# Eisenhart, Luther Pfahler

# Eisenhart, Luther Pfahler

(*b*. York, Pennsylvania, 13 January 1876; *d*. Princeton, New Jersey, 28 October 1965)

*mathematics*.

Eisenhart was the second son of Charles Augustus Eisenhart and the former Emma Pfahler. His father was a dentist, a founder Emma Pfahler. His father was a dentist, founder of the Edison Electric Light and York Telephone companies, and secretary of the Sunday school of St. Paul’ Lutheran Church. Eisenhart was taught by his mother before he entered school and completed grade school in three years. He then attended York High School until, in his junior year, he was encouraged by the principal to withdraw and devote his time to the independent study of Latin and Greek for early admission to Gettysburg College, which he attended from 1892 to 1896. Being the only upper-division mathematics student, during the last two years of college Eisenhart studied mathematics through independent guided reading.

After teaching for a year at the preparatory school of the college, he began graduate study at Johns Hopkins University in 1897 and obtained the Ph.D. in 1900 with a thesis whose topic, “Infinitesimal Deformations of Surfaces,” he had chosen himself. He was introduced to differential geometry through a lecture by Thomas Craig and studied the subject through the treatises of Gaston Darboux. According to his own testimony, the experience of independent study led Eisenhart to propose the four-course plan of study adopted at Princeton in 1923, which provides for independent study and the preparation of a thesis. Eisenhart’s scientific career was spent at Princeton; he retired in 1945.

In 1908 Eisenhart married Anna Maria Dandridge Mitchell of Charles Town, West Virginia; she died in 1913. In 1918 he married Katharine Riely Schmidt of York, Pennsylvania. He had one son, Churchill, by his first marriage and two daughters, Anna and Katharine, by his second.

Eisenhart’s work in differential geometry covers two distinct periods and fields. The first period, to about 1920, was devoted mainly to the theory of deformations of surfaces and systems of surfaces.

Modern differential geometry was founded by Gaston Darboux as a field of applications of partial differential equations. His methods were taken up by Luigi Bianchi, who created an extensive theory of the deformations of surfaces of constant negative curvature. In another direction, Claude Guichard showed between 1897 and 1899 how the partial differential equations of the deformations of triply orthogonal systems of surfaces can be interpreted in terms of the systems of lines connecting a point and its image point. These discoveries made the theory of deformations of surfaces one of the focal points of geometric research in Europe at the turn of the century. Although there were quite a number of able mathematicians working in America in the field of geometry at that time, Eisenhart was the only one to turn to the topic of deformations. His main contribution to the topic of deformations. His main contribution to the theory was a unifying principle: The deformation of a surface defines the congruence (two-parameter family) of lines connecting a point and its image (following Guichard). In general, a congruence contains two families of developable surfaces (a developable surface is formed by the tangent to a space curve). Eisenhart recognized that in all known cases, the intersections of these surfaces with the given surface and its image form a net of curves with special properties. This allows not only a unified treatment of many different subjects and a replacement of tricks by methods, but also leads to many new results that round off the theory. Eisenhart gave a coherent account of the theory in *Transformations of Surfaces* (1923). The book also contains most of Eisenhart’s previous results either in the text or in the exercises, with references. Some aspects of the theory were taken up later in the projective setting by Eduard Čech and his students. All these investigations deal with small neighborhoods for which existence theorems for solutions of differential equations are available.

Of the few papers not dealing with deformations dating from this period, a noteworthy one is “Surfaces Whose First and Second Forms Are Respectively the Second and First Forms of Another Surface” (1901), one of the first differential geometric characterizations of the sphere, a topic started by Heinrich Liebmann in 1899. Eisenhart proved that the unit sphere is the only surface whose first and second fundamental forms are, respectively, the second and first fundamental forms of another surface.

Einstein’s general theory of relativity (1916) made Riemannian geometry the center of geometric research. The analytic tools that turned Riemannian geometry from an idea into an effective instrument were Ricci’s covariant differential calculus and the related notion of Levi-Civita’s parallelism. These tools had been thoroughly explored in Luigi Bianchi’s *Lezioni di geometria differenziale*. As a consequence, the attention of geometers immediately turned to the generalization of Riemannian geometry. Most of Eisenhart’s work after 1921 was in this direction. The colloquium lectures *Non-Riemannian Geometry* (1927) contain his account of the main results obtained by him and his students and collaborators. An almost complete coverage of Eisenhart’s results, with very good references, is given in Schouten’s *Ricci Calculus*. Three directions of generalization of Riemannian geometry were developed in the years after 1920. They are connected with the names of Élie Cartan, Hermann Weyl, and Eisenhart. Cartan considered geometries that induce a geometry of a transitive transformation group in any tangent space. Weyl gave an axiomatic approach to the maps of tangent spaces by parallelism along any smooth curve. Eisenhart’s approach, inspired by Oswald Veblen’s work on the foundations of projective geometry and started in cooperation with Veblen, is the only one to deal directly with the given space. In Riemannian geometry, the measure of length is prescribed and the geodesic lines are determined as the shortest connections between nearby points. In Eisenhart’s approach, the geodesics are given as the solution of a prescribed system of second-order differential equations and the non-Riemannian geometries are obtained by asking that there should exist a Levi-Civita parallelism for which the tangents are covariant constant.

While Cartan’s and Weyl’s generalizations have become the foundations of the fiber space theory of differentiable manifolds, Eisenhart’s theory does not fit the framework of these topological theories. The reason is that the geometric objects intrinsically derived from the “paths” of the geometry, the projective parameters of Tracy Y. Thomas, have a more complicated transformation law than the generalized Christoffel symbols of Cartan and Weyl. However, there are a number of modern developments, such as the theory of Finsler spaces and the general theory of the geometric object, that fit Eisenhart’s framework but not that of the algebraic-topological approach. As far as metric geometry is concerned, the most fruitful approach seems to be to give the geodesics directly as point sets and to throw out all differential equations and analytical apparatus. On the other hand, for nonmetric geometries Eisenhart proved (in “Spaces With Corresponding Paths”[1922])that for every one of his geometries there exists a unique geometry with the same paths and for which the mapping of tangent spaces induced by the flow of tangent vectors with unit speed along the paths is volume-preserving. For the latter geometry, which would appear to give a natural setting for topological dynamics, the Cartan, Weyl, and Eisenhart approaches are equivalent.

A Number of interesting avenues of development of Riemannian geometry were opened by Eisenhart. The papers “Fields of Parallel Vectors in the Geometry of Paths”(1922) and “Fields of Parallel Vectors in a Riemannian Geometry” (1925) started the topic of recurrent fields and harmonic spaces (for a report with later references, see T. J. Willmore, *An Introduction to Differential Geometry*, ch. 7, sec. 13). The so-called Eisenhart’s theorem appears in “Symmetric Tensors of the Second Order Whose First Covariant Derivatives Are Zero” (1923): If a Riemannian geometry admits a second-order, symmetric, covariant constant tensor other than the metric, the space behaves locally like the product of two lowerdimensional spaces. Together with a theorem of Georges de Rham to the effect that a simply connected, locally product Riemannian space is in fact a Cartesian product of two spaces, the theorem is an important tool in global differential geometry. An extension of the theorem is given in “Parallel Vectors in Riemannian Space” (1938).

The basic equations for the vectors of a group of motions in a Riemannian space had been given by Killing in 1892. Eisenhart developed a very powerful analytical apparatus for these questions; the results are summarized in *Riemannian Geometry* (1926; ch. 6) and *Continuous Groups of Transformations* (1933). The later developments are summarized in Kentaro *Yano’s Group of Transformations in Generalized Spaces*(1949).

Eisenhart’s interest in mathematical instruction found its expression in a number of influential text-books—such as *Differential Geometry of Curves and Surfaces* (1909), *Riemannian Geometry*(1926),*Continuous Groups of Transformations* (1933), *Coordinate Geometry* (1939), *An Introduction to Differential Geometry With Use of the Tensor Calculus* (1940)—some in fields that until then had been dependent upon European monographs devoid of exercises and other student aids. His interest in history resulted in several papers; “Lives of Princeton Mathematicians” (1931), “Plan for a University of Discoverers” (1947), “Walter Minto and the Earl of Buchan” (1950), and the preface to “Historic Philadelphia” (1953).

## BIBLIOGRAPHY

I. Original Works. Eisenhart’s works published between 1901 and 1909 are “A Demonstration of the Impossibility of a Triply Asymptotic System of Surfaces,” in *Bulletin of the American Mathematical Society*, 7 (1901), 184–186; “Possible Triply Asymptotic Systems of surfaces,” ibid., 303–305; “Suraces Whose First and Second Forms are Respectively the Second and First Forms of Another Surfaces,” ibid., 417–423; “Lines of Length Zero on Surfaces,” *ibid*., **9** (1902), 241–243; “Note on Isotropic Congruences,” *ibid*., 301–303; “Infinitesimal Deformation of Surfaces,” in *American Journal of Mathematics*, **24** (1902), 173–204; “Conjugate Rectilinear Congruences,” in *Transactions of the American Mathematical Society*, **3** (1902), 354–371; “Infinitesimal Deformation of the Skew Helicoid,” in *Bulletin of the American Mathematical Society*, **9** (1903), 148–154; “Surfaces Referred to Their Lines of Length Zero,” ibid., 242–245; “Isothermal-Conjugate Systems of Lines on Surfaces,” in *American Journal of Mathematics*, **25** (1903), 213–248; “Surfaces Whose Lines of Curvature in One System Are Represented on the Sphere by Great Circles,” *ibid*., 349–364; “Surfaces of Constant Mean Curvature,” *ibid*., 383–396; “Congruences of Curves,” in *Transactions of the American Mathematical Society*, **4** (1903), 470–488; “Congruences of Tangents to a Surface and Derived Congruences,” in *American Journal of Mathematics*, **26** (1904),180–208; “Three Particular Systems of Lines on a Surface,” in *Transactions of the American Mathematical Society***5** (1904), 421–437; “Surfaces with the Same Spherical Representation of Their Lines of Curvature as Pseudospherical Surfaces,” in *American Journal of Mathematics*, **27** (1905), 113–172 ; “On the Deformation of Surfaces of Translation,” in *Bulletin of the American Mathematical* Society, **11** (1905), 486–494 ; “Surfaces of Constant Curvature and Their Transformations,” in *Transactions of the American Mathematical Society*, **6** (1905), 473–485; “Surfaces Analogous to the Surfaces of Bianchi,” in *Annali di matematica pura ed applicata*, 3rd ser., **12** (1905), 113–143; “Certain Surfaces With Plane or Spherical Lines of Curvature” in American Journal of Mathematics, **28** (1906), 47–70; “Associate Surfaces,” in *Mathematische Annalen*,**62** (1906), 504–538; “Transformations of Minimal Surfaces,” *in Annali di matematica pura ed applicata*, 3rd ser., **13** (1907), 249–262; “Applicable Surfaces With Asymptotic Lines of One Surface Corresponding to a Conjugate System of Another,” in *Transactions of the American Mathematical Society*, **8** (1907), 113–134; “Certain Triply Orthogonal Systems of Surfaces,” in *American Journal of Mathematics*, **29** 1907), 168–212; “Surfaces With Isothermal Representation of Their Lines of Curvature and Transformations (I),” in Their Transformations of the American Mathematical Society, **9** (1908), 149–177; “Surfaces With the Same Spherical Representation of Their Lines of Curvature as Spherical Surfaces,” in *American Journal of Mathematics*, **30** (1908), 19–42; and *A Treatise on the Differential Geometry of Curves and Surfaces* (Boston, 1909; repub. New York, 1960).

Between 1910 and 1919 he published “The Twelve Surfaces of Darboux and the Transformation of Moutard,” in *American Journal of Mathematics*, **11** (1910), 351–372; “Surfaces With Isothermal Representation of Their Lines of Curvature and Their Transformation (II),” *ibid* 475–486; “A Fundamental Parametric Representation of Space Curves,” in *Annals of Mathematics*, 2nd ser., **13** (1911), 17–35; “Sopra le deformazioni continue delle superficiereali applicabili sul paraboloide a parametro puramente immaginario,” in *Atti dell Accademia nazionale dei Lincei Rendiconti* Classe di scienze fisiche, matematiche e naturali, 5th ser., **211** (1912), 458–462; “Ruled Surfaces With Isotropic Generations” in *Rendiconti del Circolo matematico di Palermo*, **34** (1912), 29–40; “Minimal Surfaces in Euclidean Four-Space,” in *American Journal of Mathematics*, **34** (1912), 215–236; “Certain Continuous Deformations of Surfaces Applicable to the Quadrics,” in *Transactions of the American Mathematical Society*, **14** (1913), 365–402; “Transformations of Surfaces of Guichard and Surfaces Applicable to Quadrics,” in *Annali di matematica pura ed applicata*, 3rd ser., **22** (1914), 191–248; “Transformations of Surfaces of Voss,” in *Transactions of the American Mathematical Society*, **15** (1914), 245–265; “Transformations of Conjugate Systems With Equal Point Invariant,” *ibid* 397–430; “Conjugate Systems With Equal Tangential Invariants and the Transformation of Moutard,” in *Rendiconti del Circolo matematico di Palermo*, **39** (1915), 153–176; “Transformations of Surfaces ω” in *Proceedings of the National Academy of Sciences*, **1** (1915), 62–65; “One-Parameter Families of Curves,” in American Journal of Mathematics, 37 (1915), 179–191 ; “Transformations of Conjugate Systems With Equal Invariants,” in Proceedings of the National Academy of Sciences, 1 (1915), 290–295; “Surfaces ω and Their Transformations” **1** (1915), 275–310; “Sulle superficies di rotolamento e le trasformazioni di Ribacour” in *Atti dell Accademia nazionale dei Lincei. Rendiconti* Classe di scienze fisiche, matematiche e naturali, 5th ser., **242** (1915), 349–352; “Surfaces With Isothermal Representation of Their Lines of Curvature as Envelopes of Rolling” in *Annals of Mathematics* 2nd ser., **17** (1915),63–71; “Transformations of Surfaces ω” in *Transactions of the American Mathematical Society*, **17** (1916), 53–99; “Deformations of transformations of Ribaucour” in *Proceedings of the National Acadeaamy of Sciences***2** (1916) 173–177 ; “Conjugate Systems With Equal Point Invariants,” *in Annals of Mathematics*, 2nd ser., **18** (1916), 7–17; “surfaces Generated by the Motion of an Invariable Curve Whose Points Describe Straight Lines” in *Rendiconti del Circolo matematico di Palermo*, **41** (1916), 94–102; “Deformable Transformations of Ribaucour” in *Transactions of the American Mathematical Society*, **17** (1916), 437–458; “Certin Surface of Voss and Surfaces Associted With Them” in *Rendiconti del Circolo matematcio de Palemo*, **42** (1917), 145–166; “Transformations T of Conjugate Systems of Curves on a Surface” in *Transformations of the American Mathematical Society*, **18** (1917), 97–124; “Trads of Transformation of Conjugate Systems of Curves,” in *Proceedings of the National Academy of Sciences*, **3** (1917), 453–457; “Conjugate Planar Nets With Equal Invariants” in *Annals of Mathematics* 2nd ser., **18** (1917), 221–225; “transformations of Applicable Conjugate Nets of Curves on Surfaces” in *Proceeding of the National Academy of Sciences*, **3** (1917), 637–640; “Darboux’ Contribution to Geometry” in *Bulletin of the American Mathematical Society*, **24** (1918), 227–237; “Surfaces Which Can Be generated in More Than One Way by the Motion of an Invariable Curve” in *Annals of Mathematics* 2nd ser., **19** (1918), 217–230; “Transformations of Planar Nets” in *American Journal of Mathematics*, **40** (1918), 127–144; “Transformations of Applicable Conjugate Nets of Curves on Surfaces” in *Transformations of the American Mathematical Society*, **19** (1918), 167–185; “Triply Conjugate Systems With Equal Point Invariant” in *Annals of Mathematics* 2nd ser., **20** (1919), 262–273; “Transformations of Surfaces Applicable to a Quadric” in Transformactions of the American Mathematical Society, **20** (1919), 323–338; and “Transformations of Cyclic Systems of Circles” in *Proceedings of the National Academy of Sciences*, **5** (1919), 555–557.

Eisenhart’ works published between 1920 and 1929 are “The Permanent Gravitational Field in the Einstein Theory,” in *in Annals of Mathematics*, 2nd ser., **22** (1920), 86–94; “The Permanent Gravitational Field in the Einstein Theory,” *Proceedings of the National Academy of Sciences* 2nd ser., **22** (1920), 678–682; “Sully congruenze di sfere di Ribaucor che ammettono una deformazione finite” in *Atti dell’ Academianazionale dei Lincei Rendiconti* Class di scienz fisiche mathematic he e naturli, 5th ser., **292** (1920), 31–33; “Conjugate Systems of Curves R and Their Transformations,” in *Comptes rends du sixième Congrès international de mathématicians* (Strasbourg, 1920), pp. 407–409; “Darboux’s Anteil an der Geometric” in *Acta mathematica***42** (1920), 275–284; “Transformations of Surfaces Applicable to a Quadric” in *Journal de mathématiques pure et appliquées* 8th ser., **4** (1921), 37–66; “Conjugate Nets R and Their Transformations” in *Annals of Mathematics* 2nd ser., **22** (1921), 161–181; “A Geometric Characterization of the Paths of Particles in the Gravitational Field of a Mass at Rest” (abstract), in *Bulletin of the American Mathematical Society*, **27** (1921), 350; “The Einstein Solar Field,” *ibid.*, 432–434; “Sulle trasformazioni T dei sistemi triple coniugati di superficies” in *Atti dell’ Academia nazionle dei Lincei Rendiconti* Classe di scienze fisiche, matematiche e natural, **302** (1921), 399–401; “Einstein Static Fields Admitting a Group G_{2} of Continuous Transformations Into Themselves” in *Proceedings of the National Academy of Sciences*, **7** (1921), 328–334; abstract in *Bulletin of the American Mathematical Society*, **28** (1922), 34; “The Riemann Geometry and Its Generalization” in *Proceedings of the National Academy of Sciences*, **8** (1922), 19–23, abstract in *Bulletin of the American Mathematical**Society***28** (1922), 154, written with Oswald Veblen; “Ricci’s Principal Directions for a Riemann Space and the Einstein Theory,” in *Proceedings of the National Academy of Science*, **8** (1922), 24–26, abstract in *Bulletin of the American Mathematical Society*, **28** (1922), 238; “The Einstein Equations for the Solar Field From the Newtonian Point of View,” in *Science*, n.s. **55** (1922), 570–572; “Fields of Parallel Vectors in the Geometry of Paths,” in *proceedings of the National Academy of Sciences*, **8** (1922), 207–212; “Spaces With Corresponding Paths,” *ibid*., 233–238;“Condition That a Tensor Be the Curl of a Vector,” in *Bulletin of the American Mathematical Society*, **28** (1922), 425–427; “Affine Geometries of Paths Possessing an Invariant Integral,” in *Proceedings of the National Academy of Sciences*, **9** (1923),4–7;“Another Interpretation of the Fundamental Gauge-Vectors of Weyl’s Theory of Relativity,” *ibid*., 75–178; “Orthogonal Systems of Hypersurfaces in a General Riemann Space,” in *Transactions of the American Mathematical Society*, **25** (1923),259–280, abstract in *Bulletin of the. American Mathematical Society*, **29** (1923), 213; “Einstein and Soldner,” in *Science*, n.s. **58** (1923), 516–517; “The Geometry of Paths and General Relativity,” in *Annals of mathematics*,2nd ser., **24** (1923), 367–393; *Transformations of Surfaces* (Princeton, 1923; corr. reiss. New York, 1962); “Space-Time Continua of Perfect Fluids in General Relativity,” in *Transactions of the American Mathematical Society*, **26** (1924),205–220;“Spaces of Continuous Matter in General Relativity,” abstract in *Bulletin of the American Mathematical Society*, **26** (1924),378–384, abstract in *Bulletin of the American mathematical Society*, **30** (1924), 297; “Linear Connections of a Space Which Are Determined by Simply Transitive Continuous Groups,” in *Proceedings of the national Academy of Sciences*, **11** (1925),243–250“Fields of Parallel Vectors in a Riemannian Geometry,” in *Transactions of the American Mathematical Society*, **27** (1925),563–573, abstract in *Bulletin of the American Mathematical Society*, **31** (1925),292,“Einsteein’s Recent Theory of Graviataion and Electicity,” in *Proceeding of the National Academy of Sciences*, **12** (1926) 125–129; *Riemannian Geometry* (Princeton, 1926); “Geometries of Paths for Which the Equations of the Path Admit n (n + 1)/2 Independent Linear First Integrals,” in *Transactions of the American Mathematical Society*, **28** (1926),330–338, abstract in *Bulletin of the American mathematical Society*, **32** & 1926),197;“Congrruences of Parallelism of a Field of Vectors,” in *proceedings of the National Academy of Sciences*, **12** (1926),757–760; “Displacements in a Geometry of Paths Which Carry Paths Into Paths,” in *Proceedings of the National Academy of Sciences*, **13** (1927),38–42, written with M. S. Knebelman;*Non-Riemannian Geometry* (New York, 1927; 6th pr., 1968); “Affine Geometry” in *Encylopaeia Britannica* 14th ed. (1929), **1** 279–280; “Differential Geometry,” *ibid*, VII,366–367; “Contact Transformation,” in *Annals of Mathematics*, 2nd ser., **30** (1929),211–249; and “Dynamical Trajectories and Geodesics,” *ibid.*,591–606.

Between 1930 and 1939 Eisenhart published “Projective Normal Coordinates,” in *Proceedings of the National Academy of Sciences*, **16** (1930),731–740; “Lives of Princeton Mathematicians,” in *Scientific Monthly*, **33** (1931), 565–568; “Intransitive Groups of Motions,” in *Proceedings of the National Academy of Sciences*, **18** (1932), 195–2025; “Equialent Continous Groups,” in *Annals of mathematics*, 2nd ser., **33** (1932),665–676; “Spaces Admitting Complete Absolute Parallelism,” in *Bulletin of the American Mathematical Society*, **39** (1933),217–226; *Continuous Groups of Transformations* (Princeton, 1933; repr. New York, 1961); “Separable Systems in Euclidean 3-Space,” in *Physical Review, 2nd ser.*, **45** (1934), 427–428; “Separable Systems of Stackel,” in *Annuals of mathematics*, 2nd ser.,**35** (1934),284–305; “Stackel Systems in Conformal Euclidean Space,” *ibid*.,**36** (1935), 57–70; “Groups of Motions and Ricci Directions,” *ibid.*, 823–832; “Simply Transitive Groups of Motions,” in *Monatshefte für Mathematik und Physik*,**43** (1936), 448–452; “Invariant Theory of Homogeneous Contact Transformations,” in *Annuals of Mathematics*, 2nd ser., **37** (1936), 747–765, written with M.S. Knebelman; “Graduate Study and Research,” in *Science***82** (1936),147–150; “Riemannian Spaces of Class Greater Than Unity,” in *Annals of Mathematics*,2ns ser., **38** (1937), 794–808; “Parallel Vectors in Riemannian Space,” in *Annals Mathematics*, 2nd ser., **39** (1938), 316–321; and *Coordinate Geometry* (Boston, 1939; repr. New York, 1960).

In the 1940’s Eisenhart published *An Introduction to Differential Geometry With Use of the Tensor Calculus* (Princeton, 1940); *The Educational Process* (Princeton, 1945); “*The Far-Seeing Wilson,” in William Starr Myers*, ed.m,*Woodrow wilson, some Princeton Memories* (Princeton, 1946), pp. 62–68;“Plan for a University of Discoverers,” in *The Princeton University Library Chronicle*, **8** (1947), 123–139,“Enumeration of potentials for which One particle Schrödinger Equations Are Separable,” in *Physical Review*, 2nd ser., **74** (1948), 87–89;“Finsler Spaces Derived from Riemann Spaces by Contact Transformations,” in *Annals of Mathematics*, 2nd ser., **49** (1948), 227–254; “Separation of the Variables in the One-Particle Schrodinger Equation in 3-Space,” in *Proceedings of the National Academy of Sciences*, **35** (1949), 421–418; and “Separation of the Variables of the Two-Particle Wave Equation,” *ibid*., 490–494.

Eisenhart’s publications of the 1950’s are “Homogeneous Contact Transformations,” in *Proceedings of the National Academy of Sciences*, **36** (1950), 25–30;“Walter Minto and the Earl of Buchan,”in *Proceedings of the American Philosophical Society*, **94** no. 3 (1950), 282–294; “Generalized Riemann Spaces,” **37** (1951), translation of *Introduction to Differential Geometry*... “Generalized Riemann Spaces, II,” in *Proceedings of the National Academy of Sciences*, **38** (1952), 506–508; “Generalized Riemann Spaces and General Relativity,” ibid., **39** (1953), 546–550; Preface to “Historic Philadelphia,” in Transactions of the American Philosophical Society, **43** , no. **1** (1953), 3; “Generalized Riemann Spaces and General Relativity, II,” in *Proceedings of the National Academy of Sciences*, **40** (1954), 463–466; “A Unified Theory of General Relativity of Gravitation and Electromagnetism. I,” ibid., **42** (1956), 249–251; II, *ibid.*, 646–650; III, *ibid.*, 878–881; IV, *ibid.*, **43** (1957), 333–336; “Spaces for Which the Ricci Scalar *R* Is Equal to Zero,” *ibid.*, **44** (1958), 695–698; “Spaces for Which the Ricci Scalar *R* Is Equal to Zero,” *ibid.*, **45** (1959), 226–229; and “Generalized Spaces of General Relativity,” *ibid.*, 1759–1762.

The early 1960’s saw publication of the following: “The Cosmology Problem in General Relativity,” in *Annals of Mathematics*, 2nd ser., **71** (1960), 384–391; “The Paths of Rays of Light in General Relativity,” in *Proceedings of the National Academy of Sciences*, **46** (1960), 1093–1097; “Fields of Unit Vectors in the Four-Space of General Relativity,” *ibid.*, 1602–1604; “Spaces Which Admit Fields of Normal Null Vectors,” *ibid.*, 1605–1608; “The Paths of Rays of Light in General Relativity of the Non-symmetric Field V_{4},” *ibid.*, **47** (1961), 1822–1823; “Spaces With Minimal Geodesics,” in *Calcutta Mathematical Society Golden Jubilee Commemorative Volume* (Calcutta, 1961), pp. 249–254; “Spaces in Which the Geodesics Are Minimal Curves,” in *Proceedings of the National Academy of Sciences*, **48** (1962), 22; “The Paths of Rays of Light in Generalized General Relativity of the Nonsymmetric Field V_{4},” *ibid.*, **49** (1963), 18–19; and “The Einstein Generalized Riemannian Geometry,” **50** (1963), 190–193.

II. Secondary Literature. Biographical memoirs are Gilbert Chinard, Harry Levy, and George W. Corner, “Luther Pfahler Eisenhart (1876–1965),” in *Year Book of the American Philosophical Society* for 1966 (Philadelphia, 1967), pp. 127–134; and Solomon Lefschetz, “Luther Pfahler Eisenhart,” in *Biographical Memoirs. National Academy of Sciences*, **40** (1969), 69–90.

Eisenhart’s work is discussed in Luigi Bianchi, *Lezioni di geometria differenziale*, Nichola Zanichelli, ed., II, pt. 2 (Bologna, 1930); Herbert Busemann, *The Geometry of Geodesics* (New York, 1955); J.A. Schouten, *Ricci Calculus*, 2nd ed. (Berlin-Göottingen-Heidelberg, 1954); T. Y. Thomas, “On the Projective and Equi-projective Geometries of Paths,” in *Proceedings of the National Academy of Sciences*, **11** (1925), 198–203; T.J. Willmore, *An Introduction to Differential Geometry* (London, 1959); and Kentaro *Yano’s Groups of Transformations in Generalized Spaces* (Tokyo, 1949).

H. Guggenheimer

#### More From encyclopedia.com

#### You Might Also Like

#### NEARBY TERMS

**Eisenhart, Luther Pfahler**