Articles of surfaces

Parameterising the intersection of a plane and paraboloid

Suppose we have the paraboloid $z=x^2+y^2$ and the plane $z=y$. Their intersection produces a curve $C$, and certain surfaces bounded by it, for example the disc $S$ which directly fills the area of $C$ and the paraboloid $S’$ given by $z=x^2+y^2$ which extends from $C$ downwards and is bounded by $C$. My question is on […]

Orientable Surface Covers Non-Orientable Surface

I need to describe how a 4-genus orientable surface double covers a genus 5-non-orientable surface. I know that in general every non-orientable compact surface of genus $n\geq 1$ has a two sheeted covering by an orientable one of genus n-1. I tried to use the polygonal representation of these surfaces and try to get one […]

Computing fundamental forms of implicit surface

This question already has an answer here: About the second fundamental form 1 answer

Homology of orientable surface of genus $g$

I came across the problem of computing the homology groups of the closed orientable surface of genus $g$. Here Homology of surface of genus $g$ I found a solution via cellular homology. This seems to me like the natural way of calculating something of this sort although I know that it is also possible to […]

Connected sum of surfaces with boundary

The connected sum of closed surfaces (2-manifolds) is defined by removing a disk from each and gluing the exposed edges together. When defining the connected sum of surfaces with boundary, is the boundary of each surface allowed to touch its removed disk? By my intuition it seems like a bad idea to allow that, because […]

Calippo, toothpaste and milk .. packing

This post was posing the question “The Calippo™ popsicle has a specific shape … Does this shape have an official name?”, and the accepted answer was that it is a Right Conoid which in fact is much resembling. However, a conoid is not developable and so cannot be economically made out of a cardboard sheet. […]

identify the topological type obtained by gluing sides of the hexagon

Identify the topological type obtained by gluing sides of the hexagon as shown in the picture below Clearly the boundary is encoded by the word $abcb^{-1}a^{-1}c$ I do not understand how the surface is glued together – could you help me, please? Edit: This answer is a Klein bottle (see answer below) Similar problem: see […]

Union of two self-intersecting planes is not a surface

I need to show that the union of xy-plane and xz-plane, i.e. the set $S:=\lbrace (x,y,z)\in\mathbb{R}^3 : z=0 \mbox{ or } y=0\rbrace$, is not a surface. Here is my claim, $\textbf{Claim :}$ Suppose $p$ is the point $(0,0,0)$ and $U:=S\cap B(p,\epsilon)$, where $\epsilon > 0$. Then $U$ cannot be homeomorphic to any open set of […]

isometries of the sphere

There is a theorem by Pogorelov that if a $C^2$ surface $M$ in $\mathbb{R}^3$ is isometric to the unit 2-sphere, then $M$ is itself (a rigid motion of) the sphere. What is known about isometric deformations of the sphere, when the smoothness condition is relaxed? First, we need some notion of isometry for surfaces with […]

Uniqueness for antipodal points of maximum distance on closed convex surface

Let $S\subset \mathbb{R}^3$ be a closed convex surface and let $p,q\in S$ be points such that $d(p,q)=\operatorname{diam}(S)$ where $\operatorname{diam}(S)$ is the diameter of $S$ with respect to the intrinsic distance $d$ of $S$. Does $d(p,q’)=\operatorname{diam}(S)$ imply $q=q’$?