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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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Congruence relations with Pontryagin classes

I'm reading the paper Rational Analogs in projective planes by Zhixu Su. I am trying to work out the example to calculate the form $e_2$ for dimension 16 in the paper. I am however not sure how to proceed from the step before the last one to the…
Alonso Perez-Lona
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$\mathbb{R}\to\mathbb{R}$ Diffeomorphism with special property

$\phi:\mathbb{R}\to\mathbb{R}$ be a diffeomorphism with the following property $a\in\mathbb{R},|a|<\frac{1}{10}$ (i)$\phi(a)=0$ (ii) $\phi(x)=x$ when $|x|>1$ and how to generalize that in $\mathbb{R}^n$ please help.
Myshkin
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understanding involute and evolute .

I have a bit confusion about the properties of centre of circular curvature . Is there any difference between locus of centres of circle of curvature and involute ?
Wajahat iqbal
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Definitions and theorems depending on being a regular surface

Consider $S \subset \mathbb{R}^3$ a regular surface. Now, since $S$ is a regular surface, a lot of theorems, propositions and definitions arise, such as Definition. $\quad$ The quadratic form $I_p$ on $T_p(S)$, defined by $I_p(S) = |w|^2$, is called…
Maurice
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Most general $2$-form in $\mathbb R^3$

I'm currently reading "The Geometry Of Physics An Introduction" by Theodore Frankel. On page 69 in the chapter introducing the "Grassmann Algebra" he writes down this expression of a "most general $2$-form in $\mathbb{R}^3$": $$\sum_{i
ctsmd
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Frank Warner's definition of tensor product

Frank Warner defines the tensor product between two finite dimensional vector spaces $V$ and $W$ as the quotient $F(V,W)/R(V,W)$ where $F(V,W)$ is the "free vector space over $\mathbb{R}$ whose generators are points of $V\times W$ and $R(V,W)$ is…
Rub
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Geometry of the Earth

Is this true that every simple closed curve on the earth can be deformed continuously to a point without leaving the earth? Is the earth compact? Now if we consider the earth as a 2-manifold, can we say that the earth is a sphere by the…
user212905
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Deriving curvature of plane curve

I am considering curvature of a plane curve as covered in chapter 2 of Elementary Differential Geometry by Pressley. For a curve $c(t)$, we are considering the calculation of curvature and it is said that as we go from $c(t)$ to $c(t+dt)$, the curve…
Salazar_3854708
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Let $M$ be a $n$-dim manifold. If $f : M \to \mathbb{R}$ then does $\ker(df_p) = T_p(f^{-1}(c))$ for some $c \in f[M]$

This is what was written in my differential geometry class notes. Let $M$ be a $n$-dimensional manifold. If $f : M \to \mathbb{R}$ then the subspace of $T_pM$ consisting of all the tangent vectors $X_p \in T_pM$ such that $\langle df, X \rangle =…
Perturbative
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Coordinate-free definition of arc length

I'm trying to write the definition of arc length on a Riemannian manifold using fancy differential geometry language, mostly as a way to become familiarized with this language. Of course, writing this definition using just calculus is trivial. Let…
isekaijin
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Level-set of constant rank

It is well-known the regular value theorem holds. Does this generalization also hold? I can not think of a counter-example. $f:M\to N$ be a smooth map of two smooth manifolds of arbitrary dimension $m,n$. For a level set…
CO2
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Show that $\lim\limits_{n\to\infty}\left\langle N,\frac{p_{n}-p}{|p_{n}-p|}\right\rangle =0$ on a regular surface

I am trying to do the following question, because I think it is very interesting as it formalises the idea that the tangent space is practically indistinguishable from the surface if one looks in a very small neighbourhood of a point. The question…
Hans
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Diffeomorphism between the Klein bottle and the connected sum of projective spaces

I know that the Klein bottle is a $C^{\infty}$ manifold (obtained as the quotient space of the action $(x,y) \longrightarrow (x+1/2,-y)$ on the torus). I have to find the diffeomorphim between the so obained Klein bottle and the connected sum of two…
dennis
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Differential forms, pullbacks and determinants

I've seen that similar questions have been asked on mathstack, but nevertheless I would appreciate to proof my theorem using local coordinates. Given a differential $n$-form $\omega \in \Omega^n(M)$ I want to show that $$f^\star \omega = \det f'(x)…
user582360
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Image of $\gamma(t)=(\sin(t+k)\cos(t), \sin(t+k)\sin(t))$ is a circle.

I would like to show the image of $\gamma(t)=(\sin(t+k)\cos(t), \sin(t+k)\sin(t))$ is a circle. I was hinted that I should appeal to the Isoperimetric Inequality. Hence I calculate the area that the simply closed positively oriented curve…
user573025
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