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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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Cotangent bundle of a complex projective space

How does the cotangent bundle of a complex projective space looks like? Is that an Einstein manifold?
Novak Djokovic
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Understanding the definition of a differential operator on manifolds

In Christian Bar's "Geometric Wave Equations" notes it has this definition of a differential operator. I know what $\frac{\partial f}{\partial x^i}$ means when $f:M\rightarrow \mathbb{R}^n$ is a smooth function. But I don't understand what is meant…
Harmonic13
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Diffeomorphisms between factors in diffeomorphic product manifolds

Let $M$, $N$ and $P$ be three smooth manifolds such that $M \times N$ is diffeomorphic to $M \times P$. I need to know about some conditions under which one can deduce that $N$ is diffeomorphic to $P$. For example is it sufficient that $N$ and $P$…
G.J.
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Formal construction of Hodge star operator.

I'm studying differential geometry, and I'm looking for a formal construction of the Hodge star operator. For example, in the Baez and Muniain's book, the Hodge operator is defined as the unique linear operator…
Carlos Beltran
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Well-definedness of the pullback on covectors.

Basic definitions in question: Let $M,N$ be smooth manifolds, and consider a smooth map $\phi : M \rightarrow N$. The push-forward map is the map: $$\begin{align} \phi_* : & \ TM \rightarrow TN \\ & \ X \mapsto…
TheoreticalConfusion
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Curve with constant torsion and no curvature

When curvature and torsion are given a curve is fully defined (upto Euclidean motions) in 3-space. $ k=const , \tau = 0 $ represents a circle in a plane ; But what does the space curve $$ k =0 , \tau= const,$$ represent? The center line $ (u=0) $…
Narasimham
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Inner product on fiber of vector bundle over manifold.

Fix $n\in\mathbb{N}$. A vector bundle of rank $n$ is a smooth map $\pi:E\rightarrow B$ between manifolds such that $\forall p\in B: E_p := \pi^{-1}(p)$ is an $n$-dimensional vector space and $\forall p\in B$, there exists a neighborhood $U$ of $p$…
Jarne Renders
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How to show that this set isn't a regular surface?

I'm trying to solve this exercise from Do Carmo's Differential Geometry of Curves and Surfaces, and I want a hint on how to do it. The exercise is: Is the set $S =\left\{(x,y,z)\in \mathbb{R}^3 \mid z=0, \ \ x^2+y^2\leq1 \right\}$ a regular surface…
Gold
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Prove that if the chord length depends only on |s-t|, then it is a line or a part of a circle.

Let $\alpha : (a,b) \rightarrow \mathbb{R}^2$ be a smooth map(infinitely differentiable). Show that if the chord length $\Vert{\alpha(s)-\alpha(t)}\Vert$ depends only on $|s-t|$, then it is a line or a part of a circle. It comes from Shifrin's…
glimpser
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do Carmo: near isolated zeros, killing field tangent to geodesic spheres

Exercise 3.5b of do Carmo's Riemannian Geometry asks the reader to prove that given a Killing field $X$ on a manifold $M$, an isolated zero $p$ of $X$, and a normal neighborhood $U$ of $p$ in which $X$ has no other zeros, $X$ is tangent (in $U$) to…
Avi Steiner
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Help in proof from Riemannian Geometry by Docarmo.

I have been working on ${\it Lemma\,5.2}$ from Riemannian Geometry by DoCarmo which establishes the existence and uniqueness of the vector field $Zf=(XY-YX)f$, given $X$ and $Y$ as differenciable vector fields. On this proof we have expressions for…
DIEGO R.
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Taking trace of vector valued differential forms

Can anyone kindly give some reference on taking trace of vector valued differential forms? Like if $A$ and$B$ are two vector valued forms then I want to understand how/why this equation is true? $dTr(A\wedge B) = Tr(dA\wedge B) - Tr(A\wedge dB)$ One…
Student
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Isometric but differently shaped surfaces in $\mathbb{R}^3$

We have the following chain of inclusions for surfaces in $\mathbb{R}^3$ $M_1,M_2$:      $M_1,M_2$ have the same shape, i.e. are related by an ambient isometry ⇆ $M_1,M_2$'s first and second fundamental forms agree → $M_1,M_2$ are…
Hans-Peter Stricker
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Every point in S is umbilical $\rightarrow $ S is a plane or sphere.

Umbilic points on a connected smooth surface problem Here we have a proof that if every point in a surface $S\subset\mathbb{R}^3$ is umbilical then it is contained in a sphere or a plane. But this proof only works for open sets of S. In Manfredo's…
Marra
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Transition Function on the 2 - sphere

In this problem, we consider $S^2$ with the coordinate charts $(U, \phi)$, and $(U', \phi')$, where: $$U = \{(x,y,z) \in S^2 : z < 0\} \quad \phi(x,y,z) = (x,y), \ \mathrm{and}$$ $$U'\ = \{(x,y,z) \in S^2 : x > 0\} \quad \phi'(x,y,z) = (y,z)$$ My…
Solarflare0
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