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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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the de Rham poincare duals of the unit circle in the punctured plane.

I'm reading from Bott-Tu's Differential forms in algebraic topology, and came across the following exercise: (p52 Exercise 5.16(b)) Let $S$ be the unit circle in the plane, and $M := \mathbb{R}^2-\{0\}$. Show that the closed Poincare dual of the…
user355183
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Other curvatures' π

The ratio $\frac{\text{perimeter}}{\text{diameter}}$ of circles on the surface of constant curvature $0$ is constantly $3.14\dots$ and is called $\pi$. Is this ratio a constant for every other surface of constant curvature $\kappa$? How then (by…
Hans-Peter Stricker
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Existence of a coordinate system

How can we formally show that a coordinate system $(x,y)$ exists or does not exist? For instance for some given coordinate system $(r,\phi,\theta)$ defined on the manifold $M =(1,\infty)\times\mathbb{S}^2$, does there exist a coordinate system…
Aurora Borealis
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How can a cone have non-zero Riemann curvature yet can be made out of a piece of paper?

A cone with a $90^\circ$ vertex angle can be parameterized by $$(x, y, z) = (z \cos \theta, z \sin \theta, z).$$ The metric on the cone can then be found to be $$dx^2 + dy^2 + dz^2 = 2 dz^2 + z^2 d \theta.$$ The only non-zero Christoffel symbols…
user1379857
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Can I solve the Frenet-Serret formulas with the only assumption that the cirvature-torsion of the curve are constant?

I am trying to find the general equation for space curves which have constant curvatures throughout their length. In general I am interested for curves of more than 3 dimensions. Assuming that all curvatures are constant for the entire length of…
George
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Calculation of the total curvature of Jordan curves

I am looking for direct proofs that the total curvature $\int_0^{L_\gamma} \! \gamma''(s) \, \mathrm{d} s$ of any Jordan curve $\gamma$ resp. $\int_0^{L_\gamma} \! |\gamma''(s)| \, \mathrm{d} s$ of a convex Jordan curve equals $2\pi$. Direct proof…
Hans-Peter Stricker
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why a geodesic is a regular curve?

In most definitions of the geodesic, it is required to be a regular curve,i.e. a smooth curve satisfying that the tangent vector along the curve is not 0 everywhere. I don't know why.
liufu
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Showing directly that the principal directions are going to be orthogonal

So first of all, I know that the Weingarten map (which from now on I shall denote by $L$) is a symmetric linear operator, so there is an orthonormal basis of eigenvalues (Spectral Theorem). I have been trying this concrete example for a while but I…
Daniel Montealegre
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About differential of volume and area and length

The volume $V$ of a sphere with radius $r$ is given by $$V=\frac43\pi r^3,$$ and its surface area is $$S=4\pi r^2.$$ So, $$\frac{dV}{dr}=S.$$ I thought this means very thin volume of outer skin of sphere is area of the sphere. I thought that it can…
ABC
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Sections of a bundle

I would like that someone explain to me why in general $ \Gamma ( T^*M \otimes TM ) = \Gamma ( T^*M ) \otimes_{\mathcal{C}^{\infty} ( M )} \Gamma ( TM ) $ with $ \Gamma ( T^*M ) $ is a set of sections of the bundle $ T^*M $ ? Thanks a lot.
Bryan
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Given a smooth map which is open, is it a submersion?

A submersion between smooth manifolds is an open map. Is the converse true? That is, is a smooth open map $f:M\to N$ between smooth manifolds a submersion? We can additionally assume that it is surjective, if necessary, because that is the only case…
theHigherGeometer
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Understanding curvature as rate of change of angle between neighbouring tangents

When introducing the concept of curvature in "Differential Geometry of Curves and Surfaces", do Carmo makes the statement the norm $|\alpha''(s)|$ of the second derivative measure the rate of change of the angle which neighbouring tangents make…
12qu
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Is the identity map a diffeomorphism?

Unfortunately, googling this question leads to conflicting answers. According to this source, the identity map on any smooth manifold is a diffeomorphism, but it's not according to this. I appreciate it if someone gave a definitive answer.
Smith
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Covering $\mathbb{R}^n$ by countably many lower dimensional pieces?

I would like to know if it is possible to cover $\mathbb{R}^n$ by countably many immersed submanifold of dimension less than $n$. A similar version is whether it is possible to cover $\mathbb{C}^n$ by countably many analytic subsets of lower…
user27126
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Geometric interpretation of the second covariant derivative

I'm having some doubts about the geometric representation of the second covariant derivative. I know that $\triangledown^{}_a(\triangledown^{}_bv)=(\triangledown^{}_a\triangledown^{}_b)v+\triangledown^{}_{\triangledown^{}_ab}v$ So the Riemann tensor…
Johann
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