22 is neessary and suffiient for Q µ to be a gradient 3 ). In fat, due to (67), (69) the integrability onditions for the equation: IVα now read: Q µ µ log g (r ) U ωµ, if r is the degree of g. However, if r then, due to (59), IVα would take the form: µ log g, hene, g onstant, whih is not possible for a salar of first degree, exept for the trivial ase (whih is usually exluded) in whih g vanishes 32 ). From IVa, and ontration with x µ, by means of (59), (8) it follows that: (8) Q r onst. and: (82) Q µ Q µ log g µ log g Q. 7. The ovariant differential. 23. In general, there exists no ovariant differential. If one sets: (83) δ v dx µ µ v then the δv indeed transform like the values of a ovariant point, but they are not homogeneous: one has (f., (24), (78)): (84) δ v ρ r {δ v + ( P v + r Q v ) d log ρ}. A ovariant differential obviously exists when and only when the oeffiient of d log ρ vanishes, hene, for all of the points of a given degree r, when P v + r Q v ; hene, one has: 3 ) The theory of Sh. and G. belongs to the speial ase U ωµ (f., 4, 49). Sine the older presentation may be regarded as a speial ase of the one in Sh. and G., and, moreover, the newer theory of Veblen that was mentioned in 4, 47 is the same speial ase, in either event, we have U ωµ. 32 ) If g is homogeneous of r th degree and r then log g is not homogeneous, and µ log g does not atually exist. However, we use the expression as an abbreviation for µ g, and also speak of g gradients, aordingly.

23 Projetive onnetions in n-dimensional spaes 23 IIa P PA and: (85) κ κ P Q onst. r. Theorem 3. There exists a ovariant differential for quantities of eah degree when and only when both of the following onditions: P κ IIb IVb Q are satisfied. Under the assumption IIa, (77), (78) take the form: (86) x µ ( P + Q) A 2T x, (87) x v (P + r Q) v, and under the assumptions IIb, IVβ, they take the form: (88) x µ 2T x, (89) x v. These relations will be used in the alulations often.

24 Projetive onnetions in n-dimensional spaes Position displaements. 24. A displaement is essentially a means of equating quantities that exist at one position x in a spae with the quantities that exist at an infinitesimally lose position x + dx 33 ) Assuming that one has a means of equating ( simultaneously measuring ) quantities that are defined to have the same type at the same position, whih is the ase in projetive differential geometry for quantities of the same type and degree, one must then define what it means for two quantities of the same type to be alled equal at two neighboring points; i.e., the quantities at x shall be mapped to the quantities at y. However, in order for this to be the ase it suffies to map the two n+ E to eah other in a one-to-one way 34 ). In fat, suh a thing also exists in affine geometry, and indeed there is an affine map between neighboring E n ; it is given by the requirement of ovariant onstany, i.e., the vanishing of the ovariant differential. For example, the vetor v is onsidered to be equal to the vetor: (9) vɶ v + dv v v Γ dx µ at y. We now like to define a displaement in projetive differential geometry, as well, and indeed, by means of a projetive map between neighboring n+ E. Furthermore, we would not like to derive suh a thing by means of the ovariant differential that was introdued in 7 by means of the requirement of ovariant onstany, but independently of that, so we will onsider the ovariant derivative of the most general map, and, from that, derive the onditions for the Γ. These two paths reommend themselves sine they allow us to proeed in a purely geometri way (up to degree onsiderations), and beause the starting point is general, sine we will only assume a map of the n+ E, but not the E n+. A displaement for arbitrary projetors will then follow from the displaement of the position. 25. Therefore, let there be given a projetive map of the n+ E at a position x to the n+ E at a neighboring point y x + dx. It shall satisfy the following onditions: P.. As dx, it goes to the identity ontinuously. By that, we shall mean the following: If v is a position in the n+ E at x and vɶ is its image in the n+ E at y then the ratios of the oordinates of vɶ shall onverge ontinuously to the ratios of the oordinates of v whenever y onverges to x. Thus, the oordinates in both of the n+ E must be onsidered relative to two linear oordinate systems, whih are both assoiated with the same oordinate system in the n+ H. 33 ) Cf., D. van Dantzig, Die Wiederholung des Mihelson-Versuhs and Relativitätstheorie, Math. Annalen 96 (926), pp , in partiular, 7, Metrik and Physik. 34 ) Cf., J.A. Shouten [3], E. Cartan [2].

25 Projetive onnetions in n-dimensional spaes 25 P.2. Coinident positions will be mapped to oinident positions. Therefore, if u [ v µ] then one also has uɶ [ vɶ µ ]. P.3. The degree of a position is preserved by the map. This map then yields a map of the salars at x to the salars at y. Namely, if u, v are two oinident points at x, so u pv, then, beause of P.2, one an define pɶ byuɶ pv ɶɶ. We would like to make the following assumptions about this salar map: S.. As dx, it goes to the identity ontinuously. S.2. Sums and produts of two salars go to sums and produts of salars (this follows from P.2). S.3. The degree of a salar is preserved by the map (this follows from P.3). Therefore, we must onsider that for salars of null degree a salar field assoiates a number with eah position of n+ H; a salar field of higher degree maps the points at a position onto the number ontinuum in a well-defined way. Only for salars of null degree an we then infer, from the well-known theorem that the field of real numbers admits no ontinuous automorphism other than the identity, that pɶ p. If we also allow omplex numbers for funtion values then we must have pɶ p for salars of null degree, sine the only non-idential automorphism that the field of omplex numbers admits (namely, the one that takes any number to its onjugate) is not ontinuously reahable from the identity. From S.2, one then proeeds in the well-known way to infer that for salars of the same degree, p ɶ must be independent of p, i.e., that pɶ must have the form: p (9) pɶ Θ r p. Therefore, p is a funtion in the 2(n + ) arguments x, y, or briefly, a two-point funtion. Furthermore, it follows from S.2 in a well-known way that: Θ r (92) Θ r Θ r, in whih we have written Θ, instead of Θ ; (92) is valid at least for rational degrees, so we shall restrit ourselves to suh numbers. If we develop Θ as a power series in dx with oeffiients that depend only upon x, and no longer on y, and we trunate the development at the terms of seond order, then this shows that Θ must have the form: (93) Θ Q µ dx µ, sine S. implies that the first term must equal one. The trunation of the development is justified sine we all two displaements equal when their effets differ only by seond