Ds intrinsically It should be stressed how impressive all this is to one who is accustomed to physicists treatments of the matter one who is accustomed to physicists treatments of the matter the #curvature tensor figures as nothing but a black box see for instance Misner Thorne and Wheeler #tensor figures as nothing but a black box see for instance Misner Thorne and Wheeler or Robert Wald s obligatory textbooks on gravitationChapter ten proves the Gauss Bonnet formula and the remarkable Gauss Bonnet theorem relating the Gaussian curvature a local differential invariant on a surface with the Euler characteristic a global topological invariant Standard material nicely presented Chapter nine takes up Jacobi fields which describe the divergent or convergent behavior of nearby geodesics and relate it to the curvature tensor and their conjugate points an interesting global problem Also derived is the second variation formula from which Lee demonstrates that geodesics fail to be distance minimizing past their first conjugate point The last chapter is dedicated to the inferences one can make about global properties of Riemannian spaces given constraints on their curvature in the cases both of negative and of positive curvature The relevant theorems are those by Cartan Hadamard and Bonnet as well as a classification of spaces of constant curvature as obtained as a uotient of one of the three model spaces by the action of a discrete subgroup of isometries Lee manages to make these surprising statements which apply at the global level although one employs only local input seem effortless along with the basic Sturm Liouville and covering space theory on which they are basedThus the reader arrives at a satisfying conclusion having traversed a rapid but pleasant journey all the way from the elementary beginnings in a book barely over two hundred pages in length Despite its brevity Lee s derivations are full and complete and never lose sight of overarching issues There are only some seventy homework exercises but the ones included are well chosen and illustrative of the material Most of them are of moderate difficulty Lee s aim seems to be to flesh out the subject with a few select examples for instance a series of problems pursuing the existence and properties of left invariant and bi invariant metrics on Lie groups and their connections than to challenge the student with arduous computations This agrees with Lee s views on the norms of good mathematical practice as we have remarked elsewhereTo conclude Lee s elegant treatment of the elementary theory of Riemannian geometry can be warmly recommended Perhaps it is too disarming in its simplicity making a difficult subject appear easier than it really is Consider in contrast the imposing textbooks by Peter Petersen and J rgen Jost There can be no uestion of getting into functional analysis and partial differential euations in a whirlwind tour such as Lee s Maybe it is just a matter of mathematical taste Certainly there will be no cause of complaint to have in one s possession a simple and aesthetically pleasing expos that brings out the geometrical flavor of the subject very well without burdening the reader with complicated investigations into hard problems of analysis. Characterization of manifolds of constant curvature This uniue volume will appeal especially to students by presenting a selective introduction to the main ideas of the subject in an easily accessible way The material is ideal for a single course but broad enough to provide students with a firm foundation from which to pursue research or develop applications in Riemannian geometry and other fields that use its tools. ,
A very nice introduction to Riemannian geometry
Doesn T Get Bogged Down In Technicalityt get bogged down in technicality offers exercises and examples that help build intuition It s a great place to get started learning geomery The theory of curvature forms the crowning glory of geometry The ancient Greeks missed it altogether since they failed to take the differential point of view we owe to the development of the calculus during the early modern period and which by the time of Gauss had issued in a rich theory of curved surfaces in three dimensional space Later in the nineteenth century Riemann took the momentous step of generalizing our ideas of space to manifolds of arbitrarily many dimensions But the subject as we now now it in the canonical form it achieves in Einstein s general theory of relativity underwent its final refinement and polishing in the generation after Riemann at the hands of Levi Civita Bianchi Beltrami and ChristoffelThe present review is devoted to John Lee s Riemannian Manifolds An Introduction to Curvature the last of three volumes on the theory of manifolds the first two on topological and smooth manifolds respectively having been previously reviewed here by this recensionist Lee outdoes himself in this elegant little text he already establishes himself as a capable pedagogue in the first volume and his style only improves in the second but in this third he rises to dazzling heights of clarity and concision There are of course numberless books on introductory differential geometry but Lee s recent entry stands out for its modern and clean perspective Dirk Struik s classic from over half a century ago for instance confines itself to two dimensions and does everything in local coordinates thus concealing what is really going on as notably in the derivation of the second fundamental formTo be sure the student will want to approach the present text prepared with a good command of the theory of manifolds such as can be elicited from the first two of the present author s three volumes The second chapter in the present volume uickly surveys what is needed for the rest of the text but it serves mainly to establish the notation and in no sense could one learn the material adeuately from what little is skimmed off there The first chapter though sets the context could one learn the material adeuately from what little is skimmed off there The first chapter though sets the context stating what are familiar results in plane geometry such as that the interior angles of a triangle sum to 180 degrees from the point of view of describing them as local to global theorems of a ind which the full theory will supply far general and powerful versionsThe plan of organization of the work is straightforward Lee first proposes the standard examples of Riemannian metrics in the model spaces Euclidean spherical and hyperbolic The stereographic projection is worked out in detail and the euivalence of the hyperboloid Poincar ball and Poincar half space shown But the subject becomes interesting in the succeeding chapter on connections and geodesics The concept itself is well motivated and the corresponding formalism developed neatly along axiomatic lines Lee pays attention to niceties such as that the value of. This text is designed for a one uarter or one semester graduate course on Riemannian geometry It focuses on developing an intimate acuaintance with the geometric meaning of curvature and thereby introduces and demonstrates all the main technical tools needed for a advanced study of Riemannian manifolds The book begins with a careful treatment of the machinery of metrics connections and geodesics and then introduces. .
The connection applied to a vector field depends only on the near neighborhood of the point at which it is taken and that the connection can be restricted to yield a covariant derivative along a curve thereby making precise the notion of parallel translation and the meaning of a geodesic as a curve whose tangent undergoes no acceleration So far we have been dealing with uite general connections The next chapter specializes and investigates compatibility between the connection and the inner product coming from the Riemannian metric The fundamental lemma guaranteeing existence and uniueness of the Levi Civita connection associated with the metric receives an efficient proof The exponential map normal coordinates and their properties follow immediately The idea behind them is simple define coordinates around a point by moving radially outward along the geodesic flow in any given direction Lee s treatment is appealing because he uses the machinery of smooth manifold theory to show that the exponential map is in fact a diffeomorphism and behaves naturally with respect to isometries The geodesics in the three model spaces are readily obtained by arguing from symmetry Lee is characteristically precise without falling into tediousness look for instance at how he uses the Cayley transform to derive the geodesics in the upper half spaceNow geodesics are so named because of their distance minimizing property Lee shows this deftly in just a few pages by bringing in the concept of an admissible family along a curve and its variation field then deriving the first variation formula for arclength The converse follows easily from the Gauss lemma As a plus we can prove without too much difficulty the Hopf Rinow theorem characterizing when a Riemannian manifold is geodesically complete which is to say when geodesics can be continued indefinitely into the futureIn the brief seventh chapter we get at last to the idea of curvature The Riemannian curvature tensor is motivated and its tensorality and symmetry properties proved nicely without having to descend into manipulations with coordinates The main object of the chapter is to prove that a Riemannian space is flat if and only if its curvature tensor vanishes identically Once we finally have the curvature tensor at our disposal Lee s exposition takes off and the chapters from here on are nothing short of excellent Chapter eight takes up the uestion of what one can say when a Riemannian manifold is embedded as a submanifold in an ambient space with the induced metric Here is where the somewhat abstruse notion of the second fundamental form comes in abstruse notion of the second fundamental form comes in has to do with the relation the Levi Civita connection on the submanifold and the corresponding connection in the larger space Lee s presentation is completely intrinsic and renders all the operations involved luminously clear The second fundamental form along with the exponential map allow one to define sectional curvatures in arbitrary Riemannian spaces which unpack the geometrical significance of the information implicitly encoded in the Riemannian curvature tensor Again Lee procee. The curvature tensor as a way of measuring whether a Riemannian manifold is locally euivalent to Euclidean space Submanifold theory is developed next in order to give the curvature tensor a concrete uantitative interpretation The remainder of the text is devoted to proving the four most fundamental theorems relating curvature and topology the Gauss Bonnet Theorem the Cartan Hadamard Theorem Bonnet's Theorem and the.