Gaoshanyangzhi-A brief talk on the Shiing class from the angle of differential geometry

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Figure 1. Portrait of Mr. Shiing. (What seed/figure)

(in 1990, Mr. Shiing was invited to visit Tsinghua University and to make academic presentations for undergraduates in mathematics.) Old Gu for the first time to see the master, the mood is extremely excited. Mr. Chen came up to gently criticize Tsinghua University: "The huge Tsinghua, incredibly no one to talk about algebraic topology." Then, Mr. Chen painted a triangle on the blackboard with chalk and turned and asked, "What is the angle of the triangle and how much it equals." "180 degrees." "Everyone answered in unison. Mr. Chen then asked: "So outer corner and." "360 degrees." "We answer again." "Very good." Outer corner and is better than the inner angle, as it can be generalized to the curve case. "Mr. Chen drew a closed curve on the blackboard," cut the vector around the curve for a week, rotated 360 degrees. "Mr. Chen drew a curved patch on the blackboard," The starting point boundary tangent vector moves around the boundary one week in parallel and rotates 360 degrees. Thus, Mr. Chen began to explain Homology,exterior differential,de Rham cohomology,connection, curvature, characteristic class. At that time, Old Gu Mathematics Foundation is weak, English is ignorant, listen to like falling clouds, dumbfounded. After the report, Lao Gu devout to please Mr. Chen signed, get the signature after feeling extremely happy.

Now staring at the signature, old Gu sigh, past such as yesterday, vividly, the master eyebrow hidden beads, eye Ruolang star, kind and gentle, road bone fairy wind. 25 years suddenly and died, the world a few cool hot. The window of the wind bleak, leaves colorful, maple red like fire, Wan Blue sky. The people have gone, the Chen class endures. ）

Mr. Shiing, an internationally renowned master of differential geometry, combines differential geometry and algebraic topological methods to create integral differential geometries. He has completed two epoch-making important work: one is the Gauss-Boehner theorem of the Riemannian manifold and the other is the theory of the class of the vector bundle. In the history of science, Mr. Shiing and Gauss, Riemann, Jia when tied.

We discussed the topological barrier class theory of the fiber bundle class, the method used is the combination method "The savior of Moore's Law--from the combination point of view of Shiing", here we use the differential geometry method, mainly using the activity frame method to represent the contact and curvature, the use of the external differential tool in the " The origin of the Riemann geometry-a brief introduction to the Gauss wonderful theorem is described in detail.

Figure 2. The unit tangent bundle of the spherical surface.

Fiber Kongtonglun Classification We investigate the unit tangent plexus of the spherical surface, which is recorded as. The unit tangent bundle of the hemisphere is mediocre and the topology is solid tires, and we will stick two solid tires along the boundary and get it. The bonding mode is determined by the topology of the boundary tire surface and the embryo. A more rigorous, topological structure is determined by the homotopy classes of the sticky and mapped mappings. The bonding and mapping induce the homomorphism of the tire surface homology, and the function of the homomorphism on the homology group substrate is expressed as:

。

If we change the homotopy class of the sticky and mapped mappings, the resulting fiber bundle topology changes as well.

The surface is formed from the same embryo group, which is called the Surface Mapping class group (Mapping class group), recorded as. The multiplication in the mapping class group of the surface is defined as the compound of the self-identical embryo, and the unit element is the constant mapping. The mapping class group of tire surface and all the second order invertible integer matrix group isomorphism,

。

All thought that the bottom space, for the fiber bundle of fibers also constitute a group, recorded. This group of multiplication is more abstract, intuitively, is the local trivial bundle (for example, two solid tires) stick and the mapping of composite, in other words, will be "global distortion composite."

Not all of them correspond to the fiber bundle in the same embryo, and the map should maintain the fiber, i.e., while maintaining the orientation of the surface, thus having a form

，

All of these matrices constitute subgroups,. So, as discussed above, we have subgroup isomorphism of group and tyre surface mapping taxa.

By the shiing theory, the homotopy class of each fiber bundle corresponds to an indication class, which gives the isomorphism between the groups:.

By applying the combination method to calculate the bondholders of the bundle, the Shiing master proposes to use the curvature to express the class, which greatly reduces the difficulty of the calculation. The method of master Shiing can be summed up as follows: First define a contact on the fiber bundle, the curvature form on the bottom manifold is calculated by the contact, and some polynomial combinations of the curvature form give the expression class of the bundle. Here's a brief introduction to the Chen class method.

Intuitive interpretation

Figure 3. Left frame, geodesic on rabbit surface. On the right frame, the geodesic circle on the surface of the rabbit is equal to the distance between the points on the circumference and the center. (Xinsking/figure)

Geodesic imagine we are driving in the gentle terrain of the hills where the car's trajectory is a curve in three-dimensional space. The curvature of the curve has two parts, partly due to the ups and downs caused by the bending, the other part is the rotation of the steering wheel brought about by the bending, the first part is called the normal curvature, the second part is called the geodesic curvature. On a surface, a curve with a geodesic curvature of 0 is called a geodesic. In a sense, geodesic is the "most straight" curve on the surface, but also can be proved that in the local area, geodesic is the "shortest" curve on the surface. If we keep the steering wheel rotating at 0, the trajectory of the car is the geodesic.

Figure 4. The geodesic line on the face surface and the small change of the end point cause the sharp change of geodesic whole.

Parallel motion assumes that there is a geodesic line on the surface