The Gauss map of a generic immersion of a smooth, oriented surface into \({\mathbb {R}}^4\) is an immersion. But this map takes values on the Grassmanian of oriented 2-planes in \({\mathbb {R}}^4\). Since this manifold has a structure of a product of two spheres, the Gauss map has two components that take values on the sphere. We study the singularities of the components of the Gauss map and relate them to the geometric properties of the generic immersion. Moreover, we prove that the singularities are generically stable, and we connect them to the contact type of the surface and \({\mathcal {J}}\)-holomorphic curves with respect to an orthogonal complex structure \({\mathcal {J}}\) on \({\mathbb {R}}^4\). Finally, we get some formulas of Gauss–Bonnet type involving the geometry of the singularities of the components with the geometry and topology of the surface.

光滑定向表面的一般浸入\({\mathbb {R}}^4\)的高斯图是浸入。但该映射采用\({\mathbb {R}}^4\)中定向 2 平面的 Grassmanian 上的值。由于该流形具有两个球体乘积的结构，因此高斯图具有在球体上取值的两个分量。我们研究高斯图各分量的奇点，并将它们与通用沉浸的几何特性联系起来。此外，我们证明奇点是一般稳定的，并将它们与表面的接触类型和关于正交复结构的全纯曲线连接起来。J}}\)上\({\mathbb {R}}^4\)。最后，我们得到了一些涉及部件奇点几何形状与表面几何形状和拓扑结构的Gauss-Bonnet型公式。