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...
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.