In 1930s and 1940s, the projective geometry school of Zhejiang University, headed by Su, was recognized as a tripartite school of Italian school and American school at that time, and Bai Zhengguo was one of the representatives of this school. At that time, there was a problem in projective differential geometry that attracted the attention of the international mathematics community: Is there a surface whose asymptotic curves are projectively equivalent to each other? The cause of the problem comes from a theorem of W blaschke, a famous German mathematician: If the asymptotic curves of a family of ruled surfaces belong to a linear bundle, then this family is projectively equivalent. The famous Italian mathematician G.Fubini studied the inverse problem of blaschke's theorem, that is, if a family of asymptotic curves is projectively equivalent, does it necessarily belong to a linear bundle? Fubini solved the problem when the surface is ruled by itself, and got a positive answer. So Fubini asked: Apart from a family of surfaces whose asymptotic curves belong to linear bundles, are there any non-ruled surfaces whose asymptotic curves are equivalent to each other in projection? This difficult problem is called the Fubini problem. Bai Zhengguo finally solved this problem satisfactorily after painstaking research. The answer is also affirmative, that is, except for a family of asymptotic curves belonging to linear bundles, there is only a special class of overlapping projection minimal surfaces, and the family of asymptotic curves is projectively equivalent to each other [3]. For this achievement of Bai Zhengguo, G.Fubini wrote a letter of commendation, asking the magazine to publish Bai Zhengguo's paper in advance. Later, this achievement was included in the Biography of Fubini written by Tracini. Professor Su also introduced this achievement of his favorite pupil in detail in his monograph Introduction to Projective Surfaces. In addition, Bai Zhengguo has done a lot of original work in the surface theory of projective differential geometry. For example, about Moutard quadric, Godeaux quadric sequence and so on. And the projective theory of surfaces in regular space is systematically studied, and nearly 10 papers are completed and published in the relevant magazines of the American Mathematical Society in the 1940s.
Since 1950s, Bai Zhengguo began to study differential geometry in general space. 1957, he published the article "On the Total Curvature of Space Curved Polygon" [12], which extended the famous W.Fenchel theorem. The results are as follows: Let C be a space curved polygon with an internal angle of θ 1…θn, then its total curvature satisfies the following inequality:
The equal sign holds if and only if the planar curved polygons are connected by convex curved arcs. In the global differential geometry of space curves, this is a very concise inequality with distinct geometric significance. It was included in the book "Ten Years of Mathematics in China" and also mentioned in the mathematics volume of "Encyclopedia of China".
During the period of 1962- 1966, the editorial department of Journal of Mathematics was established in Hangzhou University, and Bai Zhengguo was in charge. 1965 in the formulation of the national 12-year science plan, the theory of geometry and function of the mathematics department of Hangzhou University has become one of the key implementation units of the project.
In Riemannian geometry, Bai Zhengguo perfectly solved the scale form problem of Riemannian space with several independent circle-preserving transformations proposed by the famous Japanese geometer K Kentaro [20], which is a key basic problem in circle-preserving geometry. 1980, the first "double differential" (differential geometry and differential equations) conference initiated by the famous mathematician Professor Chen Shengshen was held in Beijing. M.Berger, a famous French geometer who attended the meeting, asked Bai Zhengguo for a pamphlet about this achievement. In addition, Bai Zhengguo also studied the correlation between Codazzi-Ricci equation and Gauss equation of submanifolds in Riemannian space [2 1], the characteristics of curvature tensor of * * * flat Riemannian space and constant curvature space [23], and the * * * flat hypersurface in * * flat Riemannian space [24], and published papers in major domestic mathematical magazines successively. After the downfall of the Gang of Four, Bai Zhengguo's research direction shifted from the local properties of Riemannian manifolds to the global properties. He made a systematic study of quasi-constant curvature manifolds and got many wonderful results. For example, he proved that a Riemannian manifold can be equidistantly embedded into two manifolds with different constant curvatures, and its inverse is also true [25]. This is an interesting theorem that was not known before. Later, the famous Brazilian geometer M. do Carmo independently obtained similar results. In addition, Bai Zhengguo's global submanifold geometry also gives many good theorems [26][27].