We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
Published online by Cambridge University Press: 27 June 2025
Consider the metric ds2 = dt2 + A2 (t) dx2
+ B2 (t) dy2 + C2 (t) dz2, where t is the radial coordinate and x, y, z are “spherical coordinates” with [X, T] = [Y, T] = [Z, T] = 0, [X, Y] = 2Z, [y, Z] = 2X, and [Z, X] = 2Y. (Taking A(t) = B(t) = C(t) = t we get the standard Euclidean metric.) A straightforward computation gives and similar equalities obtained by permutation of the pairs (X, A), (y, B), (Z, C); similarly . It is also clear that the asymptotic cone is not unique.
It remains only to smooth off the vertex (t = 0), where our space is isometric to a cone over a sphere of constant curvature 100. Mike Anderson has pointed out to me that a similar construction was used earlier by Brian White, in the context of surfaces in euclidean spaces.
To save this book to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Find out more about the Kindle Personal Document Service.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.
