The Topological Gallery

This page contains fascinating pictures of various intriguing topological and geometric objects. This page also comprises all the visuals and animations that I have personally found to be useful for learning geometry and topology. It contains self-invented visualization methods as well as the ones I home come across during my journey.

Any matrix $G$ of $SL(2,\mathbb R)$ , the simple linear group of order 2, can be decomposed uniquely into the product of a rotation *$K$* , a dilation $A$ and a shear $N$ . This decomposition is called **the Iwasawa decomposition** of $SL(2,\mathbb R)$ . This gives us the following chain of diffeomorphisms
$SL(2,\mathbb R)\cong SO(2,\mathbb R)\times \mathbb R_{>0}\times \mathbb R\cong \mathbb S^1\times \mathbb R_{>0}\times \mathbb R\cong (\mathbb R^2\setminus \{0\})\times \mathbb R\cong\mathbb R^3\setminus\{z\text{-axis}\}$ and therefore allowing us to identify $SL(2,\mathbb R)$ as an embedded submanifold of $\mathbb R^3$ .

***Wente torus***, an immersed torus in $\mathbb R^3$ of constant positive *mean curvature*, discovered by *Henry C. Wente (1986)*, distinguished professor emeritus of mathematics at *University of Toledo*

*Poincaré disk model* /*conformal disk model* of the 2-dimensional hyperbolic space

*Half-plane model* of the 2-dimensional hyperbolic space

Hyperboloid model of the 2-dimensional hyperbolic space and the correspondence with the *Poincaré disk model*

A *right angled hyperbolic hexagon*: all 6 angles being right angles (the big red one in the middle of the picture); each of the 6 sides is a *geodesic*, **aka. the *straightest path* between two points**

***Hyperbolic Pants*: Obtained by gluing alternating sides of two identical right angle hyperbolic hexagons; a very useful concept in *hyperbolic geometry* and *Teichmüller Theory***

*Pants Decomposition* of a *Double Torus* (*Genus-2 Torus*): 2 pants glued in the above way gives rise to a *double torus*

*The Topologist’s Sine Curve, a very important example in topology*

*The Klein Bottle* (An Immersed Picture); **the best one can do to visualize, it cannot be embedded into a 3-dimensional Euclidean space** and there is actually no self-intersection in a *Klein-Bottle*

The parametric coordinates of the bagel / figure 8-immersion of the Klein Bottle in in $\mathbb R^3$ are as follows:
$\begin{aligned} x &= (r + \cos(\theta/2)\sin v – \sin(\theta/2)\sin 2v)\cos \theta,\\ y &= (r + \cos(\theta/2)\sin v – \sin(\theta/2)\sin 2v)\sin \theta,\\ z &= \sin(\theta/2)\sin v + \cos(\theta/2)\sin 2v, \end{aligned}$ with $0 \le \theta, v < 2\pi$ , $r>2$ . One can start with a Möbius strip and curl it to bring the edge to the midline; since there is only one edge, it will meet itself there, passing through the midline. It has a particularly simple parametrization as a figure-8 torus with a half-twist. In this immersion, the self-intersection circle (where in $\sin(v)$ is zero) is a geometric circle in the xy plane. The positive constant r is the radius of this circle. The parameter θ gives the angle in the xy plane as well as the rotation of the figure-8, and v specifies the position around the 8-shaped cross section. With the above parametrization the cross section is a **2:1 Lissajous curve.**

parametrization of the usual 3-dimensional immersion of the bottle itself is much more complicated. For $(u,v) \in [0,\pi) \times [0,2\pi)$ :
$x(u,v) = -\frac{2}{15} \cos u \Big( 3 \cos v – 30 \sin u + 90 \cos^4 u \sin u – 60 \cos^6 u \sin u + 5 \cos u \cos v \sin u \Big),\\[1em] y(u,v) = -\frac{1}{15} \sin u \Big( 3 \cos v – 3 \cos^2 u \cos v – 48 \cos^4 u \cos v + 48 \cos^6 u \cos v – 60 \sin u \\ \quad + 5 \cos u \cos v \sin u – 5 \cos^3 u \cos v \sin u – 80 \cos^5 u \cos v \sin u + 80 \cos^7 u \cos v \sin u \Big),\\[1em] z(u,v) = \frac{2}{15} \left( 3 + 5 \cos u \sin u \right) \sin v,$

**The *Hawaiian Earring Space* (A Non-example of *Semi-Locally Simply Connected* Space)**

**The *Möbius strip* and The *Cylinder* are *homotopy equivalent* (to the *circle*), but are not *homemomorphic* to each other**

Two different *embeddings* of the *genus-2-torus* in the 3-dimensional Euclidean space

*Schoen Bat-wing Eggs:* Example of a *minimal surface*; surfaces which locally minimize area

**A *helicoid* (a minimal surface) formed of soap-water film with the help of a helical wire**

**A soap-film *catenoid* (a minimal surface) formed with the help of two circular bubble wands**

*Scherk Minimal Surface* with equation $\large$e^z\cos x=\cos y$$

*Scherk Minimal Surface* with equation $\large$e^z\cos x=\cos y$$

Number of non-diffeomorphic smooth spheres $|\Theta_n|$ versus sphere dimension $n$ . The **vertical axis is logarithmic** to accommodate the wide variation in values. Markers indicate the exact counts for each dimension.

This above graph shows the number of non-diffeomorphic spheres (all homeomorphic to the usual sphere) of each dimension, from 1 through 20, notice that there is a unique smooth structure on spheres of dimensions 1, 2 and 3, for dimension 4, it is an open conjecture, for dimensions 5, 6 again it is 1, for dimension 7, there are 28 (27 of them are exotic spheres and remaining 1 is the ordinary sphere with the usual smooth structure) of them and the trend looks pretty random as the dimension increases, with sharp contrast about 11, 15 and 19