Manifolds Unfolded

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.

1. Parallel Transport and Holonomy Effect

In a flat space, a vector remains undeviated when parallel transported along a closed loop.

In a curved space, parallel transport along a closed loop causes a vector to deviate from its original direction, in general. This is called the holonomy effect.

However if the closed curve is a geodesic, then there is no deviation of the initial vector after parallel transport along itself.

Parallel Transport of a tangent vector along a non-geodesic curve along the surface of a 2-dimensional sphere; even though the vector starts off pointing along the velocity vector of the curve, it starts deviating as it is transported parallelly along the curve. This phenomenon is attributable to the presence of non-zero acceleration component along the surface i.e. covariant acceleration, which forces the vector to deviate from the velocity vector in order to maintain parallelism. For great circle, a geodesic curve, this phenomenon will be absent, and the vector will continue to align with the tangent direction without any deviation, because of zero covariant acceleration.

Note that for a given vector field along the curve , in general . So one captures the deviation through the following:

2. KAN Decomposition

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 $\mathbb R^3$.

3. The Wente Torus

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

4. Models of the hyperbolic plane

(a) The Poincaré disk model /conformal disk model of the 2-dimensional hyperbolic space

(b) Half-plane model of the 2-dimensional hyperbolic space

(c) Hyperboloid model of the 2-dimensional hyperbolic space and the correspondence with the Poincaré disk model

5. Hyperbolic Hexagon with all right angles

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

6. Hyperbolic Pants & Pants Decomposition

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), where gluing 2 pants in the above way gives rise to a double torus.

7. Topologist’s Sine Curve

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

8. The Klein Bottle

The Klein Bottle (An Immersed Picture); the best one can do to visualize (see the picture above), 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 $(u,v) \in [0,\pi) \times [0,2\pi)$:
$x(u,v) = -\frac{2}{15} \cos u \Big( 3 \cos v – 30 \sin u + 90 \cos^4 u \sin u – 60 \cos^6 u \sin u + 5 \cos u \cos v \sin u \Big),\\[1em] y(u,v) = -\frac{1}{15} \sin u \Big( 3 \cos v – 3 \cos^2 u \cos v – 48 \cos^4 u \cos v + 48 \cos^6 u \cos v – 60 \sin u \\ \quad + 5 \cos u \cos v \sin u – 5 \cos^3 u \cos v \sin u – 80 \cos^5 u \cos v \sin u + 80 \cos^7 u \cos v \sin u \Big),\\[1em] z(u,v) = \frac{2}{15} \left( 3 + 5 \cos u \sin u \right) \sin v,$

9. Universal Covering of the Figure-Eight Space

10. The Hawaiian Earring Space

The Hawaiian Earring Space is an example of a space which is NOT Semi-Locally Simply Connected.

11. Homotopy Equivalence versus Homeomorphism

The Möbius strip and The Cylinder are homotopy equivalent to a circle. In fact both strongly deformation retract onto the central circle. However, they are not homemomorphic to each other.

12. Topological Embeddings

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

13. Bat-Wing Eggs & Minimal Surfaces

Schoen Bat-wing Egg is an example of a minimal surface; surfaces which locally minimize area.

14. Exotic Spheres

The graph below 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.

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.

15. Visualize vectors and co-vectors in an Euclidean Space