Affine differential geometry
Affine differential geometry study Affine group is a larger transformation group than Euclid group, which can keep "straight line" and "parallel", but it has no concepts such as line segment length and orthogonality. In the late 1920s, Su devoted himself to studying Su, a branch of differential geometry.
It was popular all over the world at that time. One of his achievements is the introduction and determination of affine casting surfaces and affine rotating surfaces. He determined all affine casting surfaces and discussed their properties. Affine revolving surface is a special case of affine casting surface. Its characteristic is that the affine normal must intersect with a fixed straight line, so it is a very natural generalization of ordinary revolving surface. Another wonderful discovery of Su affine differential geometry is that he constructed a quartic (3rd order) affine invariant algebraic cone for general surfaces. Many remarkable covariant geometric objects in affine surface theory, including two principal tangents, three Dfarboux tangents, three Segre tangents and affine normals, can be reflected by this cone and its three vertex lines in a wonderful way, forming a very attractive composition. This cone is called Su cone. Sue's achievements in affine differential geometry made him a world-famous differential geometer in the early 1930s, and later he wrote the book Affine Differential Geometry (published in 198 1). Critics (American Mathematical Review) thought many contents were "absolutely outstanding" and said, "This beautiful and modern book.
Projection curve theory
Research on the Theory of Projective Curves The projective group is larger than the affine group, which can keep the concept of straight line, but the concept of "parallelism" no longer appears. In the 18 and 19 centuries, projective geometry has long attracted the attention of mathematicians. For example, Euclidean geometry and two other non-Euclidean geometries can be unified in the same theoretical system through subgroups. It is difficult to study its differential geometry because there is neither measurement nor parallelism. Even the curve theory, though studied by famous geometricians E. Bompiani and Xie Gu Chengyang for many years, is not satisfactory even in the three-dimensional case, let alone in the high-dimensional case. Su discovered some covariant properties of plane curves at their singularities, and clearly determined the corresponding projective frame (basic polyhedron changing with curves) of curves at normal points by using geometric structure, thus laying a sound foundation for the theory of projective curves and attracting great attention internationally. Scholars who engage in local differential geometry often throw away the singularity, and it is from the singularity that Su digs out the hidden features, which the professor appreciates very much. In this study, Su and his students also promoted the research on the singularity of algebraic curves. The related work was completed in 1930s and 1940s, and it was written as a monograph during the Anti-Japanese War, but it was never published. It was not until 1954 that it was published as his first monograph by the Chinese Academy of Sciences. Later, when the English version was published, the reviewer of Mathematical Review said, "Now projective geometry has been applied to various problems of mathematical physics and general relativity, and this book has become more important."
Projection surface theory
The study of projective surface theory is much more complicated than curve theory. In 1930s and 1940s, Su made a very in-depth and rich study on it. Only the following items are pointed out here: for the general point p on the surface, S. Lie got a covariant quadric surface, named Lie quadric surface. As the envelope of Lie quadric surface, there are four surfaces besides the original surface, so each point P has four corresponding points, which constitute the Demulin transformation of point P. At this time, the spatial quadrilateral is called Demulin quadrilateral. Starting from this quadrilateral, Su constructed a covariant quadric surface with important properties, which was later called Su quadric surface. He also studied a special kind of surface, called S-surface, which is characterized in that every point on it has the same quadric surface. This surface has many interesting properties. He completely decided them and classified them. Su also studied projective minimal surfaces, and his definition is equivalent to that introduced by G. Thomson by variational method. Sue got the "torsion theorem" of Gothic sequence of projective minimal surface, which showed beautiful geometric properties. Su also studied a kind of Laplacian sequence with period 4, which has the same diagonal confluence with another Laplacian sequence with period 4. He attributed the determination of this series to solving sine-Gordon equation or hyperbolic sine-Gordon equation, and pointed out many characteristics of this series. This kind of research is highly valued internationally, such as the Finikov School in the Soviet Union. Later, it was named Su Chain by G Bohr.
Monograph theory
Su's monograph Introduction to Projective Surfaces comprehensively summarizes his achievements in this field. Su's shackles are woven in high-dimensional space.
E. cartan, a great mathematician in this century, established the theory of external differential form. His and E. KahLer's research on the existence and freedom of solutions of general external differential form equations is one of the important achievements of modern mathematics. Gadang himself and later geometricians, such as the Finikov School of the Soviet Union, used this tool and made many important achievements in differential geometry. In 1950s, Su also used this tool to study the theory of * * * yoke net in high-dimensional projective space, and constructed many Laplacian sequences with beautiful geometric properties in high-dimensional projective space, and discussed their existence, degrees of freedom and related geometric properties respectively. His monograph Introduction to Projection Yoke Network (published in 1977) summarizes the achievements in this field. /kloc-In the 9th century, Riemannian geometry has appeared, which is a quadratic differential form based on the square of the distance between two infinite adjacent points in a defined space. Since the 20th century, stimulated by general relativity, Riemannian geometry has developed rapidly, resulting in more generalized FinSler spaces based on curve length integral, Cartesian spaces based on hypersurface area integral, path spaces based on second-order differential equations and K-spread spaces, which are collectively called generalized spaces. Since the late 1930s, Su has made many important contributions to the development of general spatial differential geometry. For Cartesian geometry, he focused on the extreme deviation theory, that is, the equation that can keep the geodesic infinitesimal deformation, which is a generalization of Jacobian equation and very important in Riemannian geometry. K-spread space is defined by fully integrable partial differential equation first proposed by J. DougLaS. Sue obtained the integrable conditions of projective forms, and studied affine isomorphism, projective isomorphism and their generalization. When discussing the geometric structure of this space, he summarized Jiadang's research on plane axioms. From 65438 to 0958, the monograph General Spatial Differential Geometry including the above results was published by Science Press. His achievements in general space geometry won the first natural science prize in China.
Hull lofting
In the early 1970s, due to the needs of shipbuilding and automobile industry and the increasing application of computers in industry, computational geometry was formed internationally. Out of concern for economic construction, Sue insisted on scientific research in adversity. Knowing the difficulty of setting out the hull with the old method, he resolutely devoted himself to this research closely related to industrial production, and introduced the affine invariant method in curve theory into computational geometry for the first time, which provided a reliable theoretical basis for some intuitive methods based on experience in the past, and made great progress in the research of cubic parametric curves and Bézier curves which were widely used. Some of these works have been successfully applied in China shipbuilding industry hull lofting, aviation industry turbine blade space modeling and related shape design, and thus won two national scientific and technological progress awards. The theoretical part of this work has been written in Computational Geometry (co-authored with Liu Dingyuan). The publication of the English version of this book has attracted international attention. In a word, Su has done a lot of outstanding research in the field of differential geometry, and has been in the international advanced ranks in various periods, which has provided valuable wealth for the future development of geometry. Because of his great achievements in mathematical research, he was elected as an academician of Academia Sinica and a member of the Standing Committee of the Academic Committee in Nanjing 1948. 1955 was elected as an academician of China Academy of Sciences (now called China Academy of Sciences). Besides doing research, he also did a lot of organization and communication work. 1935, one of the founders of chinese mathematical society, was elected as a director. He was appointed as the editor-in-chief of chinese mathematical society Annals, the earliest mathematical research publication in China. After the founding of People's Republic of China (PRC), he devoted himself to the resumption of chinese mathematical society, and served as the vice chairman of chinese mathematical society and the chairman of the Shanghai Mathematical Society. He also actively participated in the activities of Hangzhou Branch of Chinese Scientists Association and presided over the preparatory work of Zhejiang Association for Science and Technology. Later, he served as the chairman of Shanghai Science and Technology Association. He also presided over the preparatory work for the Institute of Mathematics of China Academy of Sciences, and served as the director of the preparatory office of the Institute of Mathematics until its formal establishment. At Fudan University, he not only founded the Institute of Mathematics, but also founded the national and high-quality magazine Mathematical Yearbook. This magazine enjoys an international reputation.