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Connections Between Differential Geometry And VOL. 21, 1935 MA THEMA TICS: S. B. MYERS 225 having no point of C on their interiors or boundaries, the second composed of the remaining cells of 2,,. The cells of the first class will form a sub-com- plex 2* of M,. The cells of the second class will not form a complex, since they may have cells of the first class on their boundaries; nevertheless, their duals will form a complex A. Moreover, the cells of the second class will determine a region Rn containing C. Now, there is no difficulty in extending Pontrjagin's relation of duality to the Betti Groups of 2* and A. Moreover, every cycle of S - C is homologous to a cycle of 2, for sufficiently large values of n and bounds in S - C if and only if the corresponding cycle of 2* bounds for sufficiently large values of n. On the other hand, the regions R. close down on the point set C as n increases indefinitely, in the sense that the intersection of all the RI's is precisely C. Thus, the proof of the relation of duality be- tween the Betti groups of C and S - C may be carried through as if the space S were of finite dimensionality. 1 L. Pontrjagin, "The General Topological Theorem of Duality for Closed Sets," Ann. Math., 35, 904-914 (1934). 2 Cf. the reference in Lefschetz's Topology, Amer. Math. Soc. Publications, vol. XII, end of p. 315 (1930). 3Lefschetz, loc. cit., pp. 341 et seq. CONNECTIONS BETWEEN DIFFERENTIAL GEOMETRY AND TOPOLOG Y BY SumNR BYRON MYERS* PRINCJETON UNIVERSITY AND THE INSTITUTE FOR ADVANCED STUDY Communicated March 6, 1935 In this note are stated the definitions and results of a study of new connections between differential geometry and topology. Proofs will be given elsewhere. Let S be a complete analytic 2-dimensional Riemannian space (for brev- ity, complete surface), as defined by H. Hopf and W. Rinow,I and A an arbitrary point on it. A point M on a geodesic ray issuing from A is said to be a minimum point with respect to A on g if M is the last point on g such that AM furnishes an absolute minimum (proper or improper) to the arc length of curves on S joining A to M.2 We study the locus of such points on all the geodesic rays issuing from A. A geodesic arc is said to be of class A if it furnishes an absolute minimum to the arc length of curves joining its end-points. A geodesic ray is said to be of class A if every segment of it is of class A. The order of a minimum point M with respect to A denotes the number of geodesics of class A joining it to A. Downloaded by guest on September 26, 2021 226 MA THEMA TICS: S. B. MYERS PROC. N. A. S. THEOREM 1. A minimum point of order 1 on g with respect to A must be conjugate to A, and must be a cusp of the locus offirst conjugate points to A on the geodesic rays issuingfrom A. Thus the minimum point locus and the conjugate point locus are related, and we study the latter in order to obtain information about the former. THEOREM 2. On a complete surface S, the locus offirst conjugate points to A is either (1) no points at all, (2) a single point, (3) a closed curve containing afinite number > 2 of cusps turned toward A and an equal number turned awayfrom A, (4) a set of one or more open curves, each asymptotic to a pair of geodesic rays issuing from A. Each curve of the set contains at least one cusp turned toward A, and one less cusp turned away from A. A single one of the curves cannot contain an infinite number of cusps on a finite segment of the curve. The number of curves in the set can be infinite only if a curve of the set can be found all of whose points are arbitrarily far from A along the geodesics on which they are conjugate to A. In each case the locus can be analytically parameterized in terms of 0, the angular co6rdinate ofthe geodesic rayfrom A. The following theorem gives a first indication as to how the existence of minimum points with respect to an arbitrary point A of a complete surface is connected with the topology of the whole surface. THEOREM 3. A complete surface S is closed (i.e., compact) if and only if it contains no geodesic ray of class A. It is open (i.e., not compact) ifand only if it contains a geodesic ray ofclass A issuingfrom an arbitrary point A. TiEoREM 4. Let A be an arbitrary point on the complete simply connected surface S. Then if S is closed, hence homeomorphic to the sphere, the locus m of minimum points with respect to A consists of a single point, or else a tree with a finite number of branches. If m is a single point, it is conjugate to A on all geodesics joining it to A and at equal distances from A along these geodesics. If S is open, hence homeomorphic to the plane, m consists of noth- ing at all, or a set of one or more distinct trees each of which contains one branch extending infinitely far from A. There may be an infinite number of branches extending from this branch, but not from any bounded segment of it. Whether S is open or closed, the number of arcs of m incident with a point M of m is equal to the order of M as a minlimum point with respect to A. A n arc of m containing no points conjugate to A and no interior points of order > 2 is a regular analytic arc. The region S-m-A is simply covered by the geodesic rays through A cut off at their intersections with m, and hence forms a 2-cell1 with m as its singular boundary, in which the geodesic polar co6rdinates with A as poleform a co6rdinate system. Downloaded by guest on September 26, 2021 VOL. 21, 1935 PHEYSIOLOG Y: A. G. MARSHAXK 227 THEoREm 5. Let A be an arbitrary point on a closed surface ofgenus > 0. The locus m of minimum points with respect to A is a linear graph (i.e., a finite one-dimensional complex) whose cyclomatic number is equal to the connectivity number modulo 2 of S. If S is orientable, as the geodesic ray from A sweeps out the angle from 0 to 27r the minimum point on the ray traces out each one-cell of m once in one sense and once in the opposite sense. Con- versely, this is sufficient that S be orientable. Thus the topology of S is com- pletely determined by the linear graph m and the way in which it is traced out. The statements of the preious theorem as to the order of the points of m, the regular analyticity of arcs of m, and the simple covering of the region S-m-A hold here as well. This reduction of a closed surface to a single 2-cell with a linear graph as its singular boundary is analogous to the process in analysis situs of making a similar reduction by coalescing 2-cells of a 2-dimensional mani- fold.3 Incidentally, these results clear up completely the hitherto vaguely answered question as to how long the geodesics through a point A of a Riemannian surface form a field. These methods and results present possibilities of generalization to n- dimensional Riemannian spaces, to the geometry of paths and to arbi- trary convex metric spaces. * National Research Fellow in Mathematics. 1 "Ueber den Begriff der vollstandigen differential-geometrischen Flache," Comm. Math. Helvet, 3 (1931). ' Cf. Blaschke, Vorlesungen iuber DifferentialgeometrieI, p. 231, lines 1-6. 8 See, for example, Veblen, Analysis Situs, p. 74, §§ 62, 63. THE SENSITIVE-VOLUME OF THE MEIOTIC CHROMONEMA TA OF GASTERIA AS DETERMINED B Y IRRADIATION WITH X-RA YS By A. G. MsARSHA LABORATORY OF GENERAL PHYSIOLOGY, HARvARD UNIVERSITY Communicated March 1, 1935 In studies of the lethal action of x-rays upon cells, measurements of the volume sensitive to x-rays have been subject to a serious error because populations of cells in various stages of division have been used. On the other hand no quantitative measurements of the effect of x-rays on specific stages of cell division have been made. This paper is an account of the re- sults of an experiment designed to detect varying sensitivities of different stages of the chromosome cycle and to determine the order of magnitude of the portion of the chromosome sensitive to x-rays. Downloaded by guest on September 26, 2021.
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