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Conformal Differential Geometry of a Subspace

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Conformal Differential Geometry of a Subspace CONFORMAL DIFFERENTIAL GEOMETRY OF A SUBSPACE BY AARON FIALKOW Table of contents Introduction. 310 I. Preliminary matters 1. Notation. 315 2. Riemann spaces conformai to Vm. 316 3. Conformai tensors. 318 4. Algebraic and metric properties. 319 5. Generalized covariant differentiation and the second fundamental form of V„. 323 II. Conform al tensor algebra 6. The relative conformai curvature of Vn. 325 7. The conformai measure tensors. 329 III. Conformal tensor analysis 8. Conformal differentiation. 331 9. The replacement principle. 335 10. Conformal geodesic and conformal normal coordinates and parameters. 341 11. The Weyl tensors. 342 12. The Bianchi identities for the Weyl tensors. 346 13. The deviation tensor. 350 14. The conformal Riemann tensor of V„. 352 15. Variation of the mean curvature normal. 354 16. The conformal Riemann tensor of Vm. 355 17. The Ricci identities. 359 IV. The conformal vector spaces of Vn 18. The conformal osculating and normal vector spaces of V„. 359 19. The conformal normal measure tensors. 362 20. The conformal Frenet equations. 365 V. The fundamental equations 21. The boundary, algebraic, reality and derived conditions. 370 22. The conformal Gauss-Codazzi equations. 373 23. Higher integrability conditions. 376 24. The third derivatives of the relative conformal curvature. 379 25. The second derivatives of the mean curvature. Discussion for a hypersurface. 383 26. The second derivatives of the mean curvature normal. Discussion for the general case 385 VI. SUBSPACES IN A CONFORMALLY EUCLIDEAN SPACE Rm 27. Symmetry relations. 392 28. Determination of &£$,, U$Ü¿>,£«„>I<W. 396 Presented to the Society, October 26, 1940 and February 22, 1941; received by the editors February 1, 1943, and, in revised form, December 15, 1943. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use309 310 AARON FIALKOW [September 29. The existence theorem. 412 30. Groups of conformai transformations in euclidean space Rm and in a conformally euclidean space Rm. 418 31. The conformai equivalence theorem. 421 32. Conformai differential forms. 422 VII. Exceptional subspaces 33. Conformai null subspaces. 426 34. Conformai curve theory. 427 Bibliography. 432 Introduction Scope. Conformai differential geometry (as developed in this paper) con- cerns itself with certain spaces, point transformations, and configurations. The particular spaces which are considered here are the wz-dimensional Rie- mann spaces Vm. These spaces are generalizations of and include euclidean /ra-space Rm. The basic point transformations of this paper are the conformai transformations. A conformai transformation defined on a Riemann space Vm is a mapping of Vm on itself or another Riemann space Vm so that correspond- ing angles in Vm and Vm are equal in magnitude. We do not study the con- formal transformations of the Riemannian spaces as a whole upon each other (that is, the intrinsic conformai theory of the space) but the conformai ge- ometry of configurations in these spaces. The configuration in Vm which we study is a subspace Vn. The dimensionality ra of this subspace may be any integer between 1 (curve) and raí —1 (hypersurface) inclusive. Accordingly, conformai differential geometry, as defined above, investigates those proper- ties of a curve, hypersurface or other subspace which remain unchanged when the enveloping Riemann space Vm undergoes any conformai mapping, not necessarily on itself. As an important special case, this geometry includes the inversive geometry of a euclidean space Rm. Method. Euclidean geometry may be defined as the invariant theory of the euclidean metric or congruence group. Not only do the congruence trans- formations define the geometry; they were a fundamental method in the de- velopment of the early (Greek) geometry of the plane. The picture is quite different in the classical differential geometry of two and three-dimensional euclidean space. Here, since the time of Gauss, the fundamental method in the geometry (as distinguished from its definition) has been based upon quad- ratic differential forms. These quadratic differential forms control the metric of the space and the metrics and curvatures of configurations (curves, sur- faces) in the space. What was a powerful tool for the geometric exploration of space in the hands of Gauss became the defining structure of the space in the epoch making generalization of Riemann. The geometry of Riemann spaces and the corresponding tensor calculus have since been developed by Christoffel, Ricci, Levi-Civita and others into a beautiful and elegant theory. The power of the methods of Riemannian geometry rests primarily upon License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1944] CONFORMAL DIFFERENTIAL GEOMETRY OF A SUBSPACE 311 (1) the existence of a fundamental metric tensor (whose components are the coefficients of the fundamental quadratic differential form), (2) the process of covariant differentiation, (3) the tensor character of the important geo- metric objects in the space, and of the operations performed upon these objects. In our development of conformal differential geometry, we have largely ignored the conformal transformations which define the geometry and have attempted to take maximum advantage of the methods of Riemannian geom- etry. A first necessary step in the realization of this aim is the discovery of a symmetric second order tensor in Vm (corresponding to the Riemann metric tensor) which remains unchanged by conformal transformations of Vm. In the intrinsic conformal theory of Vm, there does not exist at present any method for attaining this desirable objective. But a subspace Vn introduces new elements of structure into the envelop- ing Riemann space Vm at points of Vn. By means of these new structural properties, we discover a relative conformal scalar(1) A which exists at points of the subspace. The existence of A makes it an easy matter to define two new symmetric second order tensors which do not change when Vm undergoes any conformal transformation. These tensors, called the "conformal measure tensors," introduce a new "conformal metric" into the space. The first tensor is used to measure "conformal distance" in the subspace F„ and the second to measure "conformal distance" in the enveloping space Vm at points of V„. By making use of A, it is also possible to invent a new type of covariant differ- entiation (with respect to the subspace Vn) which plays a role analogous to ordinary covariant differentiation in Riemannian geometry. This "conformal covariant differentiation" process or "conformal derivative" enjoys all the usual properties of covariant differentiation as well as a number of others which give it its distinctive conformal character. The tensors for the measurement of conformal distance and the conformal differentiation process constitute the methodological foundation for our de- velopment of conformal differential geometry. By meansof the conformal meas- ure tensors and conformal differentiation, it is possible to apply all the usual operations to conformal tensors(2) without altering either their conformal or tensor character. Consequently our development of conformal differential geometry proceeds by a full utilization of the technical apparatus of tensor calculus and Riemannian geometry. Its formal, technical aspects are scarcely more involved than those of the ordinary (metric) theory of Riemann spaces. The same methods may be used to investigate the conformal differential ge- ometry of configurations^) in Vm other than those which are considered here. (*) This term is defined in §3 and A is defined in §6. (2) This term is defined in §3. (3) Examples of such configurations are: congruences of curves, families of subspaces, m-beins. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 312 AARON FIALKOW [September The formal calculations themselves may be shortened considerably by making use of the "replacement principle." This principle states that certain tensors which appear frequently in the sequel may be replaced by zeros in the expressions where they occur. It depends upon a change in gauge, but in its effect is somewhat analogous to the choice of special coordinate systems frequently made in classical and Riemannian geometry. Results. In the preceding paragraphs we have discussed, in general terms, the scope of our subject and the methods employed. What are the actual questions which are considered and what answers are obtained ? The conclu- clusions of this paper may be summarized under the following topics: (1) Conformai geometric objects. A number of new conformai geometric ob- jects are invented and their definition obtained in tensor form. Among these geometric objects are conformai analogues of the metric tensor, the Christoffel symbols, and the Riemann tensor, all of which are defined both for the sub- space F„ and for the enveloping space Vm at points of F„. We also construct various new tensors from the Weyl conformai curvature tensor. The existence of conformai analogues of the geodesic and normal coordinates of Riemannian geometry is proved. By successive conformai differentiation of the tangent vector space of F„, we obtain a sequence of conformai vector spaces analogous to the osculating and normal vector spaces of F„ in Riemannian geometry. The measurement of the conformai length of vectors lying in these "con- formal normal vector spaces" introduces
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