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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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How can I find the components of a metric tensor?

Suppose $\vec{\phi}(x^1,x^2) = (x^1,x^2,(x^1)^2+(x^2)^2.$ The metric tensor induced by $\vec{\phi}$ is given by: $$g = \begin{pmatrix}1+4(x^2)^2 & 4x^1x^2 \\\ 4x^1x^2 & 1+4(x^2)^2\end{pmatrix}$$ Let $x^1 = \bar{x}^1\cos\bar{x}^2$ $x^2 =…
user8290579
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coordinate system, nonzero vector field

I'm interested in the following result (chapter 5, theorem 7 in volume 1 of Spivak's Differential Geometry): Let $X$ be a smooth vector field on an $n$-dimensional manifold M with $X(p)\neq0$ for some point $p\in M$. Then there exists a coordinate…
Kelly
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Prove that a surface of revolution is regular surface.

Considering the following definitions extracted of do Carmo's book Differential Geometry of Curves and Surfaces, prove that a surface of revolution is regular surface. Def. 1: A parametrized differentiable curve is a differentiable ($C^\infty$) map…
Carlos H. Santos
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Converse of the fundamental theorem of Riemannian geometry?

The fundamental theorem of Riemannian geometry says that for a manifold with a given metric, there is a unique torsion-free connection. Suppose instead that we are given a connection. According to answers to this question, a metric exists that…
user13618
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Hessian does not depend of the curve we choice.

I am trying to understand why the hessian of a function from a surface in $\mathbb{R}$, $\left. \frac{d^2}{dt^2} \right\rvert_{0} (f \circ \alpha)(t)$ does not depend of $\alpha$. In my book you can see: ($\alpha(t)=X(u(t),v(t)), X$ is a…
davidivadful
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Hessian Matrix And Gauss Curvature Example

The connection between is written in wikipedia: "We represent the surface by the implicit function theorem as the graph of a function, $f$, of two variables, in such a way that the point $p$ is a critical point, i.e., the gradient of $f$ vanishes…
gbox
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Duality of Hodge star.

Can someone please provide a rigorous proof that: $$\star(\star A) = (-1)^{p(n-p)}sA$$ where $\star : \Lambda^{n}\to \Lambda^{n-p }$ and $s$ is the the sign of the determinant of the metric. I am using the following definition of Hodge…
MathematicalPhysicist
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Invariant vector field by group action

in a solved exercise, there is a point in the solution that I can't work out. I would be grateful if somebody could give me the detailed steps. Consider the trivial principal bundle $P = M\times U(1)$ over a $C^\infty$-manifold $M$. Let $\Phi_t$ be…
vkubicki
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Prove curves are the geodesics

Prove that the ellipsoid $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1$ always has at least three geodesics. I think these three geodesics should be cross sections of ${(x,y,0)}$, ${(x,0,z)}$ and ${(0,y,z)}$ with our ellipsoid. And…
Jack
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There is no isometry between a sphere and a plane.

How can I show that there is no isometry between a sphere and a plane? Wikipedia defines an isometry as follows: Let $(M,g)$ and $(M',g')$ be two Riemannian manifolds, and let $f:M\to M'$ be a diffeomorphism. Then $f$ is called an isometry if…
wjmolina
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Verification that $S^{n}$ is a differentiable manifold.

Setting $S^{n} := \{x\in\mathbb{R}^{n+1}: \|x\| = 1\}$, and labelling the north and south poles as $N:= (0,\ldots,0,1)$, $S:=(0,\ldots,0,-1)$, I can set the coordinate charts up as follows: Let $U_N = S^n - N$ and $U_S = S^n - S$. Taking the usual…
roo
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Boundedness of the Christoffel symbols of a connection on the normal bundle

I have the following setting: Let (M,g) be a Riemannian manifold and $\iota: M \to R^N$ some isometric embedding. This means especially that the connection $\nabla^{TM}$ of M is given by the ordinary Euklidean connection $\nabla$ on R^N composed…
AlexE
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Tangent to a fiber bundle

I am trying to prove that the kernel of a push-forward is the fiber. Let $π : E → M $ be a fiber be bundle with a fiber $F$ . What is the meaning of a tangent space to a bundle? Does it means that if we have a vector, $X$ tangent to curve $\lambda$,…
amilton moreira
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Space curve of Trefoil knot made of ideal flexible nonstretchable steel wire.

What space curve defines a Trefoil knot made of ideal flexible nonstretchable steel wire? By ideal flexible nonstretchable steel wire I mean wire that meets this requirements: a wire cannot shorten or extend but can be bend without limit a wire…
azerbajdzan
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Why continuous paths implies smooth path on the manifold?

On the page 32 of Lee's book Manifolds and differential geometry, he writes: In the definition of path connectedness..., we used continuous paths, but it is not hard to show that if two points on a smooth manifold can be connected by a…
hxhxhx88
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