An Introduction To Differentiable Manifolds And Riemannian Geometry 2nd Edition Is Written By Author Unknown And Published By Academic Press The Digital And Etextbook Isbns For An Introduction To Differentiable Manifolds And Riemannian Geometry Are 9780121160524 9780080874395 0080874398 And The Print Isbns Are 9780121160524 0121160521 Save Up To 80% Versus Print By Going Digital With Vitalsource Additional Isbns For This Etextbook Include 0121160521 012116053x 9780121160524 9780121160531 0080874398 0080574750 1281763233 9780080874395 9780080574752 9781281763235
An introduction to differentiable manifolds and Riemannian geometry 2nd Edition is written by AUTHOR, UNKNOWN and published by Academic Press. The Digital and eTextbook ISBNs for An introduction to differentiable manifolds and Riemannian geometry are 9780121160524, 9780080874395, 0080874398 and the print ISBNs are 9780121160524, 0121160521. Save up to 80% versus print by going digital with VitalSource. Additional ISBNs for this eTextbook include 0121160521, 012116053X, 9780121160524, 9780121160531, 0080874398, 0080574750, 1281763233, 9780080874395, 9780080574752, 9781281763235.
Regularity Of A Riemannian Metric
One says that the Riemannian metric g :U\to \mathbb } are continuous when given any smooth coordinate chart ( g is smooth if these functions are smooth when given any smooth coordinate chart. One could also consider many other types of Riemannian metrics in this spirit.
In most expository accounts of Riemannian geometry, the metrics are always taken to be smooth. However, there can be important reasons to consider metrics which are less smooth. Riemannian metrics produced by methods of geometric analysis, in particular, can be less than smooth. See for instance and .
Every Smooth Manifold Has A Riemannian Metric
This is a fundamental result. Although much of the basic theory of Riemannian metrics can be developed by only using that a smooth manifold is locally Euclidean, for this result it is necessary to include in the definition of “smooth manifold” that it is Hausdorff and paracompact. The reason is that the proof makes use of a partition of unity.
Proof
be a differentiable manifold and \subseteq M} are open subsets and \colon U_\to \varphi _\subseteq \mathbf ^} are diffeomorphisms.
\}_} be a differentiable partition of unity subordinate to the given atlas, i.e. such that supp \,\tau _\subseteq U_} for all
- , \tau _\cdot }_,\qquad }\qquad }_:=\varphi _^g^ }\,\,}\,\,U_,}
where is the Euclidean metric on R ^g^ }} is its pullback along
This is readily seen to be a metric on M
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The Length Of Piecewise Continuously
If is differentiable, then it assigns to each t M,} the size of which can be measured by the norm | } defines a nonnegative function on the interval ( . The length is defined as the integral of this function however, as presented here, there is no reason to expect this function to be integrable. It is typical to suppose g to be continuous and to be continuously differentiable, so that the function to be integrated is nonnegative and continuous, and hence the length of
- , ^|\gamma ‘|_\,dt,}
is well-defined. This definition can easily be extended to define the length of any piecewise-continuously differentiable curve.
In many instances, such as in defining the Riemann curvature tensor, it is necessary to require that g has more regularity than mere continuity this will be discussed elsewhere. For now, continuity of g will be enough to use the length defined above in order to endow M with the structure of a metric space, provided that it is connected.
The Metric Space Structure
Precisely, define :M\times M\to [0,\infty )} by
- a piecewise continuously differentiable curve from p } . =\inf\}p}q\}.}
It is mostly straightforward to check the well-definedness of the function d =d_,} its reflexivity property =0,} and the triangle inequality , +d_\geq d_,} although there are some minor technical complications . It is more fundamental to understand that p > 0,} and hence that satisfies all of the axioms of a metric.
The observation that underlies the above proof, about comparison between lengths measured by g and Euclidean lengths measured in a smooth coordinate chart, also verifies that the metric space topology of ( coincides with the original topological space structure of M .
Although the length of a curve is given by an explicit formula, it is generally impossible to write out the distance function d by any explicit means. In fact, if M is compact then, even when g is smooth, there always exist points where d R :M\times M\to \mathbb } is non-differentiable, and it can be remarkably difficult to even determine the location or nature of these points, even in seemingly simple cases such as when (
As in the previous section, let ( be a connected and continuous Riemannian manifold consider the associated metric space ( Relative to this metric space structure, one says that a path c is a unit-speed geodesic if for every t \in } there exists an interval
- ,c)=|s-t|\qquad \forall s,t\in J.}
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An Introduction To Differentiable Manifolds And Riemannian Geometry 2nd Edition
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