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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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Triviality of a tautological bundle

I am trying to solve the following exercise. What I know: $\tau$ is a vector bundle of dimension $n$ over $\mathbb{R}P^n$. The same is true for the trivial bundle $\mathbb{R}P^n \times \mathbb{R}^n$. Then we find surjective smooth maps $$ \pi: \tau…
Polymorph
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Confusion with what the metric gives when mapping a surface to the complex plane

Tristan Needham Visual Differential Geometry,pg-32 In the beginning, I understood the metric as the factors by which the length of displacement on surface and on the plane relate. But, in the book the following formula is given and suggests to me…
tryst with freedom
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How to efficiently compute this Pffafian of curvature form

Though one can verify this $$Pf(\Omega) = \frac{1}{8}(\vert \text{Rm} \vert^2 - 4 \vert \text{Ric}\vert^2 + R^2)$$ for a 4-dim manifold by listing out all the curvature components mechanically (I have to ask my computer to do so), is there a clever…
Math The Novice
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I'm having a problem with parallel transport which suggests the dimensionality is incorrect

I'm working in spherical coordinates and I want to transport a vector for a radial velocity over an interval with $dr\ne 0$ but all other increments zero. The formula I have found for parallel transport is $$v_{r + dr}^\mu \approx v_r^\mu - \Gamma…
Awkward
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explicit relation between interior derivative and alternation operation

Let $X$ be a vector field, $L_X$ the Lie derivative, $i_X$ the interior derivative, $A$ the alternation operation (that assign for each covariant tensor field $K$, a differential form $AK$). The definitions I am following are: $$(AK)(X_1,\cdots,…
Praphulla Koushik
  • 925
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Tangent space of the normal bundle

Let $M$ be a manifold endowed with a connection $\nabla$. Let $M_0$ be a submanifold of $M$, we denote by $N$ the normal bundle of $M_0$. The following paragraph is from the book: Heat kernels and Dirac operators (page 217) By orthogonal…
Mira
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Can a parallel transported frame be obtained from the eigenvectors of a curvature tensor?

Let $(M,g,\nabla)$ be a Riemannian manifold of dimension $n$ and let $\gamma:\mathbb{R}\to M$ be a geodesic, namely, $\nabla_{\dot{\gamma}}\dot{\gamma}=0$. It is well know that, given a basis $\{e_{j}(p)\}_{j=1}^{n}$ on the tangent space $T_{p}M$,…
RTS
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Do Carmo's discussion of how $|b'(s)|$ measures how rapidly a curve pulls away from the osculating plane at $s$

I am reading Do Carmo's Differential Geometry of Curves and Surfaces, and a passage on page $18$ is confusing me. He writes: In what fallows we shall restrict ourselves to curves $\alpha(s)$ parametrized by arc length without singular points of…
IntegrateThis
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Geodesic deviation in flat space

Suppose that $x^\mu(t,s)$ represents a family of curves. Let $v^\mu$ represents the the tangent vector to a curve $x^\mu(t,s_0)$ with $s_0$ fixed that is $v^{\mu}=\partial x^{\mu} / \partial t$ and deviation vector is given by…
amilton moreira
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Inner product of p-forms

This is a really basic question, but I cannot find a explicit answer on any book. Given a Riemannian manifold $(M,g)$, the metric induces an inner product on vectors at any point, but it also induces an inner product on $p$-forms for $1\leq p\leq…
topolosaurus
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$\frac{d}{ds} \langle T, T \rangle = 2\langle \nabla_S T, T \rangle$?

Our setting is $(M, g)$, a Riemannian manifold. Let $\Gamma(s,t) \subset M$ be a variation about curve $\gamma(t) = \Gamma(0, t)$ (Let us say that our domain of $\Gamma$ is $(a_0, a_1) \times (b_0, b_1) \subset \mathbb R^2$, and $(a_0, a_1)$…
James C
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Uniqueness of the vector in $\mathbb{R}^n$ specified by the curl, divergence and the normal component

If I know the curl, and divergence of a n-component vector in a region, and its normal component around its boundary, is the vector uniquely specified? If yes, how do I prove it? Also, is there a straightforward way to solve these equations, and get…
winawer
  • 556
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Pullback, 2-form invariant

Let $\mathbb H^2 = \{(x, y) \in \mathbb{R}^2 : y> 0\}$ and consider the $2$-form in $\mathbb H^2$ defined by $$\varphi = \dfrac{dx \wedge dy}{y^2}.$$ Show that $\varphi$ is invariant $(T^*(\varphi)= \varphi)$ under the transformation $T$ from…
Karly
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Why is the integral $C^\infty$

I am reading Differentiable manifolds from Warner. In order to prove that the dimension of the tangent space is the same as the dimension of the manifold, they use the following calculus lemma - If $g$ is of class $C^k$ ($k \geq 2$) on a convex open…
user52991
  • 708
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Ovaloid is orientable

Given a compact, connected regular surface $S$ in Euclidean space which has everywhere positive gaussian curvature = an ovaloid. It is a theorem of Hadamard that ovaloids are diffeomorphic to the sphere (discussed in Klingenberg), but the proof of…
Artin Ostal
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