Mathematics Stack Exchange
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Questions tagged [differential-geometry]

Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses on "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such structures, use (differential-topology) instead. Use (symplectic-geometry), (riemannian-geometry), (complex-geometry), or (lie-groups) when more appropriate.

Differential geometry is a branch of mathematics that uses techniques in calculus and linear algebra to study geometry. It is closely related to differential topology and PDEs (geometric analysis).

32835 questions
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Energy-momentum vector is orthogonal to itself

Let the energy-momentum covector $k$ be $k_idx^i$ in Einstein summation notation where $x^0=t$. Let ${}^3k=k_1dx^1+k_2dx^2+k_3dx^3$ be the space part of $k$. Let $Ee^{ik_{\mu}x^{\mu}}$ be the electric field of a plane wave, where $E=E_jdx^j$. The…
Tara
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An exercise of the book "Hamilton's Ricci Flow" by Bennett Chow

This is remark 1.24 on p. 13 of the book Hamilton's Ricci Flow by Bennett Chow, but how to prove this conclusion? If $\varphi (t): M^n \to M^n$ is the $1$-parameter family of diffeomorphism and $\alpha$ is a tensor, then $$…
deng ya
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why not the Ricci tensor is the contraction of first and second indices of Riemann tensor

Why the Ricci tensor is defined the contraction of first and third indices of Riemann tensor? I guess it is more natural to define it as to contract the first and the second indices? Since from Wiki : $$R_{\sigma \mu \nu}^\rho =…
ahala
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Integrating a 0-form

The Stokes theorem states: $$\int_\mathcal M d\omega =\int_{\partial \mathcal M} \omega $$ If we have that $\mathcal M$ is a one dimensional manifold with two extreme points, like a closed interval of $\mathbb R$, and $d\omega$ is a one-form, how…
MyUserIsThis
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Why is $d*F$ equal to $\partial _\mu F^{\mu \nu}$?

Given that $A = A_\nu dx^\nu$ and $F = \partial_{\mu}A_\nu dx^\mu \wedge dx^\nu$ Why does $d*F$ equal to $\partial _\mu F^{\mu \nu}$? How does all the $\frac{1}{2}\varepsilon^{abcd}F_{cd}$ fit into this picture, and how do you compute this stuff…
bobby
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Differential of a smooth function

I have seen that if $f$ is a smooth function on a smooth manifold $M$ then differential of $f$ at the point $p$ is defined by $(df)_p(X_p) = X_p(f)$ but i am not able to see that how it can be derived by the usual definition of differential map of a…
user103005
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Show that two intersecting curves on a regular surface with the same osculating plane that is not the tangent plane have the same curvature

Let $\bf{p}$ be a point on a regular surface $S$. Let $\boldsymbol{\alpha}(s)$ and $\boldsymbol{\beta}(s)$ be two curves parametrized by arc length on the surface $S$ such that $\boldsymbol{\alpha}(0) = \boldsymbol{\beta}(0) = \bf{p}$. Denote the…
Repainted
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How to show the symplectic group is a submanifold of $GL(n,\mathbb{H})$?

I am trying to show that the symplectic group $Sp(n) =\{A\in GL(n,\mathbb{H})\mid \overline{A}^TA=I\}$ is a regular submanifold of $GL(n,\mathbb{H})$ but I am stuck. Any help would be appreciated.
Galois
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Lie bracket and covariant derivatives

I came across the following equality $[\text{grad} f, X] = \nabla_{\text{grad} f} X + \nabla_X \text{grad} f$ Is this true, and how can I prove this (without coordinates)?
user09123
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Contraction between basis vectors and basis one-forms

Discretion: The title may be misleading, because I am not certain whether the one-forms are actually basis one-forms. I always thought by definition, $dx^i (e_j) =\delta^i_j $. But, I am confused because of what it says on one of the book I am…
Quantization
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Geodesics and Curves on a Plane

Show that if a curve $C ⊂ S$ is both a line of curvature and a geodesic, then $C$ is a plane curve. Give an example of a line of curvature which is a plane curve and not a geodesic. (My thoughts: Take a sphere for $S$ and let the curve $C$ be any of…
kevin
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Physical predictions using the language of manifolds?

I accept the definition of a derivative as properly motivated because it helps me make physical predictions in classical mechanics. I'm looking for something similar with manifolds. From what little symplectic geometry I have studied, it mostly…
user908123
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Local Isometry of Sphere

How does one show that there exists no neighborhood of a point on a sphere that may be isometrically mapped into a plane? I understand that I can find the first fundamental form of the sphere $(u, v, \sqrt{r^2 - u^2 - v^2})$, for fixed $a>0$, which…
kevin
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Arround Cauchy Schwarz Inequality in semi riemannian geometry

I have a question. I seen that Cauchy Schwarz inequality is not valide in the case of pseudo riemannian metric because it is not positive ( or negative) define, I would like to know if there is special cases where this inequality holds, for exemple…
zaar1992
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Understanding an immersion in $\mathbb{R}P^{2}$

Regarding the post: embedding of $\mathbb{RP}^2$ in $\mathbb{R}^4$ I want to understand why $F$ is an immersion. Since $\mathbb{R}P^{2}$ is the quotient of $\mathbb{S}^{2}$ by identifying the antipodal points and we have it suffices to show that the…
user17182
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