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More information about this seller Contact this seller Wang McKenzie Editor. Seller Inventory NEW Great condition with minimal wear, aging, or shelf wear. Seller Inventory P Condition: Used: Very Good. Very Good Hardcover, text is clean and covers are nice. Condition: UsedAcceptable.
Published by Springer International Publishing , Cham About this Item: Springer , About this Item: Condition: New. Not Signed; Intended for a one year course, this text serves as a single source, introducing readers to the important techniques and theorems, while also containing enough background on advanced topics to appeal to those students wishing to specialize in Riemannian geometry. This is one of the few Works to comb.
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Language: English. Brand new Book. This volume introduces techniques and theorems of Riemannian geometry, and opens the way to advanced topics. The text combines the geometric parts of Riemannian geometry with analytic aspects of the theory, and reviews recent research. The updated second edition includes a new coordinate-free formula that is easily remembered the Koszul formula in disguise ; an expanded number of coordinate calculations of connection and curvature; general fomulas for curvature on Lie Groups and submersions; variational calculus integrated into the text, allowing for an early treatment of the Sphere theorem using a forgotten proof by Berger; recent results regarding manifolds with positive curvature.
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Softcover reprint of hardcover 2nd ed. Seller Inventory AAV Seller Inventory LIE Seller Inventory n. Galway, Ireland. Intended for a one year course, this text serves as a single source, introducing readers to the important techniques and theorems, while also containing enough background on advanced topics to appeal to those students wishing to specialize in Riemannian geometry. This is one of the few Works to combine both the geometric parts of Riemannian geometry and the analytic aspects of the theory.
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The book will appeal to a readership that have a basic knowledge of standard manifold theory, including tensors, forms, and Lie groups. By Let e1 , e2 be an orthonormal frame of vector fields on an oriented open subset U of a surface M in R3 , and let e3 be a unit normal vector field on U. Because the Proof. The Gauss curvature equation now follows from Formula As for We can therefore take this formula to be the definition of the Gaussian curvature of an abstract Riemannian 2-manifold. It is consistent with the formula in the first version Theorem 8.
Definition For the Gaussian curvature to be well defined, we need to show that formula This we do in the next section. A point operator is also called a tensor. By Gram—Schmidt, we may assume that the frame e1 ,. Just as in the definition of the Gaussian curvature, the right-hand side of Chapter 3 Geodesics A geodesic on a Riemannian manifold is the analogue of a line in Euclidean space.
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These two properties are not necessarily equivalent on a Riemannian manifold. On a Riemannian manifold, of course, there is always the unique Riemannian connection, and so one can speak of geodesics on a Riemannian manifold. First we discuss how an affine connection on a manifold M induces a unique covariant derivative of vector fields along a smooth curve in M.
Secondly, we present a way of describing a connection in local coordinates, using the so-called Christoffel symbols. Vector field along a curve c t in M. On a framed open set U, e1 ,. By covering M with framed open sets, It suffices to check this equality locally, so let U be an open set on which an orthonormal frame e1 ,.
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In this terminology, an isometry of Riemannian manifolds preserves the Riemannian connection Problem 8. We now show that a connection-preserving diffeomorphism also preserves the covariant derivative along a curve. Choose a neighborhood U of c t on which there is a frame e1 ,. Another way, which we now discuss, is by a set of n3 functions called the Christoffel symbols.
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By the Leibniz rule and F-linearity in the first argument of a connection, the Christoffel symbols completely describe a connection on U. Elwin Bruno Christoffel — Proposition Remark Because of this proposition, a torsion-free connection is also called a symmetric connection. The geodesic is said to be maximal if its domain I cannot be extended to a larger interval. The notion of a geodesic depends only on a connection and does not require a metric on the manifold M.
However, if M is a Riemannian manifold, then we will always take the connection to be the unique Riemannian connection. The speed of a geodesic on a Riemannian manifold is constant. By Problem Velocity and acceleration vectors of a great circle. Since T u has constant length, it is never zero. Corollary Let a, b be an interval containing 0. In this section we will determine a set of differential equations on yi t for the curve to be a geodesic. By an existence and uniqueness theorem of ordinary differential equations, we have the following theorem.
Moreover, this geodesic is unique in the sense that any other geodesic satisfying the same initial conditions must agree with c t on the intersection of their domains. From Example Since a geodesic with a given point and a given velocity vector at the point is unique Theorem This proves the converse of Example By Corollary On a Riemannian manifold we always use the unique Riemannian connection to define geodesics. In this case, tangent vectors have lengths and the theory of ordinary differential equations gives the following theorem. Then Theorem Because an isometry of Riemannian manifolds preserves the Riemannian connection Problem 8.
In Example The y x Fig. We include it because of its simplicity. In fact, by a simple reparametrization, we may even assume that the speed is 1. Hence, In this calculation we never used It is in fact a redundant equation Problem The fifth postulate, called the parallel postulate, was a source of controversy.
Two lines in the plane are said to be parallel if they do not intersect. One form of the fifth postulate states that given a line and a point not on the line, there is a unique parallel line through the given point. For hundreds of years, heroic efforts were made to deduce the fifth postulate from the other four, to derive a contradiction, or to prove its independence.
A parallel vector field along a curve is an analogue of a constant vector field in Rn. By the existence and uniqueness theorem of ordinary differential equations, there is always a unique solution V t on a small interval about t0 with a given V t0.
In the next subsection we show that in fact the solution exists not only over a small interval, but also over the entire curve c. By the uniqueness theorem of ordinary differential equations, V t is uniquely determined by the initial condition V a , so if parallel translation exists along c, then it is well defined.
Parallel translation is possible from c a to c b along c, i. Let w1 ,. Since [a, b] is compact, it is covered by finitely many such open intervals. Hence, it is possible to parallel translate along c from c a to c b. While a geodesic with a given initial point and initial velocity exists only locally, parallel translation is always possible along the entire length of a smooth curve. In fact, the curve need not even be smooth. On a Riemannian manifold we will always assume that parallel translation is with respect to the Riemannian connection.
By the product rule for the covariant derivative of a connection compatible with the metric Theorem Parallel translating v along a closed piecewise smooth geodesic. Figure From this example we see that parallel translating a tangent vector of a manifold around a closed loop need not end in the original vector. This phenomenon is called holonomy. Surface of revolution in R3 In Problem 5. While these are two independent notions, when a Lie group has a bi-invariant Riemannian metric, as all compact Lie groups do, the exponential map for the Lie group coincides with the exponential map of the Riemannian connection.
On a manifold M with a connection, a geodesic is locally defined by a system of second-order ordinary differential equations. By the existence and uniqueness theorems of ordinary differential equations there is a unique geodesic through any given point q with any given direction v. By reparametrizing the geodesic by a constant factor, one can expand the domain of definition of the geodesic at the expense of shortening the initial vector. This exponential map derives its importance from, among other things, providing coordinate charts in which any isometry is represented by a linear map see Section The exponential map for a Lie group G is defined in terms of the integral curves of the left-invariant vector fields on G.
get link Unlike the exponential map of a connection, the Lie group exponential is defined on the entire Lie algebra g. It shares some of the properties of the exponential map of a connection. The problems at the end of the chapter explore the relationship between the two notions of exponential map. Now assume that M is endowed with a Riemannian metric.
A neighborhood U of p and the domain of the exponential map in the tangent bundle T M. By the naturality of the exponential map Theorem This diagram may be interpreted as follows: relative to the coordinate charts given by the inverse of the exponential map, an isometry of Riemannian manifolds is locally a linear map; more precisely, it is the linear map given by its differential.