Chinese name
many-sided
Foreign name
many-sided
It's Euclid
Curves in space
It's locally owned
Euclidean space attribute space
quick
navigate by water/air
definition
circumference
Important manifold
Development history
The concept of n-dimensional manifold has begun to take shape in J.L. Lagrangian mechanics. /kloc-in the middle of the 9th century, it was known that the N-dimensional Euclidean space was a continuum of N real variables, but the concept of general N-dimensional manifold was introduced by B.Riemann when he studied differential geometry, and he constructed it by induction. Just as the motion of a curve forms a surface, an N-dimensional manifold is formed by putting an infinite number of (n- 1)-dimensional manifolds together in a one-dimensional manifold. The research on the topological structure of manifold and its local theory began at the same time. Riemann, Betty, Poincare and others applied analytical methods. However, in order to get rid of the difficulties and disadvantages of this method, Poincare defined the n-dimensional manifold as a connected topological space, in which each point has a neighborhood which is homeomorphic to the n-dimensional Euclidean space, and studied it, thus opening up the road of combinatorial topology.
definition
In n-dimensional Euclidean space, the defined half space is expressed as. When every point p has an open neighborhood U(p) with or without homeomorphism, Hausdorff space m is called an n-dimensional topological manifold. All points of U(p)≈ (homeomorphism) p? M is called the edge of manifold M, and its complement is called the interior of M? A manifold with m = φ is called an edgeless manifold.
The edge of n-dimensional manifold m? M is an n- 1 dimensional borderless manifold. Compact borderless connected manifolds are called closed manifolds, and noncompact borderless connected manifolds are called open manifolds. There is a connected but not paracompact topological manifold. This one-dimensional manifold is called a long line. [ 1]
circumference
Circle is the simplest manifold except Euclidean space. Let's consider a circle (unit circle) with a radius of 1 and a center at the origin in a two-dimensional plane. If x and y are Euclidean coordinates on the plane, then the equation of the unit circle is.
Local coordinate card
A short segment near any point of the unit circle is like a line. Line is a one-dimensional figure, and we can mark a point on this short segment with only one coordinate. For example, any point of the unit circle on the semicircle above the X axis can be determined by the X coordinate. So there is bijective Xtop, which simply projects to the first coordinate (x) and maps the yellow part of the circle to the open interval (? 1, 1):。
Such a function is called a local coordinate graph. Similarly, there are corresponding coordinate cards on the lower semicircle, left semicircle and right semicircle of the unit circle. These four semicircles can cover the whole unit circle. We call the corresponding four local coordinate cards to form the coordinate atlas of this unit circle.
coordinate transformation
Note the overlap of the upper right coordinate card. Their intersection points are located on a quarter arc with positive X and Y coordinates on the circle. Two graphs χtop and χright project this part into the interval (0, 1). In this way, we have a function t from (0, 1) to itself. First, we take the inverse of the yellow graph to reach the circle, and then return to the interval through the green graph.
Such a function is called transformation mapping (coordinate transformation).
From the point of view of calculus, the transformation function t of a circle is only a function between open intervals, so we know that it means that t is differentiable. In fact, t is differentiable at (0, 1), and so are other transformation functions. Therefore, this map turns the circle into a differentiable manifold.