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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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Affine connections v.s. Lie derivative

It is often advertised that affine connections are a way to differentiate a vector field along another (see e.g. here). For the little I know, Lie derivatives are advertised in the same way. On a more or less intuitive level, what distinguishes the…
cocompletehippopotamus
  • 51
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Prove that $c^2=a^2+b^2-2ab\cos\left(C-\frac{KS}{3}\right)$ holds on a smooth surface.

Problem: Given a infinitely small geodesic triangle $\triangle ABC$ on a smooth surface, denote the corresponding edges as $a,b,c.$ Prove: the area of $\triangle ABC$ (denote as $S$) and the Gauss curvature on $C$ ( denote as $K$)…
LonnerT
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Gaussian curvature is affected by a conformal map

I'm studying Tu's book Differential Geometry. Problem is Two Riemannian manifolds $M$ and $M′$ of dimension 2 with a diffeomorphism $T:M→M′$ between them. For every point $p \in M$, there is a positive number $a(p)$ such…
probafds123
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How to find a path tangent to a distribution

Let $D$ be the smooth distribution on $\mathbb{R}^3$ such that $$ D_{(a,b,c)}=\{\,(x,y,z)\in\mathbb{R}^3\,:\,z-bx=0\,\}. $$ How to show that for any $p,\,q\in\mathbb{R}^3$, there exists a path $\alpha$ from $p$ to $q$ tangent to $D$?
Edson Santos
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simple calculation with Christoffel symbols on Poincare half-plane

Equip $H=\{(x,y):y>0, x,y \in \mathbb{R}\}$ with the metric $$ds^2=\frac{dx^2+dy^2}{y^2}.$$ (https://en.wikipedia.org/wiki/Poincar%C3%A9_half-plane_model). I want to show that the sectional curvature $$ K(\partial_i,\partial_j) = \frac{\langle…
Stuck
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Is $TM \cong M \times \mathbb{R}^n$ as sets?

To be clear, I know these sets are not diffeomorphic or even homeomorphic in general. However, I've been told that there doesn't even exist a bijection between these sets. But suppose $M$ is an $n$-dimensional manifold and let $\{\partial_1|_p,…
Blonge
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Extensions of diffeomorphisms from $R^3$ to $S^3$.

Is there a convenient theorem about which diffeomorphisms $f: \mathbb R^3\rightarrow\mathbb R^3$ can be extended to diffeomorphisms $\overline{f}: S^3\rightarrow S^3$? That is, given a diffeomorphism $f:\mathbb R^3\rightarrow\mathbb R^3$, when does…
George K
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Show that $|α(t)|$ is a nonzero constant if and only if $α(t)$ is orthogonal to $α'(t)$ for all $t ∈ I$ .

Let $\alpha: I → \mathbb{R^3}$ be a parametrized curve, with $α'(t) \neq 0$ for all $t \in I$ . Show that $|α(t)|$ is a nonzero constant if and only if $α(t)$ is orthogonal to $α'(t)$ for all $t ∈ I$ . my attempt: For the second implication: Suppose…
Curious
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Is the set $\{x \in \mathbb R^n : d(x, M) = c\}$ a smooth manifold for a small constant $c$ when $M$ is a smooth manifold embedded in $\mathbb R^n$?

I am a beginner of differential geometry. Let $M$ be a smooth manifold embedded in $\mathbb R^n$ and consider the subset $$ S = \{x \in \mathbb R^n : d(x, M) = c \}, $$ where $d(x, M)$ denotes the distance between a point $x$ and the manifold…
Yoshimi Saito
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Two results on the mean curvature of hypersurfaces

I am a physicist, now I consider a physically meaningful $N-1$ dimensional hypersurface $M^{N-1}$ embedding in the flat Euclidean space $R^{N}$. We have an explicit form of the hypersurface in the following parametric form: $\mathbf{Y}(u)= (x_1(u),…
Q. H. LIU
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Intuition behind Willmore energy

I read about Willmore Energy is a quantitative measure of how much a given surface deviates from a round sphere. Also, I heard that it says that things in nature tends to change their shape in such a way that they use the least energy to survive.…
Nothing
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Compatibility conditions for Maurer-Cartan forms on a homogeneous space

I am reading Maurer-Cartan forms on a homogeneous space and am unable to show that $\theta_V=Ad(h_{UV}^{-1})\theta_U+(h_{UV})^*\omega_H$. Notation : We are considering $G \to G/H$ as a homogeneous H-space, where $G$ is a Lie group, and $H$ is a…
user90041
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Find an isometry from a plane to another in $R^3$

Question If P is the plane through $(1/2, -1, 0)$ orthogonal to $(0, 1, 0)$ find an isometry F = TC such that F(P) is the plane through $(1, -2, 1)$ orthogonal to $(1, 0, -1)$. This is an exercise in O'neill's Elementary Differential Geometry. I…
Oh JoonSuk
  • 45
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Why does the tensor product appear in the codomain of a Lie Algebra valued one-form?

I'm watching a lecture series on differential geometry and a Lie-algebra valued one-form $A$ on a principle $G-$bundle $(P, \pi, M)$ is written to belong to the space $$ A \in \Omega^1(M) \otimes T_e G $$ Where $\Omega^1(M)$ is the space of…
David Feng
  • 371
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Surface area of the unit ball in $R^{n}$

I want to compute the surface area of the unit sphere in $R^{n}$, the parametrization is as follows $$x_{i}=\prod_{j=1}^{i-1}\sin{\theta_{j}}\cos{\theta_{i}}$$ where $1\leq i \leq n-1$ and $$x_{n}=\prod_{i=1}^{n-1}\sin{\theta_{j}}$$ But I am not…
Brown
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