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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 to prove a $k$-$1$ differential form is simple

I've been both trying to prove and looking for a proof in a couple of book and on the Internet, and I can't find it. How can I prove that a $k$-$1$ differential form defined on a $k$ dimensional manifold is simple? That is, it can be written as a…
MyUserIsThis
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is the stress tensor (elasticity theory) actually a pseudo tensor?

Please, is the stress tensor (elasticity theory) actually a pseudo tensor? It seems to me it must change its sign when coordinate system changes its orientation. The argument is as follows: We must integrate something over a small two dimensional…
pan potworny
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The diffeomorphism of convex set

For a convex open set $H$ in $\mathbb R^n$, is it diffeomorphic to $B = \{ x \in \mathbb R^n|\left\| x \right\| < 1\} $?
Summer
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Principal curvature for $X(u,v)=(u,v,uv(1-u))$

Let $M$ be a surface parametrized by $X(u,v)=(u,v,uv(1-u))$. Find the principal curvature $k(v)$ for all unit vectors $v\in T_{(u,0,0)}M$ (the tangent plane to $M$ at $(u,0,0)$). I find that the shape operator wrt basis $X_u$ and $X_v$…
JSCB
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Proof about covariant derivative

I'm asked to prove that if $v$ is a vector orthogonal to a family of hypersurfaces, then $v_{[a}\nabla_bv_{c]}=0$, being $\nabla$ a covarian derivative, and the $[,]$ meaning the three indices are antisymmetrised, and where $v_a=g_{ab}v^b$. I've…
MyUserIsThis
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Cross product calculation?

I can't seem to understand how this cross product is computed, maybe I am missing something obvious, so any help would be appreciated. We have $${\mathbf r}(s,\theta) =\gamma(s)+a({\mathbf n}(s)\cos\theta+{\mathbf b}(s)\sin\theta),$$ where…
p0ffer
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Terms of a $k$-form

In my differential geometry notes they use a fact that I don't know where it comes from. It says that any $k$-form can be written as: $$\omega=\sum_{i=1}^k f_idx^{\mu_1}\wedge \cdots\wedge dx^{\mu_k}$$ That is, as a sum of terms like that, where the…
MyUserIsThis
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The first 'easy exercise' in Spivak's differential geometry book

I am a 'lowly physicist' (actually, only a prospective physicist) with little to no background in formal mathematics. Recently I decided it's time to get serious, so I started reading Spivak's 'Comprehensive Introduction to Differential Geometry'.…
Danu
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Under what conditions the kernel and image of a linear bundle map are subbundles?

Under what conditions the kernel and image of a linear bundle map are subbundles?
Ramand
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Trace of $\nabla \alpha$ where $\alpha$ is a 1-form.

Let $X$ be a Riemannian manifold, $\nabla$ be a connexion and $\alpha$ be a 1-form. How do I show that $\text{Tr}(\nabla\alpha)=\sum e_{i}\alpha(e_{i})-\alpha(\nabla_{e_{i}}e_{i})$ (where $e_{i}$'s are orthonormal frame)? I have tried using the…
enoughsaid05
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Question about asymptotic direction

I'd like a hint for the following question: Show that in a elliptic point, principal directions bisects asymptotic directions. thanks.
Jr.
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Connections: Exponential Map

Given a smooth manifold. Suppose it has an (affine) connection. How is the exponential map constructed?
C-star-W-star
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(half-of a) hairy ball theorem

so I know that, by the hairy ball theorem, there does not exist a smooth global frame for the n-sphere. What about for the northern hemisphere? Can anyone comb half-of a sphere (provide a smooth global frame)?
Dave
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Are normal curvature and geodesic curvature independence of choice of curves?

By intuition, if the direction of tangent of a point $P$ is given, I think the curve passing through $P$ on the surface have only one choice (locally). So, does $T'(s)$ only depend on the given tangent direction but not the choice of curves? And if…
Y.H. Chan
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Some detail in interior product

This is a content in page 35 of foundation in differential geometry - KN For a form $r$-form $\omega $ interior product is $$ i_X\omega\doteq C(X\otimes \omega)$$ where $\{ e_i\}$ is ON, notation is $T=T_{i_1\cdots i_r}^{j_1\cdots j_s}…
HK Lee
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