Arthur Cayley's focus on invariant theory led to the establishment of matrix theory, which introduced the algebraic expression of determinant in the modern sense and became an important tool of projective geometry. Gloria's invariants theory came into being in Britain in the first half of the19th century, mainly studying algebra and its application in geometry. Matrix theory introduces the algebraic definition of vector into the study of linear transformation, which is the predecessor of tensor concept.
On the other hand, George Friedrich Ferdinand Prinz von Preu?en Bernhard Riemann put forward the concept of N-dimensional manifold, and objectively put forward the subject of further studying algebraic forms. Riemann's geometric thought not only expands geometry, but also improves the abstraction of algebra in expressing geometric objects. After Riemann, with the efforts of Christopher, Richie and Levi-Chevetta, mathematical methods such as tensor analysis were formed and Riemann geometry was established.
2. Definition, properties and application value of tensor.
Algebraically, it is a generalization of vectors. As we know, a vector can be regarded as a one-dimensional "table" (that is, the components are arranged in a row in sequence) and a matrix is a two-dimensional "table" (that is, the components are arranged in vertical and horizontal positions), so the n-order tensor is the so-called n-dimensional "table". The strict definition of tensor is described by linear mapping. Similar to vectors, when several coordinate systems change, the set of ordered numbers satisfying a certain coordinate transformation relationship is defined as tensor.
Geometrically speaking, it is a real geometric quantity, that is, it is something that does not change with the coordinate transformation of the reference system. Vector also has this property.
Scalar can be regarded as a tensor of order 0, and vector can be regarded as a tensor of order/kloc-0. Tensor has many special forms, such as symmetric tensor and antisymmetric tensor.
Sometimes, people directly use several numbers in a coordinate system (called components) to represent tensors, and the components in different coordinate systems must meet certain transformation rules (see covariant law and inversion law), such as matrix and multivariate linear form. Some physical quantities, such as stress and strain of elastic bodies, energy and momentum of moving objects, need to be expressed by tensors. In the development of differential geometry, C.F. Gauss, B. Riemann, E.B. Christophel and others introduced the concept of tensor in the19th century, and then it was developed into tensor analysis by G. Rich and his student T. Levi Zivita, and A. Einstein used tensor extensively in his general theory of relativity.
Riemannian geometry, as a non-Euclidean geometry, is essentially different from Luo Barczewski geometry. The main work of Roche Geometry is to establish a logical system different from Euclid's Elements of Geometry. The core problem of Riemannian geometry is to establish differential method in curvilinear coordinate system on the basis of differential geometry. Roche geometry is the first non-Euclidean geometry, and its basic viewpoints are as follows: first, the fifth postulate cannot be proved; Second, a series of reasoning can be carried out in the new axiomatic system, and a series of logically non-contradictory new theorems can be obtained to form new theories. The axiom system of Roche geometry is different from Euclid geometry only in that the parallel axiom of Euclid geometry is changed to: starting from a point outside a straight line, at least two straight lines can be parallel to this straight line. The parallel axioms of Riemannian geometry and Roche geometry are opposite: if you cross a point outside a straight line, you can't make a straight line parallel to a known straight line. In other words, Riemannian geometry stipulates that any two straight lines in the same plane have common points, and Riemannian geometry does not recognize the existence of parallel lines. Naturally, there is another postulate: a straight line can extend to any length, but its length is limited, which can be compared to a sphere. Riemannian geometry is established by differential geometry, so it is essentially different from Roche geometry.
The axiomatic system of Riemannian geometry introduces a curvilinear geometric space (which can be described by the curvilinear coordinate system introduced by Lame). When Riemann conceived this geometry, he tried to establish the corresponding algebraic structure. Riemann himself did not achieve this goal, but along the road he pioneered, Christopher and Richie completed the construction of new geometry. In other words, tensor analysis constitutes the core of Riemannian geometry. This is manifested in several aspects: 1. Curvature in Riemannian space is tensor, and its related operation needs absolute differential method. 2. The measurement of Riemannian space is expressed by metric tensor; 3. The parallelism of Riemannian space is defined as constant scalar product (that is, the angle with the curve is constant), which depends on Christoph symbol; 4. The establishment of linear (geodesic) equations in Riemannian space depends on covariant differential. It is precisely because of tensor analysis that Riemannian geometry has obtained the calculation function similar to calculus, thus getting rid of the shackles of staying at the level of logical structure, fundamentally inheriting differential geometry, realizing the progress of differential geometry from linear coordinate system to curvilinear coordinate system, and making the relationship between geometry and algebra closer.
In a word, tensor analysis is the generalization of vector analysis on the one hand, and the development of differential geometry on the other. Tensor analysis and Riemannian geometry develop and promote each other in interweaving.