Toponogov v a differential geometry of curves and surfaces a concise guide

ix 1 Theory of Curves in Three-dimensional Euclidean Space and in the Plane.. 1.2 Definition and Methods of Presentation of Curves 5forms a smooth regular curve more exactly, a finite numb

Victor Andreevich Toponogov with the editorial assistance of

Victor A Toponogov (deceased)

Department of Analysis and Geometry

Sobolev Institute of Mathematics

Siberian Branch of the Russian Academy

Cover design by Alex Gerasev.

AMS Subject Classification: 53-01, 53Axx, 53A04, 53A05, 53A55, 53B20, 53B21, 53C20, 53C21

Library of Congress Control Number: 2005048111

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Preface vii

About the Author ix

1 Theory of Curves in Three-dimensional Euclidean Space and in the Plane 1

1.1 Preliminaries 1

1.2 Definition and Methods of Presentation of Curves 2

1.3 Tangent Line and Osculating Plane 6

1.4 Length of a Curve 11

1.5 Problems: Convex Plane Curves 15

1.6 Curvature of a Curve 19

1.7 Problems: Curvature of Plane Curves 24

1.8 Torsion of a Curve 45

1.9 The Frenet Formulas and the Natural Equation of a Curve 47

1.10 Problems: Space Curves 51

1.11 Phase Length of a Curve and the Fenchel–Reshetnyak Inequality 56

1.12 Exercises to Chapter 1 61

2 Extrinsic Geometry of Surfaces in Three-dimensional Euclidean Space 65

2.1 Definition and Methods of Generating Surfaces 65

2.2 The Tangent Plane 70

2.3 First Fundamental Form of a Surface 74

vi Contents

2.4 Second Fundamental Form of a Surface 79

2.5 The Third Fundamental Form of a Surface 91

2.6 Classes of Surfaces 95

2.7 Some Classes of Curves on a Surface 114

2.8 The Main Equations of Surface Theory 127

2.9 Appendix: Indicatrix of a Surface of Revolution 139

2.10 Exercises to Chapter 2 147

3 Intrinsic Geometry of Surfaces 151

3.1 Introducing Notation 151

3.2 Covariant Derivative of a Vector Field 152

3.3 Parallel Translation of a Vector along a Curve on a Surface 153

3.4 Geodesics 156

3.5 Shortest Paths and Geodesics 161

3.6 Special Coordinate Systems 172

3.7 Gauss–Bonnet Theorem and Comparison Theorem for the Angles of a Triangle 179

3.8 Local Comparison Theorems for Triangles 184

3.9 Aleksandrov Comparison Theorem for the Angles of a Triangle 189

3.10 Problems to Chapter 3 195

References 199

Index 203

This concise guide to the differential geometry of curves and surfaces can berecommended to first-year graduate students, strong senior students, and studentsspecializing in geometry The material is given in two parallel streams

The first stream contains the standard theoretical material on differential etry of curves and surfaces It contains a small number of exercises and simpleproblems of a local nature It includes the whole of Chapter 1 except for the prob-lems (Sections 1.5, 1.7, 1.10) and Section 1.11, about the phase length of a curve,and the whole of Chapter 2 except for Section 2.6, about classes of surfaces, The-orems 2.8.1–2.8.4, the problems (Sections 2.7.4, 2.8.3) and the appendix (Sec-tion 2.9)

geom-The second stream contains more difficult and additional material and lations of some complicated but important theorems, for example, a proof of A.D.Aleksandrov’s comparison theorem about the angles of a triangle on a convex

sur-faces, and S.N Bernstein’s theorem about saddle surfaces In the last case, theformulations are discussed in detail

A distinctive feature of the book is a large collection (80 to 90) of nonstandard

and original problems that introduce the student into the real world of geometry.

Most of these problems are new and are not to be found in other textbooks orbooks of problems The solutions to them require inventiveness and geometricalintuition In this respect, this book is not far from W Blaschke’s well-known

1 A generalization of Aleksandrov’s global angle comparison theorem to Riemannian spaces of bitrary dimension is known as Toponogov’s theorem.

ar-viii Preface

manuscript [Bl], but it contains a number of problems more contemporary in

theme The key to these problems is the notion of curvature: the curvature of

a curve, principal curvatures, and the Gaussian curvature of a surface Almostall the problems are given with their solutions, although the hope of the author

is that an honest student will solve them without assistance, and only in tional cases will look at the text for a solution Since the problems are given inincreasing order of difficulty, even the most difficult of them should be solvable

excep-by a motivated reader In some cases, only short instructions are given In the thor’s opinion, it is the large number of original problems that makes this textbookinteresting and useful

au-Chapter 3, Intrinsic Geometry of a Surface, starts from the main notion of a

covariant derivative of a vector field along a curve The definition is based on

extrinsic geometrical properties of a surface Then it is proven that the covariantderivative of a vector field is an object of the intrinsic geometry of a surface, andthe later training material is not related to an extrinsic geometry So Chapter 3 can

be considered an introduction to n-dimensional Riemannian geometry that keeps

the simplicity and clarity of the 2-dimensional case.

The main theorems about geodesics and shortest paths are proven by methods

that can be easily extended to n-dimensional situations almost without alteration.

The Aleksandrov comparison theorem, Theorem 3.9.1, for the angles of a triangle

is the high point in Chapter 3

com-parison theorem for the angles of a triangle, or more exactly its generalization

by the author to multidimensional Riemannian manifolds, takes the place of the

Acknowledgments The author gratefully thanks his student and colleagues who

have contributed to this volume Essential help was given by E.D Rodionov,V.V Slavski, V.Yu Rovenski, V.V Ivanov, V.A Sharafutdinov, and V.K Ionin

2 The initials are in honor of E Cartan, A.D Aleksandrov, and V.A Toponogov.

About the Author

Professor Victor Andreevich Toponogov, a well-known Russian geometer, wasborn on March 6, 1930, and grew up in the city of Tomsk, in Russia During To-ponogov’s childhood, his father was subjected to Soviet repression After finish-ing school in 1948, Toponogov entered the Department of Mechanics and Math-ematics at Tomsk University, and graduated in 1953 with honors

In spite of an active social position and receiving high marks in his studies,the stamp of “son of an enemy of the people” left Toponogov with little hope ofcontinuing his education at the postgraduate level However, after Joseph Stalin’sdeath in March 1953, the situation in the USSR changed, and Toponogov became

a postgraduate student at Tomsk University Toponogov’s scientific interests wereinfluenced by his scientific advisor, Professor A.I Fet (a recognized topologistand specialist in variational calculus in the large, a pupil of L.A Lusternik) and

In 1956, V.A Toponogov moved to Novosibirsk, where in April 1957 he came a research scientist at the Institute of Radio-Physics and Electronics, thendirected by the well-known physicist Y.B Rumer In December 1958, Topono-gov defended his Ph.D thesis at Moscow State University In his dissertation, theAleksandrov convexity condition was extended to multidimensional Riemannian

be-manifolds Later, this theorem came to be called the Toponogov (comparison)

theorem.2 In April 1961, Toponogov moved to the Institute of Mathematics and

1 Aleksandr Danilovich Aleksandrov (1912–1999).

2Meyer, W.T Toponogov’s Theorem and Applications Lecture Notes, College on Differential

Ge-ometry, Trieste 1989.

x About the Author

Computer Center of the Siberian Branch of the Russian Academy of Sciences

at its inception All his subsequent scientific activity is related to the Institute

of Mathematics In 1968, at this institute he defended his doctoral thesis on the

theme “Extremal problems for Riemannian spaces with curvature bounded from

above.”

From 1980 to 1982, Toponogov was deputy director of the Institute of ematics, and from 1982 to 2000 he was head of one of the laboratories of theinstitute In 2001 he became Chief Scientist of the Department of Analysis andGeometry

Math-The first thirty years of Toponogov’s scientific life were devoted to one of the

most important divisions of modern geometry: Riemannian geometry in the large.

From secondary-school mathematics, everybody has learned something aboutsynthetic methods in geometry, concerned with triangles, conditions of theirequality and similarity, etc From the Archimedean era, analytical methods havecome to penetrate geometry: this is expressed most completely in the theory ofsurfaces, created by Gauss Since that time, these methods have played a lead-ing part in differential geometry In the fundamental works of A.D Aleksandrov,synthetic methods are again used, because the objects under study are not smoothenough for applications of the methods of classical analysis In the creative work

of V.A Toponogov, both of these methods, synthetic and analytic, are in harmoniccorrelation

The classic result in this area is the Toponogov theorem about the angles of atriangle composed of geodesics This in-depth theorem is the basis of modern in-vestigations of the relations between curvature properties, geodesic behavior, andthe topological structure of Riemannian spaces In the proof of this theorem, someideas of A.D Aleksandrov are combined with the in-depth analytical techniquerelated to the Jacobi differential equation

The methods developed by V.A Toponogov allowed him to obtain a sequence

of fundamental results such as characteristics of the multidimensional sphere byestimates of the Riemannian curvature and diameter, the solution to the Rauchproblem for the even-dimensional case, and the theorem about the structure ofRiemannian space with nonnegative curvature containing a straight line (i.e., theshortest path that may be limitlessly extended in both directions) This and othertheorems of V.A Toponogov are included in monographs and textbooks written

by a number of authors His methods have had a great influence on modern mannian geometry

Rie-During the last fifteen years of his life, V.A Toponogov devoted himself todifferential geometry of two-dimensional surfaces in three-dimensional Euclidean

space He made essential progress in a direction related to the Efimov theorem

about the nonexistence of isometric embedding of a complete Riemannian metricwith a separated-from-zero negative curvature into three-dimensional Euclidean

space, and with the Milnor conjecture declaring that an embedding with a sum

of absolute values of principal curvatures uniformly separated from zero does notexist

About the Author xi

Toponogov devoted much effort to the training of young mathematicians Hewas a lecturer at Novosibirsk State University and Novosibirsk State PedagogicalUniversity for more than forty-five years More than ten of his pupils defendedtheir Ph.D theses, and seven their doctoral degrees

V.A Toponogov passed away on November 21, 2004 and is survived by hiswife, Ljudmila Pavlovna Goncharova, and three sons

Differential Geometry of Curves and Surfaces

Definition 1.1.1 If
a = a1
i + a2
j + a3
k and
b = b1
i + b2
j + b3
k are vectors in

Definition 1.1.2 A linear transformation is a function T : V → W of vector

eigenvalue of a linear transformation T : V → V if there is a nonzero vector
a

2 1 Theory of Curves in Three-dimensional Euclidean Space and in the Plane

Definition 1.1.3 If a map ϕ : M → N is continuous and bijective, and if its

and M and N are said to be homeomorphic The Jacobi matrix of a differentiable

mapψ : N → M such that ϕ◦ψ = I (where I is the identity map) and ψ ◦ϕ = I

Theorem 1.1.1 (Inverse function theorem) Let U ⊂ Rn be an open set, P ∈ U,

and ϕ : U → R n If det J (P) = 0, then there exist neighborhoods V P of P and

V ϕ(P) of ϕ(P) such that ϕ| V P : V P → V ϕ(P) is a diffeomorphism.

y i+1, , y n ) If W ⊂ R n+1, then W = { ˜w : w ∈ W} ⊂ R n is a projection

along the i th coordinate axes.

Theorem 1.1.2 (Implicit function theorem) Let ϕ : R n+1→ R be a C k (k ≥ 1) function, P ∈ Rn+1, and (∂ϕ/∂x i )(P) = 0 for some fixed i Then there is a neighborhood W of P in Rn+1 and a C k

function f: W → R such that for

y = (y1, , y n+1) ∈ R n+1, f (y1, , y n+1) = 0 if and only if y i = f ( ˜y).

Theorem 1.1.3 (Existence and uniqueness solution) Let a map f: Rn+1 → Rn

par-tial derivatives with respect to the coordinates of
x ∈ R n

Let M over D Then the differential equation d
x/dt = f(
x, t) has a unique solution on

the interval |t − t0| ≤ min(T, b/M) satisfying
x(t0) =
x0.

1.2 Definition and Methods of Presentation of Curves

Definition 1.2.1 A connected setγ in the space R3(in the planeR2) is a regular k-fold continuously differentiable curve if there is a homeomorphism ϕ : G → γ ,

conditions:

(1) ϕ ∈ C k (k ≥ 1), (2) the rank of ϕ is maximal (equal to 1).

C k (k ≥ 1) is diffeomorphic either to a line segment or to a circle Since a

1.2 Definition and Methods of Presentation of Curves 3

any t.

y(t), z = z(t), where t ∈ [a, b], and the equations x = x(t), y = y(t), z = z(t)

called simple.

The Jordan curve theorem says that a simple closed plane curve has an interior

and an exterior.

It is often convenient to use the vector form of parametric equations of a curve:

r =
r(t) = x(t)
i + y(t)
j + z(t)
k, where
i,
j,
k are unit vectors of the axes

O X , OY, O Z If γ is a plane curve, then suppose z(t) ≡ 0.

(1)
r1(t(τ)) =
r2(τ),

(2) t (τ) = 0 for all τ ∈ (c, d).

character follow from the definition of a regular curve and from the inverse tion theorem

func-Example 1.2.1 The parameterized regular space curve x = a cos t, y = a sin t,

O X axis and the line joining the origin to the projection of the point
r(t) over the

X OY plane.

b2(x2+ y2) = a2z2and is called a (circular) conic helix.

Definition 1.2.2 A continuous curveγ is called piecewise smooth (piecewise ular) if there exist a finite number of points P i (i = 1, , k) on γ such that each

i

Example 1.2.2 The trajectory of a point on a circle of radius R rolling (without

If the circle moves along and inside of a fixed circle, then the curve is a

hypocy-cloid; if outside, then the curve is an epicycloid Parameterizations of these plane

curves are

4 1 Theory of Curves in Three-dimensional Euclidean Space and in the Plane

hypocycloids.

All cycloidal curves are piecewise regular They are closed (periodic) for m

Exer-cise 1.12.19 It has four singular points

(a) cardioid (b) astroid

Figure 1.1 Cycloidal curves

other presentations

Explicitly given curve. A particular case of the parametric presentation of a

curve is an explicit presentation of a curve, when the part of a parameter t is

f2(x); x = f1(y), y = y, z = f2(y); or x = f1(z), y = f2(z), z = z.

An explicit presentation is especially convenient for a plane curve In this case a

curve coincides with a graph of some function f , and then the equation of the

Example 1.2.3 A tractrix (see Figure 2.12 a) can be presented as a graph x =

a ln a−

√

a2−y2

y + a2− y2, 0 < y ≤ a It has one singular point P(a, 0) For a

parameterization of this plane curve see Exercise 1.12.22

Implicitly given curve. Let a differentiable map be given by

1.2 Definition and Methods of Presentation of Curves 5

forms a smooth regular curve (more exactly, a finite number of smooth regular

curves) This method is called an implicit presentation of a curve, and the system (1.2) is called the implicit equations of a curve In the plane case, an implicit

0 is a regular value

rank of the Jacobi matrix

Obviously, an explicit presentation of a curve is at the same time a

paramet-ric presentation, where the role of a parameter t is played by the x-coordinate,

say Conversely, if a regular curve is given by parametric equations, then in someneighborhood of an arbitrary point, as follows from the converse function the-orem, there an its explicit presentation Analogously, if a curve is presented byimplicit equations, then in some neighborhood of an arbitrary point it admits anexplicit presentation The last statement can be deduced from the implicit functiontheorem

Example 1.2.4 (a) The intersection of a sphere x2+ y2+ z2 = R2 of radius R

0 0.2 0.6 1 –0.4 0 0.4

–1 –0.5 0 0.5 –1

–0.5 0 0.5 1

(b) cylinder sphere

Figure 1.2 Viviani window

6 1 Theory of Curves in Three-dimensional Euclidean Space and in the Plane

1 and

y2 + z2 = R2



, 0 ≤ t ≤ 2π,

is a regular parameterization of the two curve components

(a) curve (b) cylinder cylinder

Figure 1.3 Bicylinder curve

1.3 Tangent Line and Osculating Plane

r =
r(t) = x(t)
i + y(t)
j + z(t)
k.

z (t0)
k The velocity vector field is the vector function
r (t) The speed of
r(t) at

Definition 1.3.1 The tangent line to a smooth curve γ at the point P =
r(t0) is

vector
r (t0).

One can easily deduce the equations of a tangent line directly from its

1.3 Tangent Line and Osculating Plane 7

line is given by the following equations:

is 2 at P (i.e., rows and columns of J are each linearly independent) Assume for

definiteness that the determinant

f1(x, ϕ1(x), ϕ2(x)) ≡ 0, f2(x, ϕ1(x), ϕ2(x)) ≡ 0.

equa-tion of its tangent line can be written in the form

(∂ f /∂x)(x0, y0)(x − x0) + (∂ f /∂y)(x0, y0)(y − y0) = 0. (1.7)

1.3.1 Geometric Characterization of a Tangent Line

8 1 Theory of Curves in Three-dimensional Euclidean Space and in the Plane

Theorem 1.3.1 explains the geometric characterization of a tangent line

Secondly, Theorem 1.3.1 estimates an error that we obtain from replacing a

a change we make an error of higher order than the radius d of a ball Also, this

theorem allows us to give a geometric definition of a tangent line to a curve.Denote by
τ(t0) a unit vector that is parallel to
r (t0):
τ(t0) =
r (t0)

|
r (t0)| A straight

normal line.

1.3.2 Osculating Plane

It is convenient to give a geometric definition of the osculating plane Let a plane

α with a unit normal
β pass through a point P =
r(t0) of a curve γ Denote by d

h the length of the perpendicular dropped from P1onto the planeα.

Definition 1.3.2 A planeα is called an osculating plane to a curve γ at a point

1.3 Tangent Line and Osculating Plane 9

Figure 1.4 Osculating plane

Proof First, we shall prove the second statement, assuming the existence of an

d and h it follows that

h

existence of an osculating plane, consider two cases:

(1)
r (t0) ×
r ... epicycloid Parameterizations of these plane

curves are

4 Theory of Curves in Three-dimensional... 1.3.2 A planeα is called an osculating plane to a curve γ at a point

1.3 Tangent... Geometric Characterization of a Tangent Line

8 Theory of Curves in Three-dimensional Euclidean