1. Space Geometry: This is the basis of advanced mathematical geometry, which mainly studies the properties and relationships of points, lines, surfaces and bodies in three-dimensional space. Including Euclidean geometry and non-Euclidean geometry (such as spherical geometry and hypergeometry).
2. Vector algebra: Vector is a mathematical tool to describe the position and direction of points, lines and surfaces in space. Vector algebra includes operations such as addition, subtraction, quantity multiplication, dot product and cross product of vectors, as well as concepts such as linear combination, linear correlation and linear independence of vectors.
3. Analytic geometry: Analytic geometry is a branch of mathematics that uses coordinate systems and equations to describe and study geometric figures. Including plane analytic geometry (such as equations of straight lines, circles and ellipses) and space analytic geometry (such as equations of curved surfaces and curves).
4. Differential geometry: Differential geometry is a branch of mathematics that studies the local and global properties of curves and surfaces near a point. It includes the concepts of tangent and curvature of curves, normal and Gaussian curvature of surfaces.
5. Topology: Topology is a branch of mathematics that studies the nature and structure of space, regardless of its size and shape. Including basic concepts such as connectivity, compactness and continuity of topological space, and important theorems such as homeomorphism, connectivity and compactness of topological space.
6. Projective geometry: Projective geometry is a branch of mathematics that studies graphs obtained by perspective projection transformation. Including the concepts of projective coordinate system, projective transformation and projective plane, as well as the basic theorems and problems of projective geometry.