Everand Logo

Welcome to Everand!

  • Discover millions of ebooks, audiobooks, and so much more with a free trial

    From $11.99/month after trial. Cancel anytime.

    Non-Riemannian Geometry
    Non-Riemannian Geometry
    Non-Riemannian Geometry
    Ebook264 pages2 hours

    Non-Riemannian Geometry

    By Luther Pfahler Eisenhart

    ()

    Read preview

    About this ebook

    Non-Riemannian Geometry deals basically with manifolds dominated by the geometry of paths developed by the author, Luther Pfahler Eisenhart, and Oswald Veblen, who were faculty colleagues at Princeton University during the early twentieth century. Eisenhart played an active role in developing Princeton's preeminence among the world's centers for mathematical study, and he is equally renowned for his achievements as a researcher and an educator.
    In Riemannian geometry, parallelism is determined geometrically by this property: along a geodesic, vectors are parallel if they make the same angle with the tangents. In non-Riemannian geometry, the Levi-Civita parallelism imposed a priori is replaced by a determination by arbitrary functions (affine connections). In this volume, Eisenhart investigates the main consequences of the deviation.
    Starting with a consideration of asymmetric connections, the author proceeds to a contrasting survey of symmetric connections. Discussions of the projective geometry of paths follow, and the final chapter explores the geometry of sub-spaces.
    LanguageEnglish
    PublisherDover Publications
    Release dateJan 27, 2012
    ISBN9780486154633
    Non-Riemannian Geometry

    Read more from Luther Pfahler Eisenhart

    Related to Non-Riemannian Geometry

    Titles in the series (100)

    View More
    View More

    Related ebooks

    Mathematics For You

    View More
    View More

    Related podcast episodes

    Related articles

    • UNLIMITED

      A Supermassive Lens On The Constants Of Nature

      Nov 25, 2020

      The 2020 Nobel Prize in Physics went to three researchers who confirmed that Einstein’s general relativity predicts black holes, and established that the center of our own galaxy houses a supermassive black hole with the equivalent of 4 million suns

      9 min read

    • UNLIMITED

      The Theory Of Gravity

      Jan 20, 2022

      The Theory Of Gravity
      1 min read

    • UNLIMITED

      What Quantum Gravity Needs Is More Experiments

      Feb 2, 2017

      In the mid-1990s, I studied mathematics. I wasn’t really sure just what I wanted to do with my life, but I was awed by the power of mathematics to describe the natural world. After classes on differential geometry and Lie algebras, I attended a semin

      8 min read

    • UNLIMITED

      How Wireless Charging Works

      Dec 27, 2019

      How Wireless Charging Works
      1 min read

    • UNLIMITED

      Pipe-fitting Mounts

      Jun 3, 2020

      Pipe-fitting Mounts
      2 min read

    • UNLIMITED

      A Shocking Experience

      Aug 1, 2022

      A Shocking Experience
      3 min read

    • UNLIMITED

      Sodium-based Batteries Could Be Great Alternative To Lithium

      Jun 21, 2018

      New evidence suggests batteries based on sodium and potassium hold promise as a potential alternative to lithium-based batteries. The growth in battery technology has led to concerns that the world’s supply of lithium, the metal at the heart of many

      2 min read

    • UNLIMITED

      When a Good Scientist Is the Wrong Source

      Jun 23, 2021

      Six weeks ago, a reporter, Nicholas Wade, published what seemed to be a blockbuster story, one that, if true, would expose the greatest scandal in recent history. SARS-CoV-2, he wrote, or SARS2 for short, the virus that has driven the global COVID-19

      7 min read

    • UNLIMITED

      After 500 Years, A Mathematical Principle Has Unlocked Leonardo Da Vinci’s Human Heart Mystery

      Feb 18, 2022

      RESEARCHERS FINALLY UNDERSTAND THE function of a heart feature first described by Leonardo da Vinci 500 years ago. To find the answer, scientists used fractal theory, MRIs, and a lot of computational elbow grease to shed light on muscular structures

      3 min read

    • UNLIMITED

      Welcome

      Mar 23, 2023

      Decades in the planning, ESA's Jupiter Icy Moons Explorer (JUICE for short) is scheduled for launch this April. It's the first of Europe's three planned L-class missions, the largest and most costly class of campaign in Europe's long-term plan for sp

      1 min read

    • UNLIMITED

      Record Reviews

      Nov 7, 2023

      Record Reviews
      16 min read

    • UNLIMITED

      A New Age Of Sail

      Dec 14, 2021

      A New Age Of Sail
      8 min read

    • UNLIMITED

      The Different Kinds Of Dark Energy

      Sep 5, 2024

      The leading candidate for dark energy represents vacuum energy arising from ‘virtual particles’ popping in and out of existence in empty space. The main problem with this candidate is the massive difference between its theoretical value and its obser

      1 min read

    • UNLIMITED

      Pine Bookcase

      Mar 5, 2024

      Pine Bookcase
      6 min read

    • UNLIMITED

      When Pirates Ruled Asia's Waves

      Aug 31, 2023

      When Pirates Ruled Asia's Waves
      9 min read

    • UNLIMITED

      The U.S. Is Divided Over Whether Nuclear Power Is Part Of The Green Energy Future

      Jan 18, 2022

      Nuclear power is emerging as an answer as states transition away from coal, oil and natural gas to reduce greenhouse gas emissions and stave off climate change.

      7 min read

    Related categories

    Reviews for Non-Riemannian Geometry

    0 ratings

    0 ratings0 reviews

    What did you think?

    Tap to rate

    Review must be at least 10 words

      Book preview

      Non-Riemannian Geometry - Luther Pfahler Eisenhart

      NON-RIEMANNIAN GEOMETRY

      Luther Pfahler Eisenhart

      DOVER PUBLICATIONS, INC.

      Mineola, New York

      Bibliographical Note

      This Dover edition, first published in 2005, is an unabridged republication of the work originally published by the American Mathematical Society in 1927.

      Library of Congress Cataloging-in-Publication Data

      Eisenhart, Luther Pfahler, b. 1876.

      Non-Riemannian geometry / Luther Pfahler Eisenhart.

      p. cm.

      Originally published: New York : American Mathematical Society, 1927, in series: American Mathematical Society colloquium publications, v. 8.

      Includes bibliographical references.

      eISBN 13: 978-0-486-15463-3

      1. Geometry, Differential. I. Title.

      QA641.E55 2005

      516.3'6—dc22

      2004065109

      Manufactured in the United States of America

      Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y. 11501

      PERFACE

      The use of the differential geometry of a Riemannian space in the mathematical formulation of recent physical theories led to important developments in the geometry of such spaces. The concept of parallelism of vectors, as introduced by Levi-Civita, gave rise to a theory of the affine properties of a Riemannian space. Covariant differentiation, as developed by Christoffel and Ricci, is a fundamental process in this theory. Various writers, notably Eddington, Einstein and Weyl, in their efforts to formulate a combined theory of gravitation and electromagnetism, proposed a simultaneous generalization of this process and of the definition of parallelism. This generalization consisted in using general functions of the coördinates in the formulas of covariant differentiation in place of the Christoffel symbols formed with respect to the fundamental tensor of a Riemannian space. This has been the line of approach adopted also by Cartan, Schouten and others. When such a set of functions is assigned to a space it is said to be affinely connected.

      From the affine point of view the geodesics of a Riemannian space are the straight lines, in the sense that the tangents to a geodesic are parallel with respect to the curve. In any affinely connected space there are straight lines, which we call the paths. A path is uniquely determined by a point and a direction or by two points within a sufficiently restricted region. Conversely, a system of curves possessing this property may be taken as the straight lines of a space and an affine connection deduced therefrom. This method of departure was adopted by Veblen and the writer in their papers dealing with the geometry of paths, the equations of the paths being a generalization of those of geodesics by the process described in the first paragraph.

      of the coördinates, the law connecting the corresponding functions in any two coördinate systems being fundamental. Upon this foundation general tensor calculus is built and a theory of parallelism.

      . When the paths are taken as fundamental, this is the type of connection which is derived. This restriction is not made in the first chapter, which deals accordingly with asymmetric connections.

      Vectors parallel with respect to a curve for an asymmetric connection retain this property for certain changes of the connection. This is not true of symmetric connections. However, it is possible to change a symmetric connection without changing the equations of the paths of the manifold. Accordingly when the paths are taken as fundamental, the affine connection is not uniquely defined, and we have a group of affine connections with the same paths, a situation analogous to that in the projective geometry of straight lines. Accordingly there is a projective geometry of paths dealing with that theory which applies to all affine connections with the same paths. In the second chapter we develop the affine theory of symmetric connections and in the third chapter the projective theory.

      For a sub-space of a Riemannian space there is in general an induced metric and consequently an induced law of parallelism. There is not a unique induced affine connection in a sub-space of an affinely connected space. If the latter is of order m and the sub-space of order n, each choice at points of the latter of m—n independent directions in the enveloping space but not in the sub-space leads to an induced affine connection, and to a geometry of the sub-space in many ways analogous to that for Riemannian geometry. Under certain conditions there are preferred choices of these directions, which are analogous to the normals to the sub-space. The fourth chapter of the book deals with the geometry of sub-spaces.

      A generalization of Riemannian spaces other than those presented in this book consists in assigning to the space a metric based upon an integral whose integrand is homogeneous of the first degree in the differentials. Developments of this theory have been made by Finsler, Berwald, Synge and J.H. Taylor. In this geometry the paths are the shortest lines, and in that sense are a generalization of geodesics. Affine properties of these spaces are obtained from a natural generalization of the definition of Levi-Civita for Riemannian spaces. Berwald has also obtained generalizations of the geometry of paths by taking for the paths the integral curves of a certain type of differential equations, and Douglas showed that these are the most general geometries of paths: he also developed their projective theory. References to the works of these authors are to be found in the Bibliography at the end of the book.

      This book contains, with subsequent developments, the material presented in my lectures at the Ithaca Colloquium. in September 1925, under the title The New Differential Geometry. I have given the book a more definitive title.

      In the preparation of the manuscript I have had the benefit of suggestions and criticisms by Dr. Harry Levy. Dr. J. M. Thomas and Mr. M. S. Knebelman, the latter of whom has also read the proof.

      September, 1927.

      LUTHER PFAHLER EISENHART.

      Princeton University

      CONTENTS

      CHAPTER I

      ASYMMETRIC CONNECTIONS

      1.Transformation of coördinates

      2.Coefficients of connection

      3.Covariant differentiation with respect to the L’s

      4.Generalized identities of Ricci

      5.Other fundamental tensors

      6.Covariant differentiation with respect to the I’s

      7.Parallelism. Paths

      8.A theorem on partial differential equations

      9.Fields of parallel contravariant vectors

      10.Parallel displacement of a contravariant vector around an infinitesimal circuit

      11.Pseudo-orthogonal contravariant and covariant vectors. Parallelism of covariant vectors

      12.Changes of connection which preserve parallelism

      13.Tensors independent of the choice of ψi

      14.Semi-symmetric connections

      15.Transversals of parallelism of a given vector-field and associate vector-fields

      16.Associate directions

      17.Determination of a tensor by an ennuple of vectors and invariants

      18.The invariants γ μvσ of an ennuple

      19.Geometric properties expressed in terms of the invariants γ μvσ

      CHAPTER II

      SYMMETRIC CONNECTIONS

      20.Geodesic coördinates

      21.The curvature tensor and other fundamental tensors

      22.Equations of the paths

      23.Normal coordinates

      24.Curvature of a curve

      25.Extension of the theorem of Fermi to symmetric connections

      26.Normal tensors

      27.Extensions of a tensor

      28.The equivalence of symmetric connections

      29.Riemannian spaces. Flat spaces

      30.Symmetric connections of Weyl

      31.Homogeneous first integrals of the equations of the paths

      CHAPTER III

      PROJECTIVE GEOMETRY OF PATHS

      32.Projective change of affine connection. The Weyl tensor

      33.Affine normal coördinates under a projective change of connection

      34.Projectively flat spaces

      35.Coefficients of a projective connection

      36.The equivalence of projective connections

      37.Normal affine connection

      38.Projective parameters of a path

      39.Coefficients of a projective connection as tensors

      40.Projective coordinates

      41.Projective normal coordinates

      42.Significance of a projective change of affine connection

      43.Homogeneous first integrals under a projective change

      44.Spaces for which the equations of the paths admit n(n+ 1)/2 independent homogeneous linear first integrals

      45.Transformations of the equations of the paths

      46.Collineations in an affinely connected space

      47.Conditions for the existence of infinitesimal collineations

      48.Continuous groups of collineations

      49.Collineations in a. Riemannian space

      CHAPTER IV

      THE GEOMETRY OF SUB-SPACES

      50.Covariant pseudonormal to a hypersurface. The vector-field vα

      51.Transversals of a hypersurface which are paths of the enveloping space

      52.Tensors in a hypersurface derived from tensors in the enveloping space

      53.Symmetric connection induced in a hypersurface

      54.Fundamental derived tensors in a hypersurface

      55.The generalized equations of Gauss and Codazzi

      56.Contravariant pseudonormal

      57.Fundamental equations when the determinant ω is not zero

      58.Parallelism and associate directions in a hypersurface

      59.Curvature of a curve in a hypersurface

      60.Asymptotic lines, conjugate directions and lines of curvature of a hypersurface

      61.Projectively flat spaces for which Bij is symmetric

      62.Covariant pseudonormals to a sub-space

      63.Derived tensors in a sub-space. Induced affine connection

      64.Fundamental derived tensors in a sub-space

      65.Generalized equations of Gauss and Codazzi

      66.Parallelism in a sub-space. Curvature of a curve in a sub-space

      67.Projective change of induced connection

      BIBLIOGRAPHY

      CHAPTER I

      ASYMMETRIC CONNECTIONS

      1. Transformation of coördinates. Any ordered set of n independent real variables xi, where i takes the values 1, …, n, may be thought of as coördinates of points in an n-dimensional space Vn, in the sense that each set of values of the x’s defines a point of Vn. The terms manifold and variety are synonymous with space as here defined. If φi (x¹, …, xn,…, n are real functions, whose jacobian is not identically zero, the equations

      define a transformation of coördinates in the space Vn.

      If λi and λ′i are functions of the x’s and x′’s such that

      in consequence of (1.1), λi and λ′i are the components in the respective coordinate systems of a contravariant vector. In (1.2) we make use of the convention that when the same index appears as a subscript and superscript in a term this term stands for the sum of the terms obtained by giving the index each of its n values; this convention will be used throughout the book. From (1.2) we have by differentiation

      It is assumed that the reader is familiar with relations connecting the components of a tensor in two coördinate systems.* He will observe that, because of the presence of the last term in the right-hand member of (1.3), the derivatives of λi and λ′i are not the components of a tensor.

      Consider further a symmetric covariant tensor of the second order whose components in the two coördinate systems are gij and g′αβ such that the determinant

      is different from zero. From the equations

      we get by differentiation

      A similar observation applies to these equations. However, there are n²(n+l)/2 of these equations, and they can be solved for the n²(n . We obtain

      are the Christoffel symbols of the second kind, that is,

      where gih are defined by

      from (1.3) by means of (1.5), we obtain

      where

      From (1.8) we see that λi, j and λ′α, β are the components of a tensor in the two coördinate systems. Thus we have formed a tensor by suitable combinations of the first derivatives of the components of a vector and a tensor.

      If gij is the fundamental tensor of a Riemannian space, then λi, j is the covariant derivative of λi. However, the theory of covariant differentiation in a Riemannian space has nothing to do with the fact that the tensor gij is used to define a metric. Consequently this theory can be applied to any space, if we make use of any tensor gij such that g ≠ 0.*

      2. Coefficients of connection. We have just seen that when a symmetric tensor gij from equations

      Enjoying the preview?
      Page 1 of 1
      pFad - Phonifier reborn

      Pfad - The Proxy pFad of © 2024 Garber Painting. All rights reserved.

      Note: This service is not intended for secure transactions such as banking, social media, email, or purchasing. Use at your own risk. We assume no liability whatsoever for broken pages.


      Alternative Proxies:

      Alternative Proxy

      pFad Proxy

      pFad v3 Proxy

      pFad v4 Proxy