Path Space of a Smooth Manifold

Piecewise Smooth Path

Let $M$ be a smooth manifold and let $p$ and $q$ be two (not necessarily distinct) points of $M$. By a piecewise smooth path from $p$ to $q$ will be meant a map $\omega :[0,1]\to M$ such that

• there exists a subdivision $0=t_0<t_1<\cdots <t_k=1$ of $[0,1]$ so that each $\omega [t_{i-1},t_i]$ is differentiable of class $C^{\infty }$;
• $\omega (0)=p$ and $\omega (1)=q$. The set of all piecewise smooth paths from $p$ to $q$ in $M$ will be denoted by $\Omega (M;p,q)$, briefly by $\Omega (M)$, $\Omega$.

Tangent Space

The tangent space of $\Omega$ is defined as the vector space consisting of all piecewise smooth vector fields $W$ along $\omega$ for which $W(0)=0$ and $W(1)=0$. We denote it as $T{\Omega }_{\omega }$.

Variation of Piecewise Smooth Path

A variation of $\omega$ (keeping endpoints fixed) is a function $\tilde{\alpha }:(-ϵ,ϵ)\to \Omega ,$ for some $ϵ>0$, such that:

1. $\tilde{\alpha }(0)=\omega$
2. there is a subdivision $0=t_0<t_1<\cdots <t_k=1$ of $[0,1]$ so that the map $\alpha :(-ϵ,ϵ)\times [0,1]\to M$ defined by $\alpha (u,t)=\tilde{\alpha }(u)(t)$ is $C^{\infty }$ on each strip $(-ϵ,ϵ)\times [t_{i-1},t_i]$, $i=1,\dots ,k$.
3. Since each $\tilde{\alpha }(0)\in \Omega$, $\alpha (u,0)=p,\alpha (u,1)=q$ for all $u\in (-ϵ,ϵ)$.