It is widely-known that $C^1$ manifolds are topological spaces locally homeomorphic to Euclidean spaces and possessing $C^1$ chart-converters. They have a tangent space at every point, regarding as the equivalent classes of $C^1$ paths starting from that point under tangency, that approximates smooth "neighbourhood behaviour" by linear ones. It is also well-known that a $C^1$ manifold admits a unique smoothing process.

However, the notion of "tangency" is intuitively not a concept measuring "smoothness". It just says that two curves do not intersect "transversely", and this can be precisely described by how fast the distance between a pair of points on them decreases as the points approach the intersection. So I came up with the following definition of a *quasi-smooth manifold*, and wonder **if this definiton produces just as many as smooth manifolds, or enormous exceptional structures**.

Let $(X,d)$ be a second-countable, locally compact and locally path-connected metric space. Suppose that $(X,d)$ has *enough rectifiable paths** (**explained at the end**). $\forall p \in X$, let $\Gamma_pX$ denote the space of all rectifiable paths $\gamma$ starting at $p=\gamma(0)$ and parametrized with *constant speed* (the constant path is denoted as $c_p$). Define $T_pX$, the **formal tangent space** at $p$ , to be the space of equivalent classes in $\Gamma_pX$, where $\gamma_1 \sim \gamma_2 \iff d(\gamma_1(t),\gamma_2(t)) \sim o(t) \; (t \rightarrow 0^+) $, endowed with a metric $\rho ([\gamma_1],[\gamma_2])=\overline{\lim}_{t \rightarrow 0^+} \dfrac{d(\gamma_1(t),\gamma_2(t))}{t}$. $\rho$ is well-defined since $$d(\gamma_1(t),\gamma_2(t)) \le d(\gamma_1(t),p)+d(\gamma_2(t),p) \le \mathrm{len}(\gamma_1)|^t_0+\mathrm{len}(\gamma_2)|^t_0 \le Ct$$ for some non-negative constant $C$ by definition. It can be shown that $T_pX$ admits a non-negative homogeneous structure: for $s \ge 0$, let $s[\gamma(t)]:=[\gamma(s \cdot t)] \subseteq T_pX$, then we have equalities similar to the ones in normed vector spaces like $s[c_p]=[c_p]$ and $\rho(s[\gamma_1],s[\gamma_2])=s \rho ([\gamma_1],[\gamma_2])$.

We can also give a global topology $\tau$ on $\bigsqcup_{p \in X}T_pX$. For any open set $U \subseteq X$ and continuous function $f:U \rightarrow \Bbb{R}^+$, let $V(U,f)=\{ (p,[\gamma]) 
\mid p \in U, [\gamma] \in T_pX, \rho([\gamma],[c_p])<f(p) \}$. $\tau$ is generated by such $V$. We call this space the **formal tangent bundle** of $(X,d)$ and denote it as $TX$.

While $T_pX$ has quite a nice structure, it is too large and does not coincide with the usual tangent space on smooth manifolds. So here comes the definition of $T^\delta_pX$, the **regular tangent space(s)** at $p$, being **a maximal non-negative homogeneous subspace** of $T_pX$ satisfying $\forall [\gamma_1],[\gamma_2] \in T^\delta_pX,d(\gamma_1(t),\gamma_2(t)) \sim Ct+o(t) \; (t \rightarrow 0^+)$ for some constant $C$. $T^\delta_pX$ may not be unique as well as rich in elements (in fact there are possibilities when $T^\delta_pX=[c_p]$) but it corresponds to the usual tangent spaces on smooth manifolds.

Finally, $(X,d)$ is said to be a *quasi-smooth manifold* if there exists pointwise homeomorphic $T^\delta_pX$ for every $p \in X$ such that $T^\delta_pX$ is also homeomorphic to an open neighbourhood of $p$. In fact, sometimes we can even do **better** -- If $\forall p,q \in X$ (not necessarily different), there exists a rectifiable path $\gamma : (a-\varepsilon,b+\varepsilon) \rightarrow X$ such that $\gamma(a)=p,\gamma(b)=q$ and for any $r=\gamma(t_0) \; (t_0 \in (a,b)), [\gamma(t_0 \pm t)] \in T^\delta_rX$, then we can conclude that $T^\delta_pX$ varies "quasi-smoothly" and we will say $(X,d)$ admits a **regular tangent bundle** (seen as a subspace of $TX$). Would $X$ be more probably an actual smooth manifold if it admits a regular tangent bundle?

A continuous map $f:(X,d_X) \rightarrow (Y,d_Y)$ is said to be *quasi-smooth* if $\forall \gamma_1,\gamma_2 \in \Gamma_{p}X \; (p \in X)$, $d_X(\gamma_1(t),\gamma_2(t)) \sim C_1t+o(t) \implies d_Y(f \circ \gamma_1(t),f \circ \gamma_2(t)) \sim C_2t+o(t) \; (t \rightarrow 0^+)$ for some constant $C_1,C_2$. It can be shown that quasi-smooth maps have "derivatives" that send regular tangent spaces to regular tangent spaces and thus are homomorphisms between quasi-smooth manifolds, which gives a potential categorical structure on them.

*:$(X,d)$ is said to have enough rectifiable paths if $\forall p,q \in X$ and a countable nowhere dense set $S \subseteq X$ such that $p,q$ belong to the same path-connected component of $X \backslash S$, there exists a rectifiable path $\gamma \subseteq X \backslash S$ connecting $p,q$. This property comes from that avoiding a countable subset of a smooth manifold is enough for prohibiting a smooth path going along a given direction at a given point.