The question may seem dumb at first glance. But I couldn't figure out a satisfying answer after some research. A friend of mine told me that in an interview, she was asked to explain the sliding mode control, which is a control scheme for nonlinear system. Briefly explaining, in sliding mode control we have a $\sigma(x)$ which is a scalar function of the vector $x(t)$, and $x$ represents the system states. (I think you do not need to be totally familiar with these concepts and a short glimpse might be enough to answer the question.) Then someone asked her why we call the $\sigma(x)$ a surface? Why don't we call it a sliding curve?
So this question led me to the basic question of, what is the general definition of a curve and a surface and what is the difference between them? Wikipedia says:
A plane algebraic curve is the locus of the points of coordinates $x,y$ such that $f(x,y)=0$, where $f$ is a polynomial in two variables defined over some field $F$. Algebraic geometry normally looks not only on points with coordinates in $F$ but on all the points with coordinates in an algebraically closed field $K$.
and here it says:
A complex projective algebraic curve resides in n-dimensional complex projective space $CP^n$. This has complex dimension n, but topological dimension, as a real manifold, 2n, and is compact, connected, and orientable. An algebraic curve over $C$ likewise has topological dimension two; in other words, it is a surface.
I am not an expert in math. From what I have learned previously, a curve refers to a one-dimensional object and surface is something two-dimensional (Not precise I know, intuitively speaking...) But these definitions left me confused.
I general n-dimensional space, or in topology, what is called a curve and what is a surface? And referring to the original question, what is wrong with calling the $\sigma(x)$ a sliding curve?
After perusing your Wikipedia link, "I don't know for sure", but here's the explanation that seems most likely to me (a geometer who knows next to nothing about control theory).
The state of a system under sliding mode control is modeled as a point in some phase space, a mathematical object encoding both physical configuration (position) and infinitesimal motion (velocity).
In any particular situation, a system's state traces a curve in the phase space.
The phase space itself (i.e, the set of possible states), constitutes a larger dimensional "hypersurface", which for brevity has come to be called a surface.
On the Wikipedia page, it appears the terms hypersurface and manifold are used interchangeably to speak of the locus of multiple constraints. If that's right, the meanings of those terms differs from common usage in differential geometry: In mathematics, a hypersurface is given by one constraint ("has codimension one"), and a manifold is smooth ("has a tangent space at each point").
It's certainly true that the same technical terms (particularly, curve and surface) have different definitions depending whether you ask a differential geometer or a control theorist.
Separately, a complex curve (a geometric object described locally by one complex parameter) is indeed a (special type of) real surface (described locally by two real parameters), but this appears to be a coincidence in your context.
One final take-away message: Although mathematical theorems have an absoluteness about them once notation, terminology, and logical axioms are reconciled, notation and terminology (and even logical axioms) are by no means universal.
In fact, the notational idioms in mathematics, the sciences, and engineering differ considerably. That's a fact of life, the Babel of quantitative endeavors.