(ebook - math) - Differential Geometry In Physics (Lugo, 1998), Notas de estudo de Física

Baixe (ebook - math) - Differential Geometry In Physics (Lugo, 1998) e outras Notas de estudo em PDF para Física, somente na Docsity! Differential Geometry in Phys Gabriel Ingo Department of Mathematical Sciences University of North Carolina at Wilmington si Copyright 1995, 1998 his document was reproduced by the University of North Carolina al Wilminglon from a camera ready copy supplied by le authors. Lhe lext was generated on an desktop computer using EIFX. Copyright (1992, 1995 All rights reserved. No part of this publication may be reproduced, stored in a retricval system, or transmitted, in any form or by any means, electronic. mechanical, photocopying. recording or otherwise, without the written permission of the authors. Printed in the United States of Americ dv Preface 1 Vectors and Curves 1 Tangent Vec 12 Curvesin R 1.3 Tundamental Theorem of Curves 2 Differential Forms 2.10 IForms cce e e e 2.2 Tensors and Torms of Thigher Rank .. 23 Exterior Derivatives . 24 The Todge-s Operator «cc 3 Connections 8.1 Frames 32 Curvilinear Coordinates 33 € 3.4 Cartan Equations cc. variant Derivative . cc. 4 Theory of Surfaces 41 Manifolds .lccccccccc e 42 The Tirst Tundamental Form The Second Fundamental Form .. 44 Curvalute cc eee e e Chapter 1 1.1 Tangent Vectors 1.1 Definition Euclidem n-space R?º is defincd as the set of ordered n-tuples p = (p1,...,9"), where pº ER. forcachi=1,....m. Given any two n-tuples p= (p!,..p), q=(g',...,4') and any real number c, we define two operations: p+tg o (prq +ç) (1.1) cp = (ep! cp") With the sum and the scalar multiplication of ordered n-tuples defined this way, Tuclidean space acquires the structure of a vector space of n dimensions 1.2 Definition Let x) be the rcal valued functions in Rº such that zi(p) = pi for any poimt p= (p!,...,p"). The functions 2º are then called the natural coordinates ol We the point p. Wen el =yadal=s the dimension ol the space n = 3, we olten write: x! = 1.3 Definition À real valued function in Rº is ol class C” if all the partial derivatives of the function up to order 7 exist and are continuous, “The space of infinitely differentiable (smooth) functions will be denoted by CS(R). Tn advanced caleulus, vectors are usually regarded as arrows characterized by a direction and a ion in space. Because of physical amd mathematical reasons, it is advantageous to introduce a notion of vectors which does depend em ample, 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 applicd. lu a later chapter we will also consider vectors on spaces which are curved, In Lhese cases the position of the vectors is crucial, for instance, à unit vector pointing north at the eartl's equator, Is not at all the same as a unit vector pointing north at the tropic ol Capricorn. This example should help motivate the following definition. length. Vectors as thus considered as independent of their loca: location. Tor cx 1.4 Definition A tangent vector X, in Rº, is an ordered pair (X.p). We may regard X as an ordinary advanced caleulus vector and p is the position vector of the foot the arrow. tm these notes we will use the following index conventions. Indices such as à, j,b,Lm, 1, vun from 1 to a such as p,v, 7.0, rum from O tom such as o, 0,7 8, run from | to 2. CHAPTER 1. VECTORS AND CURVE One may think of lhe parameter ! as representing time, and the curve « as representing the trajectory ol a moving point particle. 1.10 Example Let a(b) = (t+ by. ant + bo ast + ba). This equation represents a straight line passing through the point p = (b1,ba. by), in the direction of the vector v = (ay. a», a3). 1.11 Example Let a(f) = (a cosut, asim ut, bt). as the path des wrapped around a circular cylinder of radius a. This curve is called à ribed by the hypothenuse of a triangle with slope b, which The projection of the helix onto the ay-plane is a cirele and the curves risos at a constant rate in the z-direction. reular helix. Geometrically, wc may view the curve Oecasionally we will revert to the position vector notation (8) (18) shieh is more prevalent in vector caleulus and clementary physics texthooks. Of course, what this notation really means is x(4) = (2"(8) (8, (8) = (2 o 0)(1), (1.9) where x! are the coordinate slot functions in an open set in Rê. 1.12 Definition The derivative a/(t) of the curve is called the velocity vector and the derivative a”(t) is the speed of the curve. The components of the veloc dx (del da? das vg=-T-(E ma 1 = ( dt 7) (110) de PO fd Do (dus (o (D+) (111) 1 position vector given by dal da? das dx (E a a (1.12) ; called the acceleration. The length + = [Jo*(8)]| of the velocity vector is called y vector are simply given by and lhe speed is The diferential of dx of the classic a dO is called an infinitesimal tangent vector, and lhe norm ||dx]| of the infinitesimal tangent vector is called (he differential of arclenguh ds. Clearly we have ds = |ldx|| = dt (1.13) As we will see later in this text, the notion of infinitesimal objects needs to be treated in a more rigorous malhematical setting. Al lhe same time, we must not discard the great intuitive value ol this notion as envisioned by Lhe masters who invented of Calculus; even at Lhe risk of some possible confusion! Thus, shercas in the more strict sensc of modern differential geometry, the velocit vector is really a tangent vector and hence it should be viewed as a lincar derivation on the space of functions, it is helpful to regard dx as a traditional vector srhich, at the infinitesimal level, gives a lincar approximation to the curve. 12. CURVES INR? 1£ fis any smooth function on Rº, we formally define a/(8) in local coordinates by We formula (0) exit = Eua (114) “The modern nolation is more precise, since il Lakes into account that the velocity las a vector part as well as point of application. Given a point on the curve, the velocily of the curve acting on a function, yields the direcional derivative of that function in the direction tangential to the curve al the point in question. The diagram below provides a more geometrical interpretation ol the the velocity vector lor- mula (1.14). Lhe map o(!) from R to R$ induces a map au from lhe tangent space of R to the tangent space of Rê. The image o(£) in TRÊ of the tangent vector É is what we call a'(£) dl ali)= (8. Since a!(t) à function fon Rº a!(t) to the coordinate Tunclions 4”, we gel the components of the the Langent vector a langent vector in R$, it acts ou [unctions in Rê. Lhe action ol «(!) on a is the same as the action of & on Uhe composition fo a. In particular, il we apply as ilustrated a as GETROS TRÊS o'(t) 4 4 RREO R (8) (e) lares The map a, on the tangent spaces induced by the curve a is called the push-forward. Many authors use the notation da: to denote the push-forward. but we prefer to avoid this notation because most students fresh out of advanced caleulus have not yet been introduced to the interpretation of (1.15) Uhe differential as à linear isomorphism on tangent spaces. 1.13 Definition TE = 4(3) is a smooth, real valued function and a(f) is a curve in R$, wo sa 3(5) = o(t(5) is a reparametrization of a À common reparametrization of curve is obtained by using the arclength as the parameter. Using this reparametrization is quite natural, that the rate of change of Ue arclengih is seliat we call speed g that the curve nec we know from ba; =“ . v= Slot (1.16) The are length is obtained by integrating the above formula “= flotopa= f (5) (5 +) & (L17) Tn practice it is typically difficult to actually find an explicit arclength parametrization of a curve since not only does one have caleulate the integral, but also one needs to be able to find the inverse function t in terms of s. On the other hand. from a theoretical poimt of view, arclength parametrizations are ideal since any curve so parametrized. has unit speed. The proof of this fact i a simple application of the chain rule and the inversc function theorem. Bs) = [lts 6 CHAPTER 1. VECTORS AND CURV = afeto) 1 a(s) —L— ro) O st) Iron and any vector divided by its length is a unit vector. Leibnitz notation makes this even more self exident, dx dxdi de dtds dx - Cê GE 1.14 Example Let (t) = (acoswt, asincot, dt). Then VÍt) = (Cawsinwl, du cost, b), a “) = / v [sia du bo = ct, where, c> Vau b (au sin au)? + (au coswr)? + B2 du Yhe helix of unit speed is then given by su, (5) = (a cos — esmn — (9) = (oc0s E, ain E Frenet Frames Lei 3(5) be à curve paramelrized by arc length and lei (8) be lhe vector T(s) = 8'(8). (1.18) Yhe vector P'(s) is langential to the curve and it has unit lenglh. Herealter, we will call £ We unit Tangent vector, Dillerentiating We relation Fr=l (1.19) we gel 9TT'=0, (1.20) so we conelude that the vector T” is orthogonal to T. Let NX be a unit vector orthogonal to T. and let x be the scalar such that TU) = aN(s). (131) We call N the unit normal to the curve, and & thc curvature. Taking the length of both sides of last equation, and recalling that NX has unit length, wc deduee that e= Pe) (1.22) 1,2, CURVES “Then, (é) Ive = = r3(sin" t+ cos?!) (=rsint,rcost.0) sint)2 + (rcost)2 + 02 =". Therefore ds/dt = r and s = rt, which we recognize as the formula for the length of am are of circle of radius t, subtended by a central angle whosc measure is £ radians. We conclude that, EQ os E reos 5,0) his is à very simple but important example, “The fack that for à circle of radius v the curvature e has large curvalure and à large circle has small isa = 1/r could not be more intuitive, À small cir curvalure. As Uhe radius of the circle approaches infinity, the circle locally looks more and more like a straight line, and the curvature approaches to 0, 1f one were walking along a greal circle on à very large sphere (like the carth) one would be perecive the space to be locally flat. 1.18 Proposition Let o(t) be a curve of velocity V, acecleration A, speed 4 and curvature «, then V= a de 2x A = Cpyçan. 1.29 qr + ts (1.29) Proof Let s(t) be the arclengih and let 2(5) be à unit speed reparametrization. Then a(t) = (s(4)) and by the chain rule V = off) = 5"((0)() = “T A = (f) = GTA RT (st (0 = GTA (aja = GTA tax 10 CHAPTER 1. VECTORS AND CURVE Equation 1.29 is iuportant in physics. The equation states that a particle moving along a curve in space feels a component of acceleration along the direction of motion whenever there is a change ol speed, and à centripetal acceleration in lhe direction ol the normal whenever it changes direction. The centripetal acceleration and any point is e 7 where r às the radius of a circle which las imaximal tangential contact will the curve at te poiat in question. “This langential circle is called the osculating circle. “Lhe osculating circle can be envisioned by a limiting process similar to thai of the tangent 10 a curve in dillerential calculus. Lei p be point ou the curve, and let q; aud q; two nearby points. [he three points determine a cirele uniquely. This cirele is a “secant? approximation to the tangent circle. As the points qi and q» approach the point p, the “secant” cirele approaches the osculating circle. The osculating circle always lies in the the TN-plane. which by analogy, à called the osculating plane. 1.19 Example (Telix) Hs) = (acos E asin E here € EC o = (“sin beta (o) = no = a I é atu! Simplifying the last expression and substituting the vale of e, wc get dh ,r ER a = E 2 +52 ç Notice tab if b = 0) the helis collapses to a circle in lhe reduce to x = 1/a and 7 = 0. The ratio n/7 = au/b is particularly simple. Any curve where «/7 = constant is called a helix, of wlich the circular Delix is a special case, plane. lu this case the [ormulas above 1.20 Example (Plane curves) Let a(f) = (2(2),4(1),0). Them ad = (8,0) 12. CURVES INR? 11 af = (AO) O (PO) lo! x 07] E a or uyt— 1.21 Example (Comu Spiral) Let (5) = (2(4), 4(8),0), where | rj = | cosisd ) / Bo" a $2 sj) = st tt. 1.3 v(s) / sin so é (1:30) Then, using the fundamental thcorem of caleulus, wc have the Comu spiral is given by x IH The integrals (1.30) defining the coordinates of the Cormu spiral arc the classical Frenel Integrals. These functions, as well as th al itself arise in the computation of the diffraction pattern of a coherent beam of light by à straight edge. In cuses where lhe given curve o(t) is not of unit speed, the following proposition provides formulas to compute Lhe curvature and Lorsion in terms ol a 1,22 Proposition Il a(!) is a regular curve iu R$, then vo axe!b E > EA 13 ' Ei (at) om) = 1.32 = Ps (1.32) where (aaa!) is We triple vector produel [a x! 0 at”, Proof: q = 7 ofo= dP+eRa = (NUS +. = etuN'+ = werB+.. 14 CHAPTER 1 VECTORS AND CURVES Chapter 2 2.1 1-Forms One of the most puzzling ideas in clementary caleulus is the idea of the differential. In the usual definition, the differential of a dependent variable y = (2), is given im terms of the diffcrential of the independent variable by dy = F'(z)de. The problem is with the quantity de. What does dz mean? What is the difference between Ag and de? How much “smaller” than Az does de have to be? “Lhere is no trivial resolution to his question. Most introductory calculus tests evade the issue by treating de as am arbitravily small quantity (which lacks mathematical tigor) or by simply referring to du as an infinitesimal (a Lerm introduced by Newton [or au idea that could not otherwise be clearly defined at the time.) Tn this section we introduce lincar algebraic tools that veill allow us to interpret the difierential in terms of am lincar operator. 21 Definition Let p€ Rº, and let T,(Rº) be the tangent map 6 from T(R?) into R. Wo recall that such a map mus pace at p. A L-form at p is a lincar 'y the following properties a) AX)ER, VXp ER” (2.1) db) dlaMp +) = ad(Ap) + AM), Va,DER, Xp.%p E TUR!) A 1-form is a smooth choice of a lincar map é as above for cach point in the space. 2.2 Definition Let f:Rº > R be a rcal-valucd € function. We define the diffcrential df of the function as the 1-form such that dx) =*(1) (2.2) for every vector field in X in Rº. Tn other words, at any point p. the differential df of a function is an operator which assigns to à tangent vector X,, te directional derivative of the function in the direction of that vector FO) = Nil) = Ho) -X(p) (2.3) In particular, if we apply the dilferential of We coordinate Tunctions x Lo the basis vector fields, we gel (2.4) The set of all lincar functionals on a vector space is called the dual of the vector space. Tt is am standard thcorem in lincar algebra that the dual of a vector space is also à vector spacc of the 15 16 CHAPTYER 2. DIFFERENTIAL FORMS same dimension. “Lhus, the space Z/RY of all L-lorms at p is a vector space which is the dual o Lhe Langent space 7,Rº. “The space “1; =(Rº) is called the cotangent space of Rº' at the point p. Equation (2.4) indicates Lhal the sel of diferential forms tda) pre (dar) "Jp + constitutes the basis of the cotangemt space which is dual to the standard basis £ 1 ol lhe Langent space. jr s p rauges over all points ia Rº is called the cotangent bundle 'Yhe union of all the cotangent space TR). 2.3 Proposition Let f be any smooth function in Rº and let (x1,...2”] be coordinate functions in a neighborhood [ of a point p. Then. the differential df is given locally by the expression df = Ee da (2.5) i=l BL Ba ble as a Proof: The differential df lincar combination of the basis clements f(dr!),...., (dz?) ). Therefore, to prove the proposition, it sufficos to show that the expression 2.5 applied to am arbitrary tangent va with defini To sec this, consider a tangomt vector X, = q (45) and apply the expression above 3 by definition a 1-form, so, at cach point, it must be expr or, coincides om 2 É) (E (o e) (2.6) (2) (Ee (49) The definition of differentials as lincar functionals on the space of vector fields is much more sati factory than the notion of infinitesimals, since the new definition is based on the rigorous machinery of lincar algebra. Ta is am arbitrary 1-form, then locally a =a(xdel+,... + au(x)de”, ç where the cocfficients a; are CS functions. À 1-form is also called a covariant tensor of rank 1, or just simply à covector. “Lhe coellicients (a1,..., 4) are called the covariant components ol Lhe covector. We will adopt the convention to always write the covariant components of à covector with the indices down. Physicisis often reler Lo Lhe covariant components of à I-form as à covariant vector and this causes some contusion about le position of Lhe indices. We empliasize that not all one forms are obtained by taking le dilerential of à [unclion. If there esisis à function f, such that = df, then the one form «is called exact. In vector caleulus and elementary physics, exact forms are important in understanding the path independence of line integrals of conservative vector fields. As we have already noted, the cotangent space TE(R”) of |-forms at a point p has a natural vector space structure. Ne cam casily extend the operations of addition and scalar multiplication to 2.2. TENSORS AND FORMS OF HIGHER RANK 19 17 we now deline bi = gb, (2.14) we see that the equation above can be rewritten as dby= gabi. and we recover the expression for the inner product. Equation (2.14) shows that the metric can be used as mechanism to lower indices, thus Lrans- forming the contravariant components of a vector to covariant ones. Tf we let 9% be the inverse of the matrix g;;. that is ih si 5 E sÉ 9 6; (2.15) wc can also raise covariant indices by the equation bi= qb; (2.16) We have mentioned that the tangent and cotangent spaces of Tnclidean space at a particular poimt are isomorphic. In vicw of the above discussion. we see that the metric accopts a dual interpretation: one as bilincar pairing of two vectors VR) TR) SR and another as à linear isomorphism STARS T(R") Lab maps vectors Lo covectors and vice-versa. mn clementary trcatments of caleulus authors often ignore the subtleties of differential I-forms and tensor products and define the diffcrential of arclength as ds? = gide'de', although, what is really meant by such an expression is aid o del. (2.17) 2.7 Example lu cylindrical coordinates, the dilTerential of arclengih is ds? = dr + de 4 do? (2.18) Tn this case the metric tensor has components 10 0 m=/0 7 01. (2.19) 001 2.8 Example Im spherical coordinatos e = psinficosy v = psinfsinê = post, (2.20) the diffcrential of arclengkh is given by = do + de + pêsin” Odo”, (2.21) la this case the metric tensor has components 100 sm=|0 20 . (2.22) 00 ping? 20 CHAPTYER 2. DIFFERENTIAL FORMS Minkowski Space An important object in mathematical physies is the so called Minkowski space sehich is can be defined as the pair Let (M1,3,9) be the pair, where Masg= tale, a)t ci eR) and q is the bilinear map such ta HI = LA ue PA (+ (ué (2.24) Lhe matris representing Minkowski's mebric g is given by 9 =diag(=1,1,1,1), ia which case, the dilferential of arclength is given by ds = gude da” = -dtodt+ del o dal 4 de cdr 4 dr? a de? = de + (del 4 (da?)2 4 (day. (2.25) Note: “Technically speaking, Minkowski's metrie is not really a metric since g(X,X) = 0 does not imply that X = 0. Non-zero vectors with zero length are called Liglt-like vectors and they are associated with with particles which travel at the speed of light (which sec have set equal to Lin our tem of units.) The Minkowski metric gaz and its matrix inverso q” are also uscd to raisc and loser indices in the space in a manner completely analogous to Rº . Thus, for example, if À is a covariant vector with components Au= (poda, then the contravariant components of À are Wedge Products and n-Forms 2.9 Definition A map é: T(Rº) x Y(R7) — R is called alternatingil AX) = (FX) The alternaling property is reminiscent of determinants of square mabrices which change sign il auy two column vectors are switched, In fact, Lhe determinant function is à perfect example ol au alternating bilincar map on the space Max» of two by two matrices. Of course, for the definition above to apply, onc has to view Max» as the space of column vectors. 2.10 Definition A 2-form Gisa map é: T(R?) x T(Rº) — R. which is alternating and bilincar. 2.11 Definition Lei o and 3 be I-forms in Rº and let X and Y be any two vector fields, The wedge product of the two Llorms is the map AS: TR”) x T(R") — R given by the equation (en XI) = (MO) — (ICO) 2.2. TENSORS AND FORMS OF HIGHER RANK 21 2.12 Theorem If and 3 are I-lorms, then a A 2 is a 2-form, Proof: : We break up the proof into the following two lemmas. 213 Lemma The wedge product of two 1-forms is alternating. Proof: Let a and 3 be I-formsin Rº and let X and Y be any two vector ficlds. then (CAST) = (080) = e0)0T) —(0(F)800) — e(08()) (0 AB. X) 2.14 Lemma Lhe wedge product of ivo Llorms is bilinear, Proof Consider I-forms, a, 2, vector fields X1,X2,Y and Tunciions [1,62 Then, since the 1-forms are linear funciionals, we gel (en PM + = fx + PAIS) aà + PP Xo) E e(Ã) + P(A (O) — QL A) + P(A) PANIC A P(A IO) + FAQ) + P(A) PAIO + OBD] A Poa) 20) + 000 )H(X5)] Fo A XI A Po A SIX», Y) The proof of lincarity on the second slot is quite similar and it is left to the reader. 215 Corollary If q and 3 arc I-forms, then ang=-9A0 (2.27) last result tells us that secdge products have characteri similar to cross products of vectors im the sense that both of these products are anti-commutative. This means that we need to be careful to introduce à minus sign every time sec interchange the order ofthe operation. Thus, for example, we have de Adel = —de A de ifi j, vhercas de nda = dei nd =0 since any quantity which is equal Lo the negative of itself must vanislt “Lhe similarity between wedge producis is even more suiking iu the next proposition but we emphasize again that wedge products are by far much more powerful than cross products, because wedge products cam be computed in any dimension. 2.16 Proposition Leia = Ade ud d = Bida! be any two Llorms in Rº. Then ang=(ABde A de (2.28) Proof: Let X and Y be arbitrary vector fields, [hem (a A BUAS) (asd OB) — (Ad OB da (A) (As By) [da (dar 7) — dr (dai (0)] (As Bj) (dr A de (XY) 24 CHAPTYER 2. DIFFERENTIAL FORMS 2.22 Proposition a) di NS A b) d=dod=0 O) daeng)=dang+(-IJondo VacA den (2.34) Proof: a) Obvious from equation (2.32). b) Tirst we prove the proposition for a = fe A. We have d(da) = de? A de” of 7 aca de A de that Now, suppose that a is represented locally as in equation (2.32). Tt follows from d(da) = d(dAs, iu) Ada A deli. der =0 c) Leta Ee A" de A”. Lhen we can write e = As idade? A. dei 8 By jileldei A. dela, (2.35) By definition, 0n8= Ana Ba cp Ad A Adro) A (dei AoA dedo) Now we take take the exterior derivative of the last equation taking into account that d(fy) = Edy + gdf for any functions f and g. We get ond) = (MA) Bida (Aa MB cs dae ncia dei) A (dad An dai) = [As (da Ade AB, A (dra A dei] + = Asa A (det A A de A (DAR A (dit AA deio)] = dang+(-IJa nda. (2.36) Lhe (= 1)" factor comes in because Lo pass lhe term dB, 5, Uurough p L-forms of the type de”, one has to perform p transpositions. 2.23 Example Leto = Pe, gde + Qle,g)d8. Then, JP o o dos CE) ndo + (Ca 4 22) pay dy de Oy da = oP o = Pyndo + Cor a dy dy E o) àP 0 din dy. gy This example is related to Green's theorem in RZ. 2.4, THE HODGE-+* OPERATOR 25 2.24 Example Leio = M(e, y)de+ Ne, y)dy, and suppose that do = 0. Then, by the previous example, ON OM —- E yde A dy. de pg it A dy that Nº da Thus, da = 0 if? My vehich impli F and Mo for some Cl function f(a a=tdr+fydf= df The reader should also be familiar with this example in the context of exact differential cquations of first order, and conservative force fields. 2.25 Definition À dillerential form à is called exact il da = 0. 2.26 Definition À dillerential form à is called closed if there exist a [orm 2 such that a = dj. Since dod=0, it is clear Uial a closed [orm is also exact, “The converse is not at all obvious in general and we state it here srithout proof. 2.27 Poincare's Lemma Mm a simply connected space it is closed. The assumption hypothesis that the space must be simply connected is someehat subtle. The condition is reminiscent ol Cauchy's integral theorem [or [unctions of a comples variable, which uch as Rº ). if a differential is exact then slutes that if f() is holomorphie function and C is à simple closed curve, then, f fl)d:=0 er This theorem does not hold if the region bounded by the curve C is not simply connected. The standard example is the integral of the complex IHorm w = (L/:)dz around the unit circle €' bounding a punctured disk. Jo this case, 2.4 The Hodge-* Operator One of the important lessons that students learn in lincar algebra is that all vector space of finite dimension n arc isomorphie to cach other. Thus, for instance. the space Py of all real polynomials in 2 of degree 3, and the space Max» of real 2 by 2 matrices, arc basically no different than the Euclidean vector space Rá in terms of their vector space properties. We have already encountered a number of vector spaces of finite dimension in these notes. A good example of this is the tangent space T;R$. The “vector” part af + a2Z + aºZ can be mapped to a regular advanced calentus vector ali + a2j + aºk, by roplacing É by É byjand £ by kk. Of course, wo must not confuso a tangent vo h a Tuclideam vector which is just an ordered triple but as far their vector space properties. there is basically no difference. We have also observed that the tangent space TR” or which is a lincar operator w isomorphie to the cotangent space TR”. á Im this case, the vector space isomorphism maps the standard bs Vito their duals fdz'). This isomorphism then transforma a contravariant vector to a covariamt vector. Another interesting example is provided by the spaces Ai (R$) and AÇ(R?), both of which have dimension 3. Tt follows that th smorphic. Tn this casc the isomorphism i given by the map vecto » two spaces must be dr — dyAde 26 CHAPTYER 2. DIFFERENTIAL FORMS dy > —dende dz > dendy (2.38) More generally, we have seen that We dimension of the space o! m-forms in Rº is given by the (1) n nt m)=(n—-m = (um)! APR) = AR) (2:39) Lo describe the isomorphism between Lhese two spaces, we will first need to introduce the totally ymmetric Levi-Civita permutation symbol which is defined as follows Ra RO binomial coefliciem (7). Since it must be the case that + if(i, oosim) à5 am even permutation of(1,...,m) = ifi 1) is an odd permutation of(1,...,m) (2.40) 0 otherwise In dimension 3, there are ouly 3 (3!=6) nonvanishing components of esp in Cs = fis = =1 (241) The permutation symbols are us matrix, lhen, using equation (2.41), the reader can easily verify Uia ful in the theory of determinants. Tn fac =(a;)isadx3 det 4 = |A| = eqisistfaias (2.42) Vhis formula [or determinants extends in au obvious manner to 2 x à matrices. À more Lhorough discussion of the Levi-Civita symbols will appear later in these notes. lu Rº, the Levi-Civita symbol with some or all the indices up is numerically equal to the permutation symbol will all indices down since the Euclidean met On the other hand, im Minkow: ==. Thus, in calMA 3) because go = 4º iniriais Ciaivisia = —€ since any permutation of (0, 1, 2,8] must contain a O. 2.28 Definition “Lhe Hodge-+ operator is a linear map +: ALR) — AVR") defined in standard local coordinates by the equation 1 a A A der et im deimt A A datos (2.43) (em)! nto “ Since the forms dxiz A. ..Adxi» constitute a basis of the vector space AV (R?) and the +-operator umed to be a lincar map, equation (2.43) completely é ceifies the map for all m-forms. 2.4, THE HODGE-+* OPERATOR 29 4, Lea = Bidx, en ade =>V.B Tt is also possible to define and manipulate formulas of classical vector caleulus using the per- mutation symbols. For example, let a = (41.49.44) and B = (By, Bo. Bs) be any tro Euclidem vectors. Then it is casy to sec that (A x BJa = AB, and = GA; VxBh=d : ( ) + Bud To derive many classical vector identilies in this formalisim, il is necessary to first establish the following identity (see Ex. ()) EM eutm = 6,6] — 6;6] (2.54) 232 Example AX(BxCO = TA mBxCOm Am (BC) PE Am Bh) emitir Ar B; Cs) (816% — POA Bj = BA"Ca— Gba Or. reveriting in vector form Ax(BxC)=B(A-C)-C(A.B) (2.55) Maxwell Equations “Lhe classical equations of Maxwell describing electromagnetic phenomena are V.E=47p VxB=40]3 + vV.B=0 vxE=-2É õ (2.56) We would like to formulate these cquations (4,21, 22,28) be local coordinates in Minkow: in the language of differential forms. Let 2! = pace Ms. Define the Maxwell 2-form P by the equation 1 T= uodetAde”, (uv =0,1,23). (2.57) wliere —y —Ey B. By 25 dm (2.58) -B 0 Written in complete detail, Maxwell's 2-[orm is given by FP = —Rodtadel- Editada? EdtA de + Bedxl a dx? — Bydel a de 4 Bode? Ade”. (2.59) 30 CHAPTYER 2. DIFFERENTIAL FORMS We also define Lhe source current I-form 1 = Jude” = pdt+ dude! + dada? + dade”. (2.60) j are equivalent Lo the equations 2.33 Proposition Maxwell's Equations dr = 0, dep = dg4rxd, (2.61) Proof: “Lhe proof is by direct computation using the definitions of the exterior derivaive and the Hodge-* operalor. a det dia de! + de Ade? A dia de! Ade! Adin de? — E A det adia dae? + Ade! Adi A da? — Ade? n dia da? + ) 3 Coe ade dal A dal — a 0B, Cos nata del A das — dt ã Ada A dela da? — de! A del A de? + 9B Ya A dia du? + dz? JH dE, 0By 1 3 Bdzê à Jdt nda! pa” + v 0Ey OB 2 E geo gd! A dinda? Therefore, dF = O iff 08, By del À dez which is the same as v.B=0, and 24, THE HODGE+ OPERATOR st velicl means Uiat , -vxE-D=0. (2.62) Vo verily Ue second set o! Maxwell equalions, we first compute the dual of the current density I-form (2.60) using the results [rom example 2.4. We get +] =-pdel ada da? + Ide A dta de? + Jodt A del A de? 4 Jadel A dt A de? (2.63) Wo could now proceed to compute d+ P, but perhaps it is more clegant to notice that P E AM), and so, according to example (2.4), E splits into E = E, + $.. In fact, we see from (2.58) that the components of FP, are those ol —E and lhe components ol F- constitute Lhe maguetic field vector B. Using the results of example (2.4), we can immediately write the components o[ + aP = Badtndal + BydtA de 4 BodtA de + Edu! nda? — Eyde! A de? 4 Ed? A do”, (2.64) or equivalently, 0 Bo By By Ms E (2.65) “By E 0 E -B ER 0 Since the ellect ol the dual operator amounts to exchanging E -— -B Bs +E, we can infer from cquations (2.62) and (2.63) that v.E=4mp and, sa CHAPTER 3. CONN Tence Given a framo vectors e;, we can also introduce the corrosponding dual coftame forms 6; by requiving eo 2 0(e;) = 0; (3.3) since the dual coltame is a set ol Lforms, they can also be expressed in of local coordinates as linear combinations 6 = Bida Lt follows from equation( 3.3), that Ole) = Bidr(0A!,) = BiAtdr(à) = BiAtõs 8, = Lherefore we conclude that BA = £,so B = AT! = 47, In other words, wlen the frames are orthonormal we have e = WA p= Adaf (3.4) 3.3 Example Consider te translormation [rom Cartesiau to cylindrical coordinates 2 = reos0. v = sino, 2 = Using the chain rule for partial derivatives, we have à à à — = cost sing > cos dm + sin 2y 8 sô à % = -ruiafço + rosto 9. dr O Trom these cquations we casily verify that the quantiti RR “Sm o Vê O = 0 ô = 3.2, CURVILINEAR COORDINATES 35 are à lriplet of mutually orthogonal unit veclors and tus constitute an orthonormal frame. 3.4 Example For spherical coordinates( 2.20) r = psinficosó y = psinêsing 5 = pcoso, the chain rule leads to ô . Fe] . . Fe] Fe] — = sinfcosê— +sinÊsind— + cos6-— p de 3 Fe] 8 à + bai Fe] + = posfcwsd—+pcosbsind D+ E PE ÉCOS Ea | PCS CA a Fe] . . à . É Lo= cpsnfsindl + psinbcosó dó de ln this case, the vectors 78 ro psind da tutc an orthonormal frame. also const The fact that the chain rule in the two situations above leads to orthonormal frame of the level surfaces x = constant. Since one expects the gradients of the surfaces Lo also be orthogonal. Lransformatioas of Ui type are called triply orthogonal systems. coincidental. The results are related to the orthogonalit; z the level surfaces arc orthogonal whenever they intersect 3.2 Curvilinear Coordinates Orthogonal transformations such as Spherical and cylindrical coordinates appear ubiquitously in mathematical physics because Lhe geometry of a large umber of problems in Uhis area exhibit sym- metry weith respect to am axis or to the origin. Tn such situations, transformation to the appropriate tem often result in considerable simplification of the ficld couations involved in the problem. Tt has been shown that the Laplace operator sehich enters into all three of main classi ficlds, the potential, the heat and the wave equations, is separable in 12 coordinate systems. A sim- ple and cfficient method to calenlate the Laplacian in orthogonal coordinates can be implemented by the use of differential forms. coordinate 3.5 Example lu spherical coordinates the differential of arc lenguh is given by (see equation 2.21) ds? = dp? + de 4 pê sin” Gdg”, Let 6º = do o pd 08 = psintdo. (3.6)  30 CHAPTER 3. CONN CIO, Note thai these three Llorms constitute the dual coframe to the orthonormal [tame which just derived in equation( 3.5). Consider à scalar lield / = /(9,0,6). We now calculate the Laplacian ol f in spherical coordinates using lhe methods ol section 2.4. dilferential df and express the result in terms of the colrame, af 91 of To do this, we [irst compute the d= pdpt Dqdo+ ção = Cl dll, O Cl dp "pool psmide The components df in the coftame represent the gradient in spherical coordinates. Continuing with the scheme of section 2.4, we first apply the Todge-s operator. Then we rewrite the resulting 2-form in terms of wedges of coordinate differentials so that sc can apply the definition of the exterior derivative. ) df = - puorr Pe 2 - Ponllanasop EB oeo A do 1... õ = cairão Bout no of 2 of dadf = Dot smo Lap ndo ndo — E qoinaço Lda dp A da + E Jde A dp A do . of 61) 4 = l|sin6>—(p? Sl donde nd E melo + gpfsno E Ts PGE o Finally, rewriling lhe differentials back in terms of lhe the coframe, we gel Dm Temo [lg dp E | , sin é The derivalion of lhe expression [or the spherical Laplacian through the use of differential forms is elegant and leads naturally to the operator in Sturm Liouville form. The process above can also be carried out [or general orthogonal translormations. À change of coordinates 4º = 4 (uº) leads to an orthogonal trans[ormalion il in the new coordinale system vu”, the line metrie dad = 3 5 sin E) + E 4] BABA. So, lhe Laplacian of f is given by NO [,0f ap 10 [oDf Vi=20 |” 5) of, Ff Oo) + 3a de = qu(du + gold) 4 gusldus)* (3.8) ouly has diagonal entries. lu this case, we choose the colrame 1 = Vgudul = dal 8º = Vgade = hudo? 5 = Vgsadu = hadu? ecally called the wcights. Please + for one and this will cause small The quantities (Ay, fa. hs) are cl note that in the interest of comnecting to classical terminology wc have exchanged two índio discrepancies with the index sunimation convention. We will revert to using a summation symbol  COVARIANT DERIVADIVE 39 ln Euclidean space, the components of the standard [rame vectors are constant, aud thus their rates of change in auy direction vanish, Let e; be arbitrary [rame field with dual forms 8º. “The covarianl derivaives o[ te Irame vectors in lhe directions of a vector X, will in general yield new vectors. The new vectors must be linear combinations of the the basis vectors Two = u(Ne +wi(N)es Hui (Nes Vrxto = wdA)e +uN)es +wi(X)es Vres — w(A)e +uwdlX)es +wb(A es (3.14) “The coeflicients can be more succincily expressed using the compact index notation Vixe = ejui(X) (3.15) Tt follows immediately that (NX) = (VT xe). (3.16) Equivalenty, one can take the inner product of both sides of equation (3.15) with ep Lo gel <TrxeaD = <ewi(A)e> = wU(X) <ere > = (X)gj Tenee. < Fxenen >= wpi(N) (3.17) The left hand side of the last equation is the inner product of two vectors, so the expre represents an array of functions. Consequently, the right hand side also represents an array of functions. In addition, both expressions are linear on X, since by definition Vx is linear on NX. We conclude that the right land side can be interpreted as a matrix in which, each entry is a I-forims acting on the vector X to yield a function. Lhe matrix valued quantity w”, is called the conncetion form . 3.10 Let Tx be a Koszul connection and let fes) be a frame. The Christoffel ymbols associated with the connection in the given frame are the functions T£, given by Tec; = Then (3.18) The the fi Lo the Christoltel symbols as the connection once again giving rise to possíble confusion. The precise relation between the Cliristollel symbols and lhe connection I-forms is captured by the equalions “hristofrel ame vector: mbols are the cocfficients which give the representation of the rate of change of in the diveo jon of the frame vectors themselves. Many physicists therefore refer (3.19) or equivalently =14,6 (3.20) Tn a general frame in R? there arc n2 entries in the connection 1-form and n2 Chris The number of independent componen toffel symbols. is reduced if one assumes that the frame is orthonormal. 3.11 Proposition Let and e; be an orthonormal frame and Vx be a Koszul connection compatible with the metric . Then jiu; (3.21) 40 CHAPTER Proof: Since il is given Lat < ee; >= 6, we lave 0 = Va<ee> <Tyee>+<eTre> <utee>+< cute > = wÊ<ept D+ <em> = wiga + wligia = wo thus proving that w is indeed anti 3.12 Corollary The Christoffel symbols of a Koszul connection in an orthonormal frame are aulisynumetric on the lower indices; that is (3.22) We conclude that in an orthonormal frame in R the number of independent coefficients of the connection Iform is (L/2)n(n — 1) since by anlisymmetry, the diagonal entries are zero, and one only needs lo count (he number of entries in We upper triangular part of he nx n matrix vo; Similarly, Ule number of independent Christollel symbols gets reduced to (1/2)n?(n— 1). In We case of au orthonormal frame in Rº, where g, is diagonal, w”, is also antisymmetric, so the connection 5 equations become fa 0 si (O) vl(N) ei Tx lc [=| =ub(X) wi (N) e» |. (3.23) es = UN) 0 es Comparing the Trenct frame equation( 1.27), ec notice the obvious similarity to the general frame cquations above. Clearly, the Trenet frame is a special case im vhich the basis vectors have been adapted to a curve resulúng iu a simpler connection in which some of the coeflicients vanish. A further simplification occurs in the Frenet frame since here Lhe equalions represent the rate of change of the frame only along (he direction of the curve rather than an arbitrary direction vector X. 3.4 Cartan Equations Perhaps, the most important contribution to the development of Differential Gicometry is the work of Cartan culminating into funous equatioas of suructure which we discuss in this chapter. First Structure Equation 3.13 Theorem Let (c;] be a frame with connection w É and dual coframe 6º. Then dE und =0 (3.24) Proof; Lei be a frame, and let 6º be the corresponding coframe. Since 0 (c;). we have — -1 pi (= (AY dad, 3.4, CARTAN EQUATIONS Let X be au arbitrary vector field, Then al Tra = Tr(ó4,) (O) = (4) = dat WD(X) = MATA (A) (A) (ATA Tonee, af = (ATYA(A!). or in matrix notation wu = ATÍgA (3.25) On the other hand, taking the exterior derivative of 9º. sec find dé = dAyjAda = dATY;A Ao de = dATANO. But, since ATIA = 7, we have (ATA = —ATÍdA = —w, hence d=—w AB. (3.26) Tn other words dê wi AR = Second Structure Equation Let 8º be a colrame in Rº with connection w*,. Taking the exterior derivative ol the first equation ol structure and recalling the properties (2.34) we gel ala) + dll; A 0) =0 di; A Bi win doi =0, Substituting recursively from the first equation of structure, we get dan win (ow Ad) =0 dan pule Nu AGi=0 (dot; + wi nula =0 i É e di, udp Aus =0, We now introduce the following 3.14 Definition The curvature Q of a connection w is the matrix valued 2-form di gi Á & 2 9= DÊ, = dai, ot Noob, (2.27) Thcorem Let 8 be a colrame with connection « in Rº . Lhe lhe curvature form vanishes (3.28) 3.15 O=dutwunw=o  44 CHAPTYER 4, PHEORY OF SUREAC 4.3 Definition A dilferentiably smooth surface in R$ is a set of points M iu Rº such tlrat 1. Tfp E M then p belongs to some CX chart. 2. 4 p E M belongs to two dillerent charts x and y, Wen lhe two charts are CS equivalent. Intuitively, wc may think of a surface as consisting locally of number of patches 2 sewn? to cach other so as to form a quilt from a global perspec The first condition in the definition the second differentiabi fatos that cach local pateh looks like a piece of RZ, whercas joined together smoothly. An- other way to state this idea is to say that a surface a space that is locally Euclideam and it has a condition indicates that the patches am differentiable structure so that the notion of differentiation makes sense. Tf the Euclidean space is of dimension n, the “surface is called au r-dimensional manifold 4.4 Example Consider the local coordinate clarl x(m0) = in te cos 2, sin sin 2, cos +). tor equation is equivalent to three scalar functions in two variables e = sinucose, y = sinusino, 2 = cos (4.5) +! 42º =, Uhe chart can ibly represent the whole sphere because, although a sphere is locally Euclidean, (Uhe earth Clearly, the surface represented by this chart is part of the sphere not po: is locally lat) there is certainly à topological dillerence between a sphere and a plane, Indeed, if one analyzes the coordinate chart carefully you will note that at the North pole (4 =0. 2= |. the coordinat 4 = 0 regardles e number of labels. The fact that it às required to ation of the 3-dimensional becom of the ngular. This happens because 4 = 0 implies that à value of, so that the North pole has am infin have two parameters to des nature of of the surfacs ibe a patch on a surface in R$ is a manife constant while varying the other, then the resulting I-parameter equations describe a curve on the surlace. Thus, [or example, letting “= constant in equation (4. s. Fone holds onc of the parameter: ) we gel the equalion of a meridian great circle, 4.5 Notation Given a parametrizatiou of a surface in a local clart x(u,0) = x(u!,u?) (ue), we will denote the partial derivatives by any of We following notations: dx Ku =X =. Xuu =X du Ox MM = Ka =X = dy Ox Xy = a Kd EE Que Bud 4.2 The First Fundamental Form Let 2i(uº) be a local parametrization of a surface. Then, the Tuelidean inner produet in R$ induces an inner product in the space of tangent vectors at cach point in the surface. This metric on the surface is obtained as follow  4.2. THE FIRST PUNDAMENTAL FORM 45 2 ds ddr da dx! Oui Jus Bu dudu”, Thus, where a natural metrie We conclude that the surface, by virtuc of being embedded in R$. inheri sehich awe weill call the induced metric. À pair (44,9). sehero M is amanifold and 9 = gu ade sdu? is a metrie is called a Riemannian manifold if considered as an entity im itsclf, and Ricmannian submanifold of Rº if viewed as an object embedded in Tinclidean space. An equivalent version of the metrie (1.6) can be obtained by using a more traditional caleulus notation dx Xu du + xudu ds? = dx-dx = (xudu + xude) -(xudu + xede) (x Xu du? 4 Mx Ra )dude(x, ed? We cam rewrite the last result as = Edu? + 2Fdudo + Gr”, (4.8) where E = =X Xu Fo = ga=XKuXu = da = Ko Xu = g22=K Xu. “That is Gug = Xu Xg =€ XX > 4.6 bdf The clement of arclength ds” = gasdu” o du? (4.9) is also called the first fundamental form. We must caution the reader that this quantity is not a form in the sense of diffcrential gcometry since ds? involvos the symimetric tensor product rather than the wedge product. The first fundamental form pla: that will find it convenient to introduce yet a third more modem version. Tollowing the same development as in the theory of curvos, consider a surface M defined locally by a function (ul, 12) — a(ul, 42). We say that a quantity X, is a tangent vector at a pointp e M.if X, is a líncar derivation on the space of € real-valued functions [/|/: AM — RJ on the surf: The set ol all tangent vectors at a point p E M is called the tangent space 1,M. As before, a vector field X on the surface is à smooth such a crucial role in the theory of surfaces choice ol a tangent vector at eacl point on lhe surface and the union of all tangent spaces is called the tangent bundle LM. Lhe coordinate chart map a:RÉ— MER? 46 CHAPTYER 4. VHEORY OF SUREAG induces à push-forward map aiIRS UM defined by 0) Inquey= Va o 7) Just as in the case of curvos, when we revert back to cla xi (4º). sehat we really mean is (r/c 0) (4º), where 2! examples of tangent vectors on M are given by the pusl-forward of the standard basis of TR? us ical notation to describe a surface as are the coordinate functions in Rê. Particular These tangent vectors which earlier we called xo are defined by à à NO luis az (ao) “fe us Tn this formalism, the first fundamental form 7 is just the symmetrie bilincar tensor defined by induced metric MAES UN,V)=<AYS, (4.10) where X and Y are any pair of vector ficlds in TM. Orthogonal Parametric Curves Lei Y and W be vectors tangent to a surface AL defined locally by a chart x(uº). Since the vectors Xalpha Spam the tangent space of M at cach point, the vectors V and F7 can be written as lincar combinations Voxa Wo = Wºxa. The functions Vº and Fº are called the curvilincar coordinates of the vectors. We cam caleulate the length and the inner product of the vectors using the induced Riemannian metrie, [HIP = <yyD>=< Vez vêxa >= CV? Coxa xs > VIP = gastov? IP = gesWowo, and <vW> = <VixaWêx >= VW? < xx > = ganvtWo, The angle 8 subtended by the the vectors « and His the given by the equation <LW> AA cos) = (4.11) Let uº = 6º(t) and 4º = “S(t) be two curves on the surface. Then the total differentials dy » dt tó . due = CC gr amd duto a THE SECOND FUNDAMENTAL FORM 49 Tigure 4.2: Normal Curvature by the constrainl ol the curve to lie on a surface. “The geodesic curvature vector , measures Lhe “sideward” component ol the curvature in the tangent plane to the surface. Thus, il one draws a straight line on a flat piece of paper and then smoolhly bend the paper into à surface, ten the traight line would now acquire some curvature. Since the line vas originally straight, there is no sideward component of curvature so A, = Qin this case. This means that the entire contribution to the curvature comes from the normal component, reficeting the fact that the only reason there is eurvature here is due to the bend im the surface itself. s a point p € M and a direction vector X, E TM, one can geometrically ss of all unit speed curvos in M vlúch contain the poiat p and whose tangent vectors line up with the direction of X. Of course, but al au infinitesimal level, all Lhese curves can be obtained plane containing the vector X and the normal to M. Similarly. if one specifi envision the normal curvature by considering the equivalence cla Were are infinitely many such curve by intersecting (he surface will a ” verti AI curves in Uhis equivalence class lave lhe same normal curvature and their geodesic curvatures vamish. 1u Uis sense, the normal curvature is more of à property pertuining to a direction on the surface at a point, whercas, the geodesic enrvature really depends on the curve itself. Tt might be impossible for a hiker vealking on the ondulating hills of the Ozarkts to find a straight line trail, since the rolling hills of the terrain extend in all directions. TIowever, it might be possible to walk on a path with zero geodesie curvature as long as the hiker can maintain the samc compass direction. To find an explicit formula for the normal eurvature wc first differentiate equation (4.15) dt ds d du = ea r = ) dus “da x) duê du” du? *escgs ds Taking the inner product of the last equation with the normal and noticing that < x,,n >= 0, we get au du” du? Ka <f,uD>=<X,0>—>——— ds ds basdutdu? +“ 417 Goadurdu? (417) sehere bu =< Xu. > (4.18) 4.9 Definition “Lhe expression HH = bogdutS du” (4.19) is called the second fundamental form . 50 CHAPTYER 4. VPHEORY OF SURA 4.10 Proposition “The second fundamental form is syummetric. Proof: In the classical formulation of the second fundamental form the proof is lrivial. We have dos = ba since for a (º patch x(u”), we have xag = X5a because the partial derivalives conmute, We will denote the coeflicients of lhe second fundamental form by = bn =<X%en> f = bo=<xen> = by=<myn> 9 = by=<xn> + so that equation (4.19) can be written as TI = edu? + 9fdudo + gor? (4.20) and equation (4.17) as CTT Pdul+2Fdudo+ MOTO Ceni +afdado + gd We would also like to point out that just as the first fundamental form can be represented as 1=< dx dx >, so cam we represent We second fundamental form as H=-<dedu> To see this it suffices to note that differentiation of the identity < x. >= O implies that <Xyg.n >> — <XDy >. Yherefore, <dxdn> = <xodu, ndo” > < xadut,ngdu” > < xong > dudu” —<xgn > dutd? =IT From a computational point a view, a more useful formula for the coelficients of We second fundamental formula cam be derived by first applying We classical vector identity 4 JACA. 1 (4x 8) (C «D=| BG 55] (4.22) to compute xs xxo PO = (uu xx) (ru x) Xu Xu Xu Xu des | Eu Xu Xu Xu Xu Xu Xu Ko = EG-Fê Consequentiy, the normal vector can be written as Xu X Xy Xu X Xy n=> DD = “xxx VECSF? 4.4 CURVAPURE “Thus, we can wrile We coeficients bos direclly as Lriple products involving derivatives of ( expressions for Uese coelficients are (XX Xau) (ru xu ue ) (x xuxa) s = 4 VEG— + Jhe first fundamental form on à surface measures the (square) of the distance between tivo infinitesimally separated points, There is à similar interpretation of We second fundamental form as we show below. “The second fundamental form measures the distance from a point on the surface Lo the tangent plane at à second infinitesimally separated point. To see Uis simple geometrical interpretation, consider a point xy = x(ug) E M and a nearby point x(us + dus). Txpanding on a Taylor serics, sec get x(ui + du") = xp + (xpJadu” + Jogdutdu 4... We recall that the distance formula [tom a point x to a plane which contains xq is just Lhe scalar projection of (x — xy) onto the normal. Since the normal to the plane at xy is the same as the unit, normal to the surface and < x,,n >= 0, we find that the distance D is D = <x-xn> 1 ; =3< (xoJas nm > dutdu? 1 Sta sia he first fundamental form (or rather, its determinant) also appears in caleulus in the context of caleulating Ve area of a parametrized surface. the reason is that if one considers an infinitesimal parallelogram subtended by the vectors xudu ad x,do, then the dillerential of surface area is given by the length of the cross product of these two infinitesimal tangent vectors. That is ds bre x xe] dede ff VEG- P? dude information about the shape of the surface at a point. Tor example, the discus basl=e9— $? > 0 then all the neighboring points lic on the same side of the tangent plane, and hence, the surface is concave im one direction. Hat a point on a surface é > 0, the poimt is called am elliptic poimt, if 6 < O. the point is called hyperbolic or a saddle point, and if 6 = 0, the point is called parabolic. The second fundamental form contain ion above indicates that if b 4.4 Curvature Curvature and all related questions which surround curvature, constitute the central object of study 1 geometry. One would like to be able to answer questions such as, what quantilies remain invariant as one surface is smoothly changed into another? There is certainly something in differenti intrinsically different from a cone, which wc can construct from a flat piece of paper and a sphere vehich we can not. What is it that makes these two surfa the shortest path between tsvo objects sehen the path is constrained to be om a surfac o different? ow does one caleulate ? 54 CHAPTYER 4. VPHEORY OF SURA 4.18 Thcorem “Lhe Gauss map is à self adjoint operator on EM. Proof: We have already shown that L: TM — TM is a lincar map. Recall that am operator L on a linear space is sell adjoint il < LX,Y >=< X,LY >, so that the theorem is equivalent to proving Uat Wal the second fundamental for is symmetric (LLX, Y] = H[Y, X). Computing the diflerence of lhese two quantities, we gel MXOT-H[Y,X] = <LXYD>-<LYX> <VyNYD>-<FyrN, X>. Since < X,N >=< Y,N >= 0, and lhe connection is compatible with We metric, we know that <FxNV> 0 NNxY> <FENXD — -<NSçX>, hence, XY] TI(y.X] <NVyXD-<NVxy >, <NVEX -VxY > < NX, Y]> =0 (UT [XY] E (MO) One of the most important topies in an introductory course linear algebra deals with the spectrum of self adjoint operators. “The main result in tis area states Uial if one considers the eigenvalue equation EX = 8X (4.30) then the cigenvalues arc always real and cigenvectors corresponding to different cigenvalues arc or thogonal. In the current situation, the vector spas in question are the tangent spaces at cach point of a surface in R$, so the dimension is 2. Tlence. we expeet two cigenvalues and two cigenvectors LX = mN (4.31) LX» = NX. (4.32) 4.19 Definition The cigenvalues x, and xy of the Gauss map £ arc called the principal cur- vatures and the cigenvectors Xy and Xy are called the principal di uations may occur depending on the classification of the cigenvalues at cach point p on the surface: tions. Several possible 1. Tf a, É xo and both cigenvalues are positive, then p is called an clliptic poimt 2. Al kyio < 0, then p is called à hyperbolic point. 3. Tay = 40 £ 0, then p is called an umbilic point. 4. if 81x = 0. then p is called a parabolie point Tt is also well known from lincar algebra, that the the determinant and the trace of a self adjoint. operator are the only invariants under a adjoint (simila invariants ansformation. Clearly thes are important in the case of the operator L.. and they deserve to be given special names. 4.20 Definition The determinant A = det(7) is called the Gaussian curvature of M and HF = (1/2)Tr(£) is callod the mean curvature. 4.4 CURVAPURE Since any self-adjoint operator is diagonalizable and in a diagonal bas Lis diag(ur, 12), E follows inmediately Uiab , lhe matrix representing Ko = ans = dl tr) (4.33) 4.21 Proposition Lei X and Y be any linearly independent vectors in 1(M). Then LXxXLY = EX xY) (EXXDA(X x 17) = 2X xy) (4.34) Proof: Since EX,LY € T(M). they can be expresses as lincar combinations of the basis vectors X and Y, EX = 0X+hY LO = aX+Hboy. computing the cross product, sec get ixxty = [O blxxy aa by = deMIX xY). Similary EXXIDAM XE) = (ntbX XY) = IUL(X x 7) (HA x 4). 4.22 Proposition K = (4.35) Proof: Starting with equations (4.34) take the dot product of both sides with X x Y and use the vector identity (1.22). We immediately ger | <LX.X> <EXY > | K= <INX> <EX,X> <XND> <XY> <YX> <yr> <LX> o <FY> <IVX> <IrY> | <XX> <XY> | [RS SP | | SNÃS <A Y> | 2H = <YÃ> <YY> 56 CHAPTYER 4, VHEORY OF SURFACES Yhe result follows by taking X = xu and Y = xy 4.23 Thcorem (Euler) Let X; and X be unit eigenvectors of L andei X = (cos) X1+(sin 6) X. Yhen 1HA,X) = nicos?6 + nosin” 6 Proof: Easy. Just compute LH(X,X) =< LX,X >, using the fact the LXG = 11X, LXo = 9X, aud noting that the eigenveclors are orthogonal. We gel, <LXXND> = <(cos0)m Xi + (sinO)aa Ko», (cos0)X, + (sin 0) Xa > ricos? 0 < XX > +rgsin? 6 < No, Xa > 2a. = «cos? 0 + nysin