# Tagged Questions

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With metric tensors of the unit sphere in normal coordinates, their Taylor series for $p\in S$ near the north pole $N$ can be written as follows. $$g_{rr}(p) \equiv 1; g_{r\theta}(p) = g_{\theta ... 2answers 158 views ### A non orientable closed surface cannot be embedded into \mathbb{R}^3 Can someone please remind me how this goes? Here's the idea of proof I'm trying to recall: let S be a closed surface (connected, compact, without boundary) embedded in \mathbb{R}^3. Then one can ... 1answer 242 views ### Restriction of a differentiable map R^3\rightarrow R^3 to a regular surface is also differentiable. This is again an excercise from Do Carmo's book. Prove: if f:R^3 \rightarrow R^3 is a linear map and S \subset R^3 is a regular surface invariant under L, i.e, L(S)\subset S, then the ... 1answer 116 views ### About zeros of vector fields in compact surfaces I'm studying compact surfaces and in particular the relationship between zeros of vector fields defined on them and Euler characteristic of the surface herself. Let be S a compact (smooth) surface ... 1answer 90 views ### Bounded vector field on a closed surface Let S\subseteq\mathbb{R}^3 a closed surface and let X\in\mathfrak{X} (S) a vector field on S such that \mid\mid X_p\mid\mid \le M \forall p\in S for some constant M>0. Prove that X ... 1answer 273 views ### Surfaces are homeomorphic iff are diffeomorphic. I have read this statement in several places: "Two surfaces are homeomorphic iff are diffeomorphic". I think the nontrivial implication follows in this manner: First, we triangulate the surface and ... 2answers 208 views ### Embedded surface in \mathbb{R}^3 Let U \subseteq \mathbb{R}^2 be an open set and let \sigma : U \rightarrow \mathbb{R}^3 be a parametrization of an oriented surface S embedded in \mathbb{R}^3 whose unit normal in \sigma ... 0answers 204 views ### Morse theory and homology of an algebraic surface (example) Let T_n denote the n-th Chebyshev polynomial and define f_n(x,y,z)\!:=\!T_n(x)\!+\!T_n(y)\!+\!T_n(z) and$$Z_n:=\mathcal{Z}(f_n) \subseteq \mathbb{R}^3,$$the Bachoff-Chmutov surface, where in ... 2answers 138 views ### Transform flat surface into paraboloid Is it possible to transform a flat surface into a paraboloid$$z=x^2+y^2 such that there is no strain in the circular in the circular cross section (direction vector A)? If the answer is yes, is ...
I am trying to detect when a closed parametric surface intersect itself. My surface is described as a triplet of parametric functions $x(u,v)$, $y(u,v)$ and $z(u,v)$ where $u,v\in[0,1]$. For that ...
From Hirsch's Differential Topology, p. 180. The first of the isotopy extension theorems says; Let $A\subset M$ be a compact submanifold and $F:V\times I \rightarrow S^{3}$ an isotopy of $A$. If ...