0
$\begingroup$

The surface of such an object would be $\dfrac{C^2}\pi$, where $C$ is the circumference, which should be derivable from the definition.

  • If defined as a Riemann manifold, what would it's metric be? (A smoothly varying inner product of tangent vectors)
  • What would other defining properties, such as Gaussian curvature, be when described in a coordinate-free intrinsic way? (Without radii of curvatures or local coordinate charts)

Edit: Further clarification : A metric is just a section of the tensor product of the cotangent bundle with itself . It is a geometric object. I am asking for a geometric definition of the manifold and thus its metric.. not a representation in coordinates or some other made up structure. The question is what is the intrinsic , invariant definition of the sphere as a manifold I am new to differential geometry and I woul like to learn it properly...

$\endgroup$
  • $\begingroup$ For future posts, using mathjax will make your questions much more readable and, in turn, solicit more answers. $\endgroup$ – Victoria M Feb 10 at 17:12
  • $\begingroup$ Welcome the Mathematics Stack Exchange community. A quick tour of the site will help you get the most of your time here. For typesetting your equations, please use MathJax. Here is a great reference. $\endgroup$ – dantopa Feb 10 at 17:13
  • $\begingroup$ The title of the question refers just to a differentiable manifold structure of a sphere whereas the content of the question refers to a particular Riemannian metric on the sphere, which is an additional structure. In particular the $2$-sphere (the differentiable manifold) admits many distinct (nonisometric) Riemannian metrics. $\endgroup$ – Travis Feb 10 at 21:12
  • 1
    $\begingroup$ @Schaurberger Stop insulting people. $\endgroup$ – Michael Greinecker Feb 11 at 8:53
0
$\begingroup$

Here are two possible definitions of the round metric on the 2-sphere of radius 1.

Using polar coordinates (strictly speaking not defined at the poles) $(\theta,\phi)\in[0,\phi]\times [0,2\pi)$ \begin{equation} g = \mathrm{d}\theta^2 +\sin^2\theta \, \mathrm{d}\phi^2 \end{equation}

Pullback with respect to the inclusion $\iota:S^2\hookrightarrow\mathbb{R}^3$ of the Euclidean metric $g_E$ \begin{equation} g=\iota^* g_E, \quad g_E=\mathrm{d}x^2+\mathrm{d}y^2+\mathrm{d}z^2. \end{equation}

Volume form, volume (up to a sign until you specify an orientation) curvature and so forth can all be derived from the metric. But these are are the topics of a basic course in differential/Riemannian geometry and there is no much point in summarising them here. You should probably find a book (there are many threads on this) and read about it if you are interested.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.