Classical Differential Geometry - UCLA Mathematics

1 Classical Differential GeometryPeterPetersenPrefaceThis is an evolving set of lecture notes on the Classical theory of curves andsurfaces. Pictures will be added eventually. I recommend people download3DX-plorMathto check out the constructions of curves and surfaces with this app. Itcan also be used to create new curves and surfaces in parametric form. Other usefuland free apps are Geogebra, Grapher (on Mac), and a minimum a one quarter course should cover sections , , , , , and chapters 4, 5. In a semester course it d be possible to cover more fromchapter 2 and also delve into chapter 6. Chapters 6 and 7 can be covered in asecond quarter class. Note that section is a necessary prerequisite for provingthe general Gauss-Bonnet in section excellent reference for the Classical treatment of Differential Geometry is thebook by Struik [2].

2 The more descriptive guide by Hilbert and Cohn-Vossen [1]isalso highly recommended. This book covers both Geometry and Differential geome -try essentially without the use of calculus. It contains many interesting results andgives excellent descriptions of many of the constructions and results in text is fairly Classical and is not intended as an introduction to abstract2-dimensional Riemannian Geometry . In fact we do not discuss covariant differen-tiation or parallel translation. Most proofs are local in nature and try to use onlybasic linear algebra and multivariable calculus. The only sense in which the text ismore modern is in not using the language of differentials and infinitesimals as mostof the Classical texts standard topics are not covered in the text.

3 However, I hope most ofthem can be found among the exercises. As such, they can easily be incorporatedinto lectures as the instructor sees d like to thank Chadwick Sprouse and Michael Williams for trying out thesenotes and providing valuable feedback. Reading your notes is like reading poetry, and I don t under-stand that either. Reed Douglas, UCLA 1. General Curve Arclength and Linear Integral Curves29 Exercises34 Chapter 2. Planar The Fundamental The Rotation Three Interesting Convex Curves56 Exercises58 Chapter 3. Space The Fundamental Characterizations of Space Closed Space Curves74 Exercises77 Chapter 4. Basic Surface Tangent Spaces and The First Fundamental Special Maps and Parametrizations99 Exercises103 Chapter 5.

4 Curvature of Curves on The Gauss and Weingarten Maps and The Gauss and Mean Principal Ruled Surfaces144 Exercises151 Chapter 6. Surface Generalized and Abstract Curvature on Abstract The Gauss and Codazzi The Gauss-Bonnet Topology of Closed and Convex Surfaces190 Exercises192 Chapter 7. Geodesics and Metric Mixed Shortest Short Distance and Constant Comparison Results220 Chapter 8. Riemannian Geometry224 Appendix A. Vector Vector and Matrix Geometry of Differentiation and Differential Equations233 Appendix B. Special Coordinate Cartesian and Oblique Surfaces of Monge Surfaces Given by an Geodesic Chebyshev Isothermal Coordinates247 Bibliography250 CHAPTER 1 General Curve TheoryOne of the key aspects in Geometry isinvariance.

5 This can be somewhat difficultto define, but the idea is that the properties or measurements under discussionshould be described in such a way that they they make sense without referenceto a special coordinate system. This idea has been a guiding principle since theancient Greeks started formulating Geometry . We ll often take for granted that wework in a Euclidean space where we know how to compute distances, angles, areas,and even volumes of simple geometric figures. Descartes discovered that thesetypes of geometries could be described by what we call Cartesian space throughcoordinatizing the Euclidean space with Cartesian coordinates. This is the generalapproach we shall use, but it is still worthwhile to occasionally try to understandmeasurements not just algebraically or analytically, but also purely descriptively ingeometric terms.

6 For example, how does one define a circle? It can defined as aset of points given by a specific type of equation, it can be given as a parametriccurve, or it can be described as the collection of points at a fixed distance from thecenter. Using the latter definition without referring to coordinates is often a veryuseful tool in solving many CurvesThe primary goal in the geometric theory ofcurvesis to measure their shapesin ways that do not take in to account how they are parametrized or how Euclideanspace is coordinatized. However, it is generally hard to measure anything withoutcoordinatizing space and parametrizing the curve. Thus the idea will be to see ifsome sort of canonical parametrization might exist and secondly to also show thatour measurements can be defined using whatever parametrization the curve comeswith.

7 We will also try to make sure that our formulas do not necessarily referto a specific set of Cartesian coordinates. To understand more general types ofcoordinates requires quite a bit of work and this will not be done until we introducesurfaces later in these traveling in a car or flying an airplane. The route traveled will trace acurve. You can easily keep track of time and distance traveled. The goal of curvetheory is to decide what further measurements are needed to retrace the precisepath traveled. Clearly one must also measure how one turns and that becomes theimportant thing to describe fundamental dynamical vectors of a curve whose position is denoted byqare thevelocityv=dqdt,accelerationa=d2qdt2,a ndjerkj= lineto a curveqatq(t)is the line throughq(t)with directionv(t).

8 The goal is to find geometric quantities that depend on velocity (or CURVES2lines), acceleration, and jerk that completely determine the path of the curve whenwe use some parametertto travel along LineMost of the curves we study will be given asparametrized curves, ,q(t)=264x(t)y(t)..375:I!Rn,whereI Ris an interval. Such a curve might be constant, which is equivalent toits velocity vanishing it is never stationary. In otherwords, the speed is always positive, or the velocity never curves are given to us in a more implicit form. They could comeas solutions to first order Differential equationsdqdt=F(q (t),t).In this case we obtain a unique solution (also called an integral curve) as long aswe have an initial positionq(t0)=q0at some initial (q)only depends on the position we can visualize it as a vector field as it givesa vector at each position.

9 The solutions are then seen as curves whose velocity ateach positionqis the vectorv=F(q).Very often the types of Differential equations are of second (or even higherorder)d2qdt2=F q(t),dqdt,t .In this case we have to prescribe both the initial positionq(t0)=q0and velocityv(t0)=v0in order to obtain a unique solution next result shows how Differential equations can be used to following conditions are equivalent for a regular curveq(t):(1)The curve travels along a line:q(t)=q0+ (t)v0, where (t)is a scalarvalued function andq0,v0are fixed vectors.(2)The velocities are all parallel to each other:v(t)= (t)v0, where (t)isa scalar valued function andv0is a fixed CURVES3(3)The velocity and acceleration at each point are parallel to each other:a(t)= (t)v(t), where (t)is a scalar valued (1))(2):Use (t)= (t).

10 (2))(3):Sincethecurveisregular (t)6=0. Thus we can use (t)= (t) (t).(3))(1): The equationa(t)= (t)v(t)can be written as a differentialequationdvdt= (t) shows thatv(t)=v(t0)exp Ztt0 ,since the right hand side solves the equation and has the same initial value att0asthe left hand side. Thus we obtain a new Differential equationdqdt= (t)v0,which shows thatq(t)=q(t0)+v0 Rtt0 ,sincetherighthandsidesolvestheequation and has the same initial value att0as the left hand side. (t),c (t):I!Rkbe two vector valued curves.(1)d(c c )dt=dcdt c +c dc dt.(2)ddt 12|c|2 =c c.(3)ddt(|c|)=c c|c|as long asc6=0.(4)ddt 1|c| = c c|c|3as long asc6= (1) follows from the product rule for differentiation.(2) follows by using (1) withc =cand that|c|2=c c.(3) follows from (2) by observing that we also haveddt 12|c|2 =|c|d|c|dt.