# Some illustrations of dynamical systems

The gradient field defines the steepest descent direction. The gradient flow dynamic defines a segmentation of the space into attraction bassins of the local minimizers.  pic.twitter.com/wW0flPEWor

— Gabriel Peyré (@gabrielpeyre) 16/6/2018.

# The Gaussian curvatures of spheres

1. An example of the Gaussian curvature

Example 1 Compute the Gaussian curvature of sphere

$\displaystyle S = \{(x, y, z) \in \mathbb{R}^3 | x^2 + y^2 + z^2 = R^2\}.$

Parametrizing ${X:}$

$\displaystyle U \subset \mathbb{R}^2 \rightarrow S$

$\displaystyle (u, v)\mapsto (u, v, \sqrt{R^2 - u^2 - v^2}).$

we have

$\displaystyle X_u = (1, 0, -\frac{u}{\sqrt{R^2 - u^2 - v^2}}), X_v = (0, 1, -\frac{v}{\sqrt{R^2 - u^2 - v^2}})$

The coefficients of the second fundamental form:

$\displaystyle E = \langle X_u, X_u \rangle = 1 + \frac{u^2}{R^2 - u^2 - v^2}$

$\displaystyle G= \langle X_v, X_v \rangle = 1 + \frac{v^2}{R^2 - u^2 - v^2}$

$\displaystyle F = \langle X_u, X_v \rangle = \frac{uv}{R^2 - u^2 - v^2}$

$\displaystyle \Rightarrow EG - F^2 = (1 + \frac{u^2}{R^2 - u^2 - v^2})(1 + \frac{v^2}{R^2 - u^2 - v^2}) - \frac{u^2v^2}{(R^2 - u^2 - v^2)^2}$

$\displaystyle = 1 + \frac{u^2}{R^2 - u^2 - v^2}+ \frac{v^2}{R^2 - u^2 - v^2}$

$\displaystyle EG - F^2= \frac{R^2}{R^2 - u^2 - v^2}$

we compute coefficients of the first fundamental form:

$\displaystyle N= -\frac{X_u \wedge X_v}{\|X_u \wedge X_v\|}$

$\displaystyle X_u \wedge X_v = (\frac{u}{R^2 - u^2 - v^2}, \frac{v}{R^2 - u^2 - v^2}, 1)$

$\displaystyle \|X_u \wedge X_v\| = \sqrt{\frac{R^2}{R^2 - u^2 - v^2}}$

$\displaystyle = \frac{R}{\sqrt{R^2 - u^2 - v^2}}$

$\displaystyle \Rightarrow N = -\frac{X_u \wedge X_v}{\|X_u \wedge X_v\|} = -(\frac{u}{R}, \frac{v}{R}, \frac{\sqrt{R^2 - u^2 - v^2}}{R})$

$\displaystyle X_{uu} = (0, 0, - \frac{R^2 - v^2}{\sqrt{(R^2 - u^2 - v^2)^3}})$

$\displaystyle X_{vv} = (0, 0, -\frac{R^2-u^2}{\sqrt{(R^2-u^2-v^2)^3}})$

$\displaystyle e = \langle N, X_{uu} \rangle = \frac{v^2 - R^2}{R}\cdot \frac{1}{R^2 - u^2 - v^2}$

$\displaystyle g = \langle N, X_{vv} \rangle = \frac{u^2 - R^2}{R}\cdot \frac{1}{R^2 - u^2 - v^2}$

$\displaystyle f = - \langle N_u, X_v \rangle = \frac{uv}{R(R^2 - u^2 - v^2)}$

where

$\displaystyle N_u = (-\frac{1}{R}, 0, \frac{u}{R\sqrt{R^2 - u^2 - v^2}})$

$\displaystyle X_v = (0, 1, -\frac{v}{\sqrt{R^2 - u^2 - v^2}})$

These imply that

$\displaystyle eg - f^2 = \frac{1}{R^2(R^2 - u^2 - v^2)^2}[(u^2 - R^2)(v^2 - R^2) - u^2v^2]$

$\displaystyle = \frac{1}{R^2(R^2 - u^2 - v^2)^2}[- (u^2 + v^2)R^2 + R^4] = \frac{R^2[R^2 - u^2 - v^2]}{R^2(R^2 - u^2 - v^2)^2}$

$\displaystyle = \frac{1}{R^2 - u^2 - v^2}.$

By the above computation, the curvature of sphere is

$\displaystyle K = \frac{eg - f^2}{EG - F^2} = \frac{1}{R^2}$.

# MATH-F-420: Differential geometry of Verbitsky

## MATH-F-420: Differential geometry

##### Monday 16:00-18:00, P.OF.2058

Announcement for this course.

Slides:

Handouts:

Miscellanea: test problems, exam, etc.

# On the diffetential of a mapping 2

In the case ${\psi : \mathbb{R}^n \rightarrow \mathbb{R}^m}$ case, there is a linear map, which is “linear approximation” of ${\psi}$. In the manifold case, there is a similar linear map, but now it acts between tangent spaces. If ${M}$ and ${N}$ are smooth manifolds and ${\psi \colon M \rightarrow N}$ is a smooth map then for each ${m \in M}$, the map

$\displaystyle d\psi \colon T_mM \rightarrow T_{\psi(m)}N$

is defined by

$\displaystyle d\psi(v)(f) = v(f \circ \psi)$

is called the pushforward. Actually,

$\displaystyle d\psi \colon TM \rightarrow TN.$

Suppose that ${\dim{M} \ge \dim{N}}$ and ${f \colon M \rightarrow N}$ is a differentiable mapping. We have

Definition 1 The mapping ${f}$ is called a trivial fibration (differentiable) on ${N}$ if there exists a differential manifold ${F}$, is called fibre of ${f}$, and a diffeomorphism

$\displaystyle \phi \colon M \rightarrow N \times F$

such that the following diagram is commutative

# On the differential of a mapping

From on Warner’s book.

Smooth curve on manifold ${M}$:

A ${C^\infty}$ mapping ${\alpha : (a, b) \rightarrow M}$. Let ${t \in (a, b)}$, we define the tangent vector of the curve ${\alpha}$ at ${t}$ is the vector

$\displaystyle d\alpha\bigg(\dfrac{d}{dt}\bigg|_{t = 0}\bigg) \in T_{\alpha(0)}M.$

we apply the formula

$\displaystyle d\psi(v)(g) = v(g \circ \psi),$

where ${g}$ is an any function on ${M}$.

Put ${\psi = \alpha(t)}$ and ${v = \dfrac{d}{dt}\alpha(t)\big|_{t = 0}}$, the above formula implies

$\displaystyle d\alpha(\frac{d}{dt}\big|_{t=0})(f) = (\frac{d}{dt}\big|_{t=0})(f \circ \alpha) = \frac{d}{dt}(f \circ \alpha)\big|_{t=0}.$

This is directional derivative.

# Một ví dụ về mặt chính quy

BT. Cho hàm $f(x,y,z)=z^{2}$. CMR $0$  không là giá trị chính qui của hàm $f$ thế nhưng $f^{-1}(0)$ vẫn là mặt chính qui.

Lời giải.

Ta có ma trận $(f_x \ f_y \ f_z) = (0 \ 0 \ 2z)$ suy biến tức $rankA < 1$ khi và chỉ khi $z=0$, do đó tại điểm $(x, y, 0), \forall x, y \in \mathbb{R}$, ma trận trên suy biến và do đó là điểm kì dị. Mà $f(x,y,0)=0$ nên $0$ là giá trị tới hạn, hay ko phải giá trị chính qui. Nhưng $f^{-1}(0)$ là mặt phẳng Oxy, đây là mặt trơn nên chính qui.

# The tangent plane of a surface and the tangent space of a manifold (at one point)

The tangent plane of a surface
By a tangent vector to $S$ at a point $p \in S$, we mean the tangent vector $\alpha'(0)$ of a differentiable parametrized curve $\alpha: (-\epsilon, \epsilon) \to S$ with $\alpha(0) = p$.

What is the tangent plane of a surface? That is a plane which containes all of the tangent vectors of this surface at point $p \in S$.

Proposition
Let $\varphi: U \subset \mathbb{R}^2 \to S$ be a parametrization of a regular surface $S$ and let $q \in U$. The vector subspace of dim $2$,

$d\varphi_q(\mathbb{R}^2) \subset \mathbb{R}^3,$

coincides with the set of tangent vectors to $S$ at $\varphi(q)$

There is a similar situation in here. In the case of manifold, the tangent space is also built from the set of all of tangent vectors of a manifold.

Tangent spaces of a manifold

In Milnor’s book (\cite{Milnor}), the tangent space $TM_x$ at $x$ for arbitrary smooth manifold $M \subset \mathbb{R}^k$ is defined:

Choose a parametrization $g : U \to M \subset \mathbb{R}^k$ of a neighborhood $g(U)$ of $x$ in $M$, with $g(u) = x$. We have $dg_u: \mathbb{R}^m \to \mathbb{R}^k$. So the image $dg_u(\mathbb{R}^m)$ of $dg_u$ is equal to $TM_x$.

References

J. W. Milnor, Topology from differentiable viewpoint, 1965.

M. do Carmo, Differential geometry, curves and surfaces, 1976.