Analytic surface

in a Euclidean space

An arbitrary two-dimensional analytic submanifold $X$ in the space $\mathbf R^n$, $n>2$. However, the term "analytic surface in $\mathbf R^n$" is often employed in a wider sense as a manifold which is (locally) analytically parametrizable. This means that the coordinates of the points $x=(x_1,\dots,x_n)\in X$ can be represented by analytic functions $x_i=x_i(t)$ of a real parameter $t=(t_1,\dots,t_k)$ which varies in a certain range $\Delta\subset\mathbf R^k$, $1\leq k<n$. If the rank of the Jacobi matrix $|\partial x/\partial t|$, which for an analytic manifold is maximal everywhere in $\Delta$, is equal to $k$, then the dimension of the analytic surface $X$ is $k$.

In the complex space $\mathbf C^n$ the term "analytic surface" is also employed to denote a complex-analytic surface $X$ in $\mathbf C^n$, i.e. a manifold which allows a holomorphic (complex-analytic) parametrization. This means that the complex coordinates of points $z=(z_1,\dots,z_n)\in X$ can be expressed by holomorphic functions $z_i=z_i(\tau)$ of a parameter $\tau=(\tau_1,\dots,\tau_k)$ which varies within a certain range $\Delta\subset\mathbf C^k$, $1\leq k<n$ (it is also usually assumed that $\rank\|\partial z/\partial\tau\|\equiv k$). If $\Delta=\mathbf C^k$ and all the functions $z_i(\tau)$ are linear, one obtains a complex-analytic plane (cf. Analytic plane). If $k=1$, the term which is sometimes employed is holomorphic curve (complex-analytic curve); if all functions $z_i(\tau)$ are linear, one speaks of a complex straight line in the parametric representation:

$$z_i=a_i\tau+b_i;\quad a_i,b_i\in\mathbf C,\quad i=1,\dots,n,\quad(a_1,\dots,a_n)\neq0.$$

References

[1]	B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1976) (In Russian)
[2]	V.S. Vladimirov, "Methods of the theory of functions of several complex variables" , M.I.T. (1966) pp. Chapt. 2 (Translated from Russian)