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Did you know how Riemannian manifolds redefine our understanding of geometry?
In many areas of mathematics, the definition of geometry continues to evolve over time. However, when it comes to Riemannian manifolds, the geometric meaning we understand is redefined, and the depth of understanding is increasingly improved. The Riemannian manifold is not only a mathematical abstract concept, but also the cornerstone of geometry, which affects the world we see, including applications in physics, computer graphics, and machine learning.
Basic concepts of Riemannian manifolds
Riemannian manifolds can be thought of as geometric spaces in which we can seamlessly define geometric concepts such as distance, angle, length, volume, and curvature. In fact, Euclidean spaces, n-spheres, hypergeometric spaces, and smooth surfaces such as ellipsoids and paraboloids can all be considered examples of Riemannian manifolds.
The term Riemannian manifold is named after the German mathematician Bernhard Riemann, who first conducted basic thinking on this concept.
Formally speaking, the Riemannian metric (or simply metric) is an inner product chosen for each tangent point on a smooth manifold. It helps us extract geometric data from Riemannian metrics, allowing us to perform integration and differential calculations to define curvature and parallel transport methods. Any smooth surface in three-dimensional Euclidean space can be thought of as a Riemannian manifold, and its metric comes from the way it appears in this space. This is true even for submanifolds of Euclidean space of any dimension.
Intrinsic viewpoint of Riemannian manifold
Although John Nash proved that every Riemannian manifold can be proposed as a submanifold of Euclidean space, the definition of Riemannian manifold emphasizes the intrinsic point of view, which directly defines geometric concepts on abstract space, while There is no need to refer to the environment space. In many cases, such as hypergeometric spaces and projective spaces, the definition and construction of Riemannian metrics more naturally uses this intrinsic perspective. Furthermore, many metrics on Lie groups and homogeneous spaces are defined in an intrinsic way, such that the inner product can be generalized from a single tangent space to the entire manifold via group action.
The study of Riemannian geometry has a profound connection with mathematical fields such as geometric topology, complex geometry and algebraic geometry.
The application range of this subject is extremely wide, from general relativity and gauge theory in physics, to computer graphics and machine learning, and even cartography. Extensions of Riemannian manifolds include pseudo-Riemannian manifolds, Finsler manifolds and sub-Riemannian manifolds.
The history of Riemannian manifolds
The ideological roots of Riemannian manifolds can be traced back to 1827, when Carl Friedrich Gauss discovered that the Gaussian curvature of a surface embedded in three-dimensional space depends only on local measurements within the surface. This result, known as the "saliency theorem", shows that Gaussian curvature is an intrinsic property of surfaces.
The concept of Riemannian manifolds and their curvature was first introduced by Bernhard Riemann in 1854. Although it was not rigorously defined at the time, subsequent developments formalized it.
With the advancement of mathematics, Eli Catan introduced the concept of Catan connection in 1936, while Levi-Civita defined a special connection on Riemannian manifolds. When Einstein developed his general theory of relativity, he used the theory of pseudo-Riemannian manifolds, which are geometric structures that describe four-dimensional space-time.
Riemannian metric and its dynamics
The Riemannian metric assigns a positive definite inner product to the tangent space at each tangent point. This metric allows us to calculate the length of curves within a manifold, as well as quantify various geometric properties on the manifold. Its importance lies in the fact that it is not just a tool, but a cornerstone for understanding the entire Riemannian manifold structure.
The smooth use of Riemannian metrics is central to the geometric analysis and organization of different types of manifolds.
By describing these manifolds and their metrics, we can not only analyze current geometric phenomena, but also properly predict and operate possible future situations. This connection makes the Riemannian flow an irreplaceable part of mathematics and natural science, and also triggers endless mathematical thinking.
Conclusion
Ultimately, the Riemannian manifold is not only a mathematical construct, but also a substantial sublimation of geometric thought. Through it, we can more accurately understand the working laws of the world and challenge traditional geometric concepts. With the deepening of research, what new revelations and discoveries will the Riemannian manifold bring us?
