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 5h answered Angle form, 1-form, proof verification. 5h comment What is the volume and surface area of the 1-Sphere? If $n$ is a non-negative integer, the usual definitions are: The unit $n$-sphere is the set $S^{n} = \{x \in \mathbf{R}^{n+1} : \|x\| = 1\}$, and the unit $(n+1)$-ball is $B^{n+1} = \{x \in \mathbf{R}^{n+1} : \|x\| \leq 1\}$. The $n$-sphere has "$n$-volume", and the $(n+1)$-ball has "$(n+1)$-volume". ($1$-volume is "length", $2$-volume is "area", etc. "Area" of an $n$-sphere is meaningless unless $n = 2$.) Could you please formulate your question carefully using these definitions? As is, it's difficult to understand what you're asking. :) 9h comment n points can be equidistant from each other only in dimensions $\ge n-1$? Let $E^{N}$ denote $N$-dimensional Euclidean space, $n$ a positive integer, and $r$ a positive real number. Assuming the question is: "If $E^{N}$ contains a set of $n$ points, any two of which are at distance $r$, is $n \leq N + 1$?", the answer is "yes". The accepted answer to this question sketches a stronger result. 10h comment Circumference of hyperbolic circle is $2\pi \sinh r$ Good question, but seems to be a sub-question of your earlier query...? 10h comment Is it possible to have a sphere $S^m$ equidistant to sphere $S^n$ in $R^k$? (+1) Your final assertion is correct: If $S^{n}$ is a sphere centered at $p$ in $\mathbf{R}^{N}$, let $V$ be the smallest containing affine subspace ($\dim V = n + 1$), and let $W$ be the maximal orthogonal affine subspace through $p$, a.k.a. the set of points equidistant from each point of $S^{n}$. Any "complementary" sphere must be contained in $W$, and so has dimension$$m \leq \dim W - 1 = N - \dim V - 1 = N - n - 2.$$That is, $m + n + 2 \leq N$. :) 10h answered Why is a surface of revolution injective? 11h comment Differntial Geometry A strikingly similar question was migrated here recently.... 20h answered Why do disks on planes grow more quickly with radius than disks on spheres? 1d comment Compute the volume element in a differentiable manifold. I'm no longer sure what you're trying to express: You have an equation for $dV$ that requires a continuous unit normal field on $M$; the gradient (or its negative) gives you a suitable normal field at each point. If you have a basis for $T_{p}M$, you can calculate the value of $dV$ on this basis. Does that not answer the title question? 1d comment Compute the volume element in a differentiable manifold. 1. Yes, $M = g^{-1}(0)$ is a level set. (It's a level curve only if $n = 2$.) 2. If the normalized gradient gives the "wrong" orientation, use $-\nabla g/\|\nabla g\|$. (One of the two must "work" in your formula for the volume element.) 1d answered When are the eigenvalues of the second fundamental form equal to the principal curvatures? 1d comment When are the eigenvalues of the second fundamental form equal to the principal curvatures? My oops: When I typed "second fundamental form" earlier my brain was thinking "shape operator". I'll type up an answer shortly.... 1d comment Compute the volume element in a differentiable manifold. The gradient field of a differentiable function is orthogonal to the level sets. 1d comment When are the eigenvalues of the second fundamental form equal to the principal curvatures? Something sounds odd with the first exercise solution. Are you sure the solution didn't say "the coordinate directions are not the principal directions"? 1d comment When are the eigenvalues of the second fundamental form equal to the principal curvatures? The second fundamental form depends only on the geometry of the surface, not on a choice of coordinates. As you say, its eigenvalues are the principal curvatures and its eigenvectors the principal directions. If that's not what you're asking, could you please explain the purported dependence on the coordinate vectors of a parametrization? 1d comment Covering spaces of $S^1$ @Sergio: Under the stated hypotheses, it's not even generally true that $f$ is a covering map onto its image. (For example, take $f(t) = e^{t}$.) Could you please check the hypotheses you're given? 1d comment Is $[X,Y] \neq 0$ the sufficient condition of $e^{X+Y} \neq e^Xe^Y$? In fact, there exist non-commuting $3 \times 3$ real matrices $X$ and $Y$ for which $e^{X} = e^{Y} = e^{X+Y} = I$. 1d comment Compute the volume element in a differentiable manifold. Yes, that's right. 1d answered vector bundle associated to the representation of a lattice 1d comment Compute the volume element in a differentiable manifold. The ambient space is $A$ (or $\mathbf{R}^{n}$). :)