Di erential Geometry in Physics

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This document was reproduced by the University of North Carolina at Wilmington from a camera. ready copy supplied by the authors The text was generated on an desktop computer using LATEX. c 2006 2019, All rights reserved No part of this publication may be reproduced stored in a retrieval system. or transmitted in any form or by any means electronic mechanical photocopying recording or. otherwise without the written permission of the author Printed in the United States of America. These notes were developed as a supplement to a course on Differential Geometry at the advanced. undergraduate first year graduate level which the author has taught for several years There are. many excellent texts in Differential Geometry but very few have an early introduction to differential. forms and their applications to Physics It is the purpose of these notes to bridge some of these gaps. and thus help the student get a more profound understanding of the concepts involved We also. provide a bridge between the very practical formulation of classical differential geometry and the. more elegant but less intuitive modern formulation of the subject In particular the central topic of. curvature is presented in three different but equivalent formalisms. These notes should be accessible to students who have completed traditional training in Advanced. Calculus Linear Algebra and Differential Equations Students who master the entirety of this. material will have gained enough background to begin a formal study of the General Theory of. Relativity,Gabriel Lugo Ph D,Mathematical Sciences and Statistics. Wilmington NC 28403,lugo uncw edu,Preface iii,1 Vectors and Curves 1. 1 1 Tangent Vectors 1,1 2 Curves in R3 8,1 3 Fundamental Theorem of Curves 17. 2 Differential Forms 25,2 1 1 Forms 25,2 2 Tensors and Forms of Higher Rank 27.
2 3 Exterior Derivatives 34,2 4 The Hodge Operator 40. 3 Connections 49,3 1 Frames 49,3 2 Curvilinear Coordinates 51. 3 3 Covariant Derivative 54,3 4 Cartan Equations 57. 4 Theory of Surfaces 63,4 1 Manifolds 63,4 2 The First Fundamental Form 66. 4 3 The Second Fundamental Form 74,4 4 Curvature 78.
4 4 1 Classical Formulation of Curvature 79, 4 4 2 Covariant Derivative Formulation of Curvature 81. 4 5 Fundamental Equations 85,4 5 1 Gauss Weingarten Equations 85. 4 5 2 Curvature Tensor Gauss s Theorema Egregium 91. 5 Geometry of Surfaces 99,5 1 Surfaces of Constant Curvature 99. 5 1 1 Ruled and Developable Surfaces 99,5 1 2 Surfaces of Constant Positive Curvature 102. 5 1 3 Surfaces of Constant Negative Curvature 105,5 1 4 Ba cklund Transforms 108.
5 2 Minimal Surfaces 117,5 2 1 Minimal Area Property 117. 5 2 2 Conformal Mappings 121,5 2 3 Minimal Surfaces by Conformal Maps 126. 0 CONTENTS,6 Riemannian Geometry 137,6 1 Riemannian Manifolds 137. 6 2 Geodesics 147,6 3 Gauss Bonnet Theorem 156,Vectors and Curves. 1 1 Tangent Vectors, 1 1 Definition Euclidean n space Rn is defined as the set of ordered n tuples p hp1 pn i.
where pi R for each i 1 n, We may associate p with the position vector of a point p p1 pn in n space Given any two. n tuples p hp1 pn i q hq 1 q n i and any real number c we define two operations. p q hp1 q 1 pn q n i 1 1,cp hcp cp i, These two operations of vector sum and multiplication by a scalar satisfy all the 8 properties needed. to give Rn a natural structure of a vector space1, 1 2 Definition Let xi be the real valued functions in Rn such that xi p pi for any point. p hp1 pn i The functions xi are then called the natural coordinates of the the point p When. the dimension of the space n 3 we often write x1 x x2 y and x3 z. 1 3 Definition A real valued function in Rn is of class C r if all the partial derivatives of the. function up to order r exist and are continuous The space of infinitely differentiable smooth. functions will be denoted by C Rn or F R, In calculus vectors are usually regarded as arrows characterized by a direction and a length. Vectors as thus considered as independent of their location in space Because of physical and. mathematical reasons it is advantageous to introduce a notion of vectors that does depend on. location For example if the vector is to represent a force acting on a rigid body then the resulting. equations of motion will obviously depend on the point at which the force is applied. In a later chapter we will consider vectors on curved spaces In these cases the positions of the. vectors are crucial For instance a unit vector pointing north at the earth s equator is not at all the. same as a unit vector pointing north at the tropic of Capricorn This example should help motivate. the following definition, 1 4 Definition A tangent vector Xp in Rn is an ordered pair x p We may regard x as an.
ordinary advanced calculus arrow vector and p is the position vector of the foot of the arrow. 1 In these notes we will use the following index conventions. Indices such as i j k l m n run from 1 to n,Indices such as run from 0 to n. Indices such as run from 1 to 2,2 CHAPTER 1 VECTORS AND CURVES. The collection of all tangent vectors at a point p Rn is called the tangent space at p and. will be denoted by Tp Rn Given two tangent vectors Xp Yp and a constant c we can define new. tangent vectors at p by X Y p Xp Yp and cX p cXp With this definition it is clear that. for each point p the corresponding tangent space Tp Rn at that point has the structure of a vector. space On the other hand there is no natural way to add two tangent vectors at different points. The set T Rn consisting of the union of all tangent spaces at all points in Rn is called the tangent. bundle This object is not a vector space but as we will see later it has much more structure than. just a set,Figure 1 1 Tangent Bundle, 1 5 Definition A vector field X in U Rn is a smooth function from U to T U. The difference between a tangent vector and a vector field is. that in the latter case the coefficients ai are smooth functions of. xi Since in general there are not enough dimensions to depict a. tangent bundle and vector fields as sections thereof we use abstract. diagrams such as shown Figure 1 1 In such a picture the base. space M in this case M Rn is compressed into the continuum. at the bottom of the picture in which several points p1 pk are. shown To each such point one attaches a tangent space Here the. tangent spaces are just copies of Rn shown as vertical fibers in the. diagram The vector component xp of a tangent vector at the point. p is depicted as an arrow embedded in the fiber The union of all. such fibers constitutes the tangent bundle T M T Rn A section of. the bundle that is a function from the base space into the bundle Figure 1 2 Vector Field. amounts to assigning a tangent vector to every point in the the base. It is required that such assignment of vectors is done in a smooth way so that there are no major. changes of the vector field between nearby points, Given any two vector fields X and Y and any smooth function f we can define new vector fields. X Y and f X by,X Y p Xp Yp 1 2,f X p f Xp, Remark Since the space of smooth functions is not a field but only a ring the operations above.
give the space of vector fields the structure of a ring module The subscript notation Xp to indicate. the location of a tangent vector is sometimes cumbersome but necessary to distinguish them from. vector fields, Vector fields are essential objects in physical applications If we consider the flow of a fluid in. a region the velocity vector field indicates the speed and direction of the flow of the fluid at that. point Other examples of vector fields in classical physics are the electric magnetic and gravitational. 1 1 TANGENT VECTORS 3, fields The vector field in figure 1 2 represents a magnetic field around an electrical wire pointing. out of the page, 1 6 Definition Let Xp x p be a tangent vector in an open neighborhood U of a point p Rn. and let f be a C function in U The directional derivative of f at the point p in the direction of. x is defined by,Xp f f p x 1 3, where f p is the gradient of the function f at the point p. The notation, is also used commonly This notation emphasizes that in differential geometry we may think of a.
tangent vector at a point as an operator on the space of smooth functions in a neighborhood of the. point The operator assigns to a function the directional derivative of that function in the direction. of the vector Here we need not assume as in calculus that the direction vectors have unit length. It is easy to generalize the notion of directional derivatives to vector fields by defining. X f X f f x 1 4, where the function f and the components of x depend smoothly on the points of Rn. The tangent space at a point p in Rn can be envisioned as another copy of Rn superimposed. at the point p Thus at a point p in R2 the tangent space consist of the point p and a copy of. the vector space R2 attached as a tangent plane at the point p Since the base space is a flat. 2 dimensional continuum the tangent plane for each point appears indistinguishable from the base. space as in figure 1 2, Later we will define the tangent space for a curved continuum such as a surface in R3 as shown. in figure 1 3 In this case the tangent space at a point p consists of the vector space of all vectors. actually tangent to the surface at the given point. Figure 1 3 Tangent vectors Xp Yp on a surface in R3. 1 7 Proposition If f g F Rn a b R and X X Rn is a vector field then. X af bg aX f bX g 1 5,X f g f X g gX f, Proof First let us develop an mathematical expression for tangent vectors and vector fields that. will facilitate computation,4 CHAPTER 1 VECTORS AND CURVES. Let p U be a point and let xi be the coordinate functions in U Suppose that Xp x p. where the components of the Euclidean vector x are hv 1 v n i Then for any function f the. tangent vector Xp operates on f according to the formula. Xp f vi p 1 6, It is therefore natural to identify the tangent vector Xp with the differential operator.
Notation We will be using Einstein s convention to suppress the summation symbol whenever. an expression contains a repeated index Thus for example the equation above could be simply. written as, This equation implies that the action of the vector Xp on the coordinate functions xi yields the com. ponents ai of the vector In elementary treatments vectors are often identified with the components. of the vector and this may cause some confusion,The quantities. form a basis for the tangent space Tp Rn at the point p and any tangent vector can be written. as a linear combination of these basis vectors The quantities v i are called the contravariant. components of the tangent vector Thus for example the Euclidean vector in R3. x 3i 4j 3k, located at a point p would correspond to the tangent vector. x p y p z p, Let X v i be an arbitrary vector field and let f and g be real valued functions Then. X af bg vi af bg,v i i af v i i bg,av i i bv i i,1 1 TANGENT VECTORS 5.
X f g vi f g,v i f i g v i g i f,f v i i gv i i,f X g gX f. Any quantity in Euclidean space which satisfies relations 1 5 is a called a linear derivation on. the space of smooth functions The word linear here is used in the usual sense of a linear operator. in linear algebra and the word derivation means that the operator satisfies Leibnitz rule. The proof of the following proposition is slightly beyond the scope of this course but the propo. sition is important because it characterizes vector fields in a coordinate independent manner. 1 8 Proposition Any linear derivation on F Rn is a vector field. This result allows us to identify vector fields with linear derivations This step is a big departure. from the usual concept of a calculus vector To a differential geometer a vector is a linear operator. whose inputs are functions and the output are functions which at each point represent the directional. derivative in the direction of the Euclidean vector. 1 9 Example Given the point p 1 1 the Euclidean vector x h3 4i and the function f x y. x2 y 2 we associate x with the tangent vector,3 2x p 4 2y p. 3 2 4 2 14, 1 10 Example Let f x y z xy 2 z 3 and x h3x 2y zi Then. X f 3x 2y z,3x y 2 z 3 2y 2xyz 3 z 3xy 2 z 2,3xy 2 z 3 4xy 2 z 3 3xy 2 z 3 10xy 2 z 3. 1 11 Definition Let X be a vector field in Rn and p be a point A curve t with 0 p is. called an integral curve of X if 0 0 Xp, In local coordinates the expression defining integral curves constitutes a system of first order.
differential equations so the existence and uniqueness of solutions apply locally. 6 CHAPTER 1 VECTORS AND CURVES, 1 12 Definition Let F Rn Rm be a vector function defined by coordinate entries F p. hf 1 p f 2 p f m p i The vector function is called a mapping if the coordinate functions are all. differentiable If the coordinate functions are C F is called a smooth mapping If hx1 x2 xn i. are local coordinates in Rn and hy 1 y 2 y m i local coordinates in Rm a mapping F is represented. in advanced calculus by m functions of n variables. y j f j xi i 1 n j 1 m 1 9, For each point p Rn a mapping F Rn Rm induces a linear transformation F from the. tangent space Tp Rn to the tangent space TF p Rm This map is called the Jacobian map or the. push forward If we let X be a tangent vector in Rn then the tangent vector F X in Rm . Di erential Geometry in Physics These notes were developed as a supplement to a course on Di erential Geometry at the advanced 4 Theory of Surfaces 43

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