2020年12月21日 · The way they did geometry from the ancient Greeks to Descartes. Coordinates and their maps are the foundation of standard differential geometry. The theory is coordinate free, but riddled with non geometric objects, and with the need to prove that geometrical objects are not just coordinate nonsense.

Differential geometry doesn't see rational points. Neither does algebraic geometry over $\mathbb C$. EDIT Considering the comments of the OP, I think I should be more explicit in my answer.

2009年10月20日 · The property you are interested in is known as being normal.For affine varieties, the definition of normal is just that the coordinate ring is integrally closed, and the operation on varieties that corresponds to taking the integral closure of the coordinate ring is known as normalization (a general variety is said to be normal if it is locally isomorphic to a …

2024年9月11日 · What is true is that there always exists one such coordinate system where the expansion holds. These are called "Bochner coordinates" or "K-coordinates". You can just search these terms on the internet to find references with the statement and proof.

2016年10月22日 · Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

I'm trying to motivate a bit of algebraic geometry in an abstract algebra course (while simultaneously trying to learn a bit of algebraic geometry), and I thought that it might be nice to present an example of the analogy between Riemann surfaces and algebraic integers (since one might say that the entire subject originates from this analogy ...

2017年11月16日 · $\begingroup$ @BigbearZzz, I have yet to find a rigorous and readable introduction to differential (and Riemannian) geometry. I advise against Chavel's cited book, and avoid Berger's "A panoramic view of Riemannian geometry" at all costs! If you want rigour, but are ready to sacrifice intuition, try the classic "Kobayashi & Nomizu". $\endgroup$

On an oriented Riemannian 2-manifold, every harmonic function u has a harmonic conjugate function v, defined by the relation dv = * du, where * denotes the Hodge star operator. Near a point where du is non-zero, the pair (u,v) then provides an isothermal coordinate system, and every isothermal coordinate system arises this way. $\endgroup$

Note also that you can push forward the standard coordinate fields in the plane by an arbitrary diffeomorphism and obtain random-looking vector fields that Lie-commute by construction. From the Riemannian viewpoint, it is odd to insist that one of them "does not change" along the other.

1.The Euler-Lagrange equations are coordinate independent, but if the Lagrangian is a bit more sophisticated it is difficult to come-up with an invariant form. One convenient way to do this is via the Legendre transform leading to Hamiltonian equations. However, this can be used only for certain classes of Lagrangians.