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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).

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Definition of Null Hypersurface

I am a physics student confused with the notion of null hypersurface, so sorry if this question is very simple. Given a manifold $M$ and a hypersurface $H$ defined on it, we can always take the hypersurface to be locally equal to a level set of a…
LeastSquare
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Find the tangent and normal lines to the curve $\gamma(t)=(2\cos(t)-\cos(2t), 2\sin(t)-\sin(2t))$ at $t=\frac{\pi}{4}$

The normal line to a curve in the plane at a point $\mathbf p$ is the straight line passing through $\mathbf p$ perpendicular to the tangent line at $\mathbf p$. Find the tangent and normal lines to the curve $\gamma(t)=(2\cos(t)-\cos(2t),…
Emily
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Orientation on $S^2$

I'd like to make sure I'm getting the proof of the following statement right: Let $S^2\subset \mathbb{R}^3$ be the unit sphere and define a vector field $N(x,y,z)=(x,y,z)$. Define a 2-form $\omega\in \Omega^2(\mathbb{S}^2)$ as…
user54631
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A representation of a 1 form

Let $x, y, z$ be the usual coordinates on $\mathbb{R}^{3}$. Consider the 1-form on $\mathbb{R}^{3}$ given by $\phi = dx+ydz$. Do there exist smooth functions $u$ and $v$ such that $\phi=u\ dv$? Why? Attempt at answer: Suppose there existed…
TheManWhoNeverSleeps
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Mean curvature of even order

I read Antonio Ros, Compact Hypersurfaces with Constant Higher Order Mean Curvatures,1987. I don't understand following sentence from the second page 6th line. From the Gauss equation, we have that if $r$ is even, $H_r$ is a intrinsic invariant of…
Donghwi Seo
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Codifferential of a $p$-vector in components

I'm learning differential geometry from a textbook, and I got stuck on a problem. I'm supposed to calculate this for a $p$-vector $F$ in $n$ dimensions: $(\mathrm{div}_\omega F)^{i...j} = F^{ki...j}\ _{,k}$ where $\mathrm{div}_\omega F =…
dom_miketa
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The number of variables that parametrize a particular curve or surface.

It is possible to parametrize a line in $\Bbb{R^n}$ using one variable. For example, $(t,2t)$ is a line in $\Bbb{R^2}$ for $t\in\Bbb{R}$. However, it is also possible to parametrize it using two variables. The same line in $\Bbb{R^2}$ can also be…
user67803
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Why is $\frac{d}{d \mu} \big|_{\mu=0} \, F^*_\mu F^*_\lambda t = F^*_\lambda \frac{d}{d \mu} \big|_{\mu=0} \, F^*_\mu t $?

In the book "Manifolds, Tensor Analysis, and Applications" by Marsden, Ratiu, Abraham the following relation (see the proof of 6.4.1, third edition) is used: $$\frac{d}{d \mu} \bigg|_{\mu=0} \, F^*_\mu F^*_\lambda t = F^*_\lambda \frac{d}{d \mu}…
Yrogirg
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Differential of a multi-variable map

This is something that I find is always a bit vague in differential geometry and would be very glad if someone could give me a definite rule. Here is a prototype example of what I want to compute. Let $A,B,C,D$ some smooth manifolds and $$ f:A\to…
Stan
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two problems to finding the regular value of matrix group

$1$. Let $F:M_2(\mathbb{R}) \to M_2(\mathbb{R})$ given by $F(X)=X^TX$. $2$.$F:M_2(\mathbb{R}) \to S_2(\mathbb{R})$ given by $F(X)=X^TX$ where $S_2(\mathbb{R})=$ {$X \in M_2(\mathbb{R}): X^T=X$}. Does $O$ and $I$ are regular values of 1 and 2? My…
user118413
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Functional of Einstein tensor

What does this equal to, and how do I actually calculate this correctly? $$ \frac{\delta G_{ab}}{\delta g_{cd}}=? $$
BinaryBurst
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Compute the (surface) curvature

Consider the manifold $P:=S^2 \times \mathbb{R}^2$ equipped with the product metric $g((x,y),(x',y'))=g_{S^2}(x,x')+g_{\mathbb{R}^2} (y,y')$ where $S^2$ has a constant curvature that is 1. Let $z_1,z_2,z_3,z_4$ be an orthonormal basis of the tangent…
unterbrause
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Gradient in spherical coordinates?

The gradient in spherical coordinates is given by $$\left(\partial_r f, \frac{1}{r} \partial_\theta f, \frac{1}{r \sin \phi}\partial_\phi f\right)$$ However, I get a wrong answer if I try to compute it a different way, by lowering the index of the…
user182973
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Local theory of curves using Taylor expansion

Consider a $C^m$ curve $\gamma$ in $\mathbb{R}^3$ (with $m\geq 3$). Locally, at $0$ for convenience, one can express the curve as $$ \gamma(s) = \gamma(0)+sT+(s^2/2!)kN+(s^3/3!)(\dot{k}N-k^2T+k\tau B)+O(>3) $$ ($k,\tau$ and the vectors $T,N,B$ are…
user14174
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Confusion about canonical basisvectors of the tangentspace of a manifold

Let $M$ be a $n$-dimensional manifold. We want to deduce a basis of $T_xM$ for $x\in M$. for $x\in M$ we can find an open neighbourhood $U$ such that $x\in U$ and an homeomorphism onto an open set $\varphi(U)$ in $\Bbb{R}^n$, thus…
Issus55
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