Construction of Functions on Manifolds in Euclidean Space

Let $M\subset {\mathbb{R}}^n$ be a manifold of dimension $k<n$, differentiably embedded in ${\mathbb{R}}^n$.

Endpoint Map

Let $NM\subset M\times {\mathbb{R}}^n$ defined by $NM=\{(q,v):q\in M,v\text{perpendicular to}M\text{at}p\}$ i.e. $NM$ is the normal vector bundle of $M$, so equivalently $NM=\{(q,v):q\in M,v\in (T_{p}M{)}^{\perp }\}.$ Let $E:NM\to {\mathbb{R}}^n$ be $E(q,v)=q+v$.

Focal Point

$e\in {\mathbb{R}}^n$ is a focal point of $(M,q)$ with multiplicity $\mu$ if $e=q+v$ where $(q,v)\in N$ and the Jacobian of $E$ at $(q,v)$ has nullity $\mu >0$( that is, $\dim (\ker d{E}_{(q,v)})=\mu$). The point $e$ will be called a focal point of $M$ if $e$ is a focal point of $(M,q)$ for some $q\in M$.

Intuitively a focal point of $M$ is a point in ${\mathbb{R}}^n$ where nearby normals intersect.

Sard

If $M_1$ and $M_2$ are differentiable manifolds having a countable basis, of the same dimension, and $f:M_1\to M_2$ is of class $C^1$, then the image of the set of critical points has measure $0$ in $M_2$.

Corollary

For almost all $x\in {\mathbb{R}}^n$, the point $x$ is not a focal point of $M$.

Let $u^1,\dots ,u^k$ be coordinates for a region of the manifold $M\subset {\mathbb{R}}^n$. Then the inclusion map $x:M\to {\mathbb{R}}^n$ determines $n$ smooth functions $x_1(u^1,\dots ,u^k),\dots ,x_n(u^1,\dots ,u^k)$, denoted as $\vec{x}=(x_1,\dots ,x_n)$.

First Fundamental Form

The first fundamental form associated with the coordinate system is defined to be the symmetric matrix of real valued functions $(g_{ij})=(\frac{\partial \vec{x}}{\partial u_i}\cdot \frac{\partial \vec{x}}{\partial u_j})$ or equivalently $I={\sum }_{i,j}{g}_{ij}d{u}^i\otimes d{u}^j$

Second Fundamental Form

The vector $\frac{{\partial }^2\vec{x}}{\partial u_i\partial u_j}$ at a point of $M$ can be expressed as the sum of a vector tangent to $M$ and a vector normal to $M$. Define $\overrightarrow{l_{ij}}$ to be the normal component of $\frac{{\partial }^2\vec{x}}{\partial u_i\partial u_j}$. Given any unit vector $\vec{v}$ which is normal to $M$ at $\vec{q}$ the matrix $\left(\vec{v}\cdot \frac{{\partial }^2\vec{x}}{\partial u_i\partial u_j}\right)=(\vec{v}\cdot \overrightarrow{l_{ij}})$ can be called the second fundamental form of $M$ at $\vec{q}$ in direction $\vec{v}$.

Second Fundamental Form

Let $(M,g)$ be a Riemannian manifold isometrically immersed (or embedded) into an ambient Riemannian manifold $(\bar{M},\bar{g})$. Let $\bar{\nabla }$ denote the Levi-Civita connection on the ambient manifold $\bar{M}$, and let $\nabla$ denote the induced Levi-Civita connection on the submanifold $M$. For any smooth tangent vector fields $X,Y\in \Gamma (TM)$, the ambient covariant derivative ${\bar{\nabla }}_{X}Y$ can be uniquely decomposed into a tangential component and a normal component with respect to $M$: ${\bar{\nabla }}_{X}Y=({\bar{\nabla }}_{X}Y{)}^{\top }+({\bar{\nabla }}_{X}Y{)}^{\perp }$ By the definition of the induced connection, the tangential part is exactly ${\nabla }_{X}Y$. The normal part is defined as the second fundamental form, denoted by $II$. This relationship is known as the Gauss formula: ${\bar{\nabla }}_{X}Y={\nabla }_{X}Y+II(X,Y)$ Equivalently, the second fundamental form is given by: $II(X,Y)=({\bar{\nabla }}_{X}Y{)}^{\perp }$ The second fundamental form $II$ is a symmetric $C^{\infty }(M)$-bilinear map $II:\Gamma (TM)\times \Gamma (TM)\to \Gamma (NM)$ that takes two tangent vector fields and returns a normal vector field in the normal bundle $NM$. Given a unit normal vector field $\nu \in \Gamma (NM)$, the scalar second fundamental form associated with $\nu$ is defined by taking the inner product: $I{I}_{\nu }(X,Y)=\bar{g}(II(X,Y),\nu )=\bar{g}({\bar{\nabla }}_{X}Y,\nu )$

Lemma

The eigenvalues of the matrix $(\vec{v}\cdot {\vec{l}}_{ij})$ are called the principal curvature $K_1,\dots ,K_n$ of $M$ at $\vec{q}$ in the normal direction $\vec{v}$. Consider the normal line $\ell$ consisting of all $\vec{q}+t\vec{v}$, where $\vec{v}$ is a fixed unit vector orthogonal to $M$ at $\vec{q}$. The focal points of $(M,\vec{q})$ along $\ell$ are precisely the points $\vec{q}+K_i^{-1}\vec{v}$, where $1\le i\le k$, $K_i\ne 0$. Thus there are at most, $k$ focal points of $(M,\vec{q})$ along $\ell$, each being counted with its proper multiplicity.

Corollary

$\vec{q}+t\vec{v}$ is a focal point of $(M,\vec{q})$ with multiplicity $\mu$ if and only if the matrix $(g_{ij}-t\vec{v}\cdot \overrightarrow{l_{ij}})$ is singular, with nullity $\mu$.

Lemma

For $p\in {\mathbb{R}}^n$, the function $L_p$ is defined as following: $L_p:M\to \mathbb{R}$ defined as $L_p(q)=\Vert p-q{\Vert }^2,$ The point $\vec{q}\in M$ is a degenerate critical point of $L_p$ if and only if $\vec{q}$ is a focal point of $(M,\vec{q})$. The nullity of $\vec{q}$ as critical point is equal to the multiplicity of $\vec{p}$ as focal point.

Proof Fix $\vec{p}\in {\mathbb{R}}^n$, let $f=L_{\vec{p}}:M\to \mathbb{R}$, $f(\vec{x}(u^1,\dots ,u^k))=\vec{x}\cdot \vec{x}-2\vec{x}\cdot \vec{p}+\vec{p}\cdot \vec{p}$, we have $\frac{\partial f}{\partial u^i}=2\frac{\partial \vec{x}}{\partial u^i}\cdot (\vec{x}-\vec{p}),$ so $f$ has a critical point at $\vec{q}$ if and only if $\vec{q}-\vec{p}$ is normal to $M$ at $\vec{q}$. The second partial derivative at a critical point are given by $\frac{{\partial }^{2}f}{\partial u^i\partial u^j}=2\frac{\partial \vec{x}}{\partial u^i}\cdot \frac{\partial \vec{x}}{\partial u^j}+2\frac{\partial \vec{x}}{\partial u^i\partial u^j}\cdot (\vec{x}-\vec{p}).$ Setting $\vec{p}=\vec{x}+t\vec{v}$, we get $\frac{{\partial }^{2}f}{\partial u^i\partial u^j}=2(g_{ij}-t\vec{v}\cdot \overrightarrow{{\ell }_{ij}}).$

Theorem

For almost all $p\in {\mathbb{R}}^n$ (all but a set of measure 0), the function $L_p:M\to \mathbb{R}$ defined as $L_p(q)=\Vert p-q{\Vert }^2$ has no degenerate critical points.

Corollary

On any manifold $M$ there exists a differentiable function, with no degenerate critical points, for which each $M^a$ is compact.

Theorem

A differentiable manifold has the homotopy type of a CW-complex.

Theorem

On a compact manifold $M$ there is a vector field $X$ such that the sum of the indices of the critical points of $X$ equals $\chi (M)$, the Euler characteristic of $M$.

Proof This can be seen as follows: for any differentiable function $f$ on $M$ we have $\chi (M)=\sum (-1{)}^{\lambda }{C}_{\lambda }$ where $C_{\lambda }$ is the number of critical points with index $\lambda$. But $(-1{)}^{\lambda }$ is the index of the vector field grad $f$ at a point where $f$ has index $\lambda$.

Corollary

Any bounded smooth function $f:M\to \mathbb{R}$ can be uniformly approximated by a smooth function $g$ which has no degenerated critical points. Furthermore $g$ can be chosen so that the $i$-th derivative of $g$ on the compact set $K$ uniformly approximate the corresponding derivatives of $f$, for $i\le k$.

Index theorem for $L_p$

The index of $L_p$ at a non-degenerate critical point $q\in M$ is equal to the number of focal points of $(M,q)$ which lie on the segment from $q$ to $p$; each focal point being counted with its multiplicity.