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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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Fixed point set defined by an isometry is a geodesic

The question is asking to me prove that: Consider a fixed point set $F=\{x \in S : f(x)=x\}$ in a smooth Riemannian surface with $f:S \rightarrow S$ be an isometry. If $F$ is a smooth 1-manifold, then we can define a smooth curve (parametrized by…
SamC
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Principal directions

Consider a catenoid $C$ parametrized by $$r(u,v)= (u, \cosh u \cos v, \cosh u \sin v), u\in \mathbb{R}, v\in(-\pi, \pi)$$ I am required to show that the principal directions are the same as the coordiante curves, u= constant, v=constant. The…
Chulumba
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Expressing Basis Vectors in Polar Coordinates

Consider the polar coordinate transformation $$ x = r\cos \theta $$ $$ y = r\sin \theta $$ I am trying to find the most direct way to compute the coordinate basis vectors $$ \frac{\partial}{\partial x}, \frac{\partial}{\partial y} $$ The…
ItsNotObvious
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Orientation on a Manifold

Let M be an (n-1)-manifold in R^n . Let M(e) be the set of end-points of normal vectors (in both directions) of length e and suppose e is small enough so that M(e) is also an (n-1)-manifold. Show that M(e) is orientable (even if M is not)
JimJones
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Hopf fibration is a submersion

Hopf Fibration is $F:S^3\to S^2$ given by formula $F\left(z_1,z_2\right)=\left(\left(\phi^+\right)^{-1}\left(\frac{z_1}{z_2}\right)\right)$ for $z_2 \ne0$ and $F\left(z_1,0\right)=\left(1,0,0\right)$ where $\phi^+$ is stereographic projection from…
Adrian Panasiuk
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A question on generalized Gauss-Bonnet theorem

I am trying to learn about basic characteristic classes and Generalized Gauss-Bonnet Theorem, and my main reference at the moment is From Calculus to Cohomology by Madsen & Tornehave. I know the statement of the theorem is as follows: Let $M$ be an…
Qiao
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Symmetric $k$-tensor

I've searched on Google but I could not find an example of a symmetric tensor. I've found this blog post but I cannot construct any example of a symmetric tensor. I know that a tensor $T$ is symmetric iff $T= \operatorname{Sym} T$. Could you give…
user20353
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Orthogonal transformation and vector product

I found these thing in an exercise 1.5.6 in the book Differential Geometry of curves and surfaces - Do Carmo. "Show that the vector product of 2 vectors is invariant under orthogonal transformation with positive determinant. Is the assertion still…
genov4
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The differential $\text d F_p$ is injective iff the pullback $F_p^*$ is surjective.

I'm trying to prove the following claim: Let $F\colon M \to N$ be a differentiable application beetween $C^\infty$ manifolds. Then the differential $\text dF_p\colon T_p M \to T_{F(p)}N$ is injective if and only if the pullback $F_p^*\colon…
pppqqq
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Parameter Curves are Geodesics

So let's suppose we have a surface $M$ that is embedded in $\mathbb{R}^3$ with an orthogonal parametrization. Further, assume that the parameter curves (i.e., $X(u$0$, v)$ and $X(u, v$0$)$ ) are geodesics that are unparametrized (i.e., not…
P. Sinclair
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Submersions and induced homomorphism on fundamental groups

Suppose that $M$ and $B$ are two smooth manifolds and $\Pi:M\rightarrow B$ a submersion (and onto). Fixed $x\in M$ and $b\in B$, is the induced homomorphism $\Pi_{\#}:\pi_{1}(M,x)\rightarrow \pi_{1}(B,b)$ also onto? I think so if we assume that the…
benji
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Using a helical vector field to foliate $\mathbb{R}^3$

Having finished an introductory course to GR as part of my physics undergrad degree, I decided to look at some differential geometry over the summer, and picked up Schutz's Geometrical methods of mathematical physics. On page 82 (in case you have…
dom_miketa
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Relation between the integral of geodesic curvature and Gaussian Curvature

I need help with an exam question: Let $S$ be a regular oriented surface such that for any simple, closed, and positively oriented curve in $S$ the value of the integral of the geodesic curvature along this curve is always the same( that is,…
cryptow
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Question about line of curvature

If $\alpha$ is a planar geodesic on surface $M$, show that $\alpha$ is a line of curvature. My try: $\alpha$ planar imply torsion=0, and binomial vector is constant. Since $0=\kappa_g=\kappa_\alpha B\cdot U$, ($U$ is normal vector of $M$), so B…
JSCB
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Orientation double cover

Let $M$ be a manifold and let $\bigwedge^\text{top}TM$ be the top exterior product of the tangent bundle. Then this becomes a line bundle. Let $g$ be any metric on $\bigwedge^\text{top}TM$ and define…
Rempe5555
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