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.
30,707
questions
1
vote
1
answer
36
views
Showing the differential of a map $f:S\to\Bbb R^m$ is linear?
I've been asked an exercise where I have to prove that the differential, at a point, of a map defined on surface, is a linear map. I have this definition and then this lemma where they prove this in ...
- 23.2k
1
vote
1
answer
56
views
Is the spherical coordinate system a vector space?
There are basis in spherical coordinate system but I think it does not satisfy vector axioms.
Is calling them basis an abuse of notation?
- 11
0
votes
0
answers
18
views
principal symbol of a differential operator composed with its adjoint
Consider a linear differential operator with injective (principal) symbol from the space of sections of a (real) vector bundle to that of another. Suppose that the adjoint of this differential ...
- 145
2
votes
1
answer
40
views
Intuitive understanding of inward and outward vector definitions
Let $M$ be a smooth manifold with boundary and $p \in \partial M$. In Lee's Introduction to Smooth Manifolds he gives the following definition:
If $p \in \partial M$, a vector $v \in T_pM \setminus ...
- 2,745
2
votes
1
answer
28
views
Isometric invariance of riemannian distance function
I need proof verification on the following theorem (Lee intro to Riemannian Manifolds Prop 2.51):
Let $\varphi : (M,g) \longrightarrow (\tilde{M},\tilde{g})$ be an isometry between connected ...
- 211
0
votes
0
answers
26
views
Parallel transport on hyperboloid
I have the formula for parallel transport along a geodesic in the hyperboloid model of hyperbolic space but how is it derived?
$$
P_{\mathbf{x} \rightarrow \mathbf{y}}(\mathbf{v})=\mathbf{v}-\frac{\...
- 11
0
votes
0
answers
19
views
mean curvature flow for non-mean-convex surfaces
There are many results for mean curvature flow starting from mean-convex surfaces: Huisken and coauthor even proved long-time behavior in this case https://ems.press/journals/jems/articles/15560
I was ...
- 93
1
vote
2
answers
49
views
Compactly supported harmonic forms
If $(\mathcal{M},g)$ is a compact Riemannian manifold (without boundary), then it is well-known that a $k$-form $\alpha$ is harmonic, i.e. $\Delta\alpha=0$, where $\Delta$ is the Laplace-Beltrami ...
- 1,982
1
vote
0
answers
31
views
Differentiable way of detecting characteristic classes of odd (real) dimension vector bundles?
I learned that Chern classes work for complex vector bundles (corresponding to a real bundle of even dimensions). Equivalently, Pontryagin classes work for even dimensions.
Is there a characteristic ...
- 231
0
votes
1
answer
50
views
Geodesics in hyperboloid model of hyperbolic space
What is the proof of
$$\gamma_{\mathbf{x} \rightarrow \mathbf{u}}(t) = \cosh(t)\mathbf{x} + \sinh(t)\mathbf{u}$$
being the formula for a geodesic on the hyperboloid model of hyperbolic space? I haven'...
- 11
0
votes
0
answers
30
views
prove $\frac{1}{2\pi i}dz/z$ is the generator of $H^1(\Bbb{P}^1\setminus \{0,\infty\},\Bbb{Z})$
I was trying to prove that the singular cohomology $H^1(\Bbb{P}^1\setminus \{0,\infty\},\Bbb{Z})$ has a deRham representative $dz/z$. That is $\frac{1}{2\pi i}dz/z \in H^1(\Bbb{P}^1\setminus \{0,\...
- 4,284
2
votes
0
answers
49
views
The definition of smooth sections given in Introduction to smooth manifolds by John M. Lee
In p. 176 of John M. Lee's Introduction to smooth manifolds, 2nd edition, Lee defines vector fields along subsets $A \subseteq M$ of a smooth manifold with boundary $M$.
The part that intrigues me is ...
- 3,383
0
votes
0
answers
21
views
Solving a system of equation with 3 variables in cylindrical symmetry
I want to find the vector potential component's assuming cylindrical symmetry. I expand the Poisson’s equation in cylindrical coordinates which yields the three equations as in the picture. The ...
- 1
3
votes
1
answer
46
views
Is there a canonical coordinate representation for Lie algebras?
If we are given a Lie algebra of vector fields $\{X_i\}_{i = 1} ^N$ on a manifold $M ^n$, is it possible
to determine in local coordinates the $X_i$'s if we know the value of
$m := \dim \text{span} \{...
3
votes
2
answers
107
views
Can every Lie group be realized as conformal group of smooth manifolds
I was reading a paper
Saerens, Rita; Zame, William R., The isometry groups of manifolds and the automorphism groups of domains, Trans. Am. Math. Soc. 301, 413-429 (1987). ZBL0621.32025.
Here they ...
- 152
0
votes
0
answers
71
views
How can I get the following estimate?
I am having a hard time getting the following lower estimate. Is it obvious? At first, I thought it was, but now I do not see it that way, so any hint would be appreciated.
$$\frac{\int_{M}|\nabla \...
- 47
0
votes
0
answers
39
views
On kronecker product representation of $AA^t$
Proposition: Prove that orthogonal matrices $\mathrm O_n(\mathbb{R})$ is a regular submanifold of $\operatorname{GL}_n(\mathbb{R})$.
Proof: Denote $f:\operatorname{GL}_n(\mathbb{R})\to \operatorname{...
- 31
1
vote
0
answers
30
views
Second fundamental form of isometric embedding
There is this well-known Weyl Embedding theorem which guarantees that a metric of positive Gaussian curvature can be realized as a convex surface in $R^3$, which is unique up to rigid motion in $R^3$. ...
- 93
2
votes
1
answer
74
views
Lie derivative of a differential form
I have a differential $1$-form $\omega = x\mathrm{d}x + x\mathrm{d}y$ and I need to find its Lie derivative along $X = (x+y)\partial_{x} - 2y\partial_{y}$. The first approach is by using Cartan ...
- 173
1
vote
0
answers
46
views
Hyperboloid model exponential map
I am working with the hyperboloid model of hyperbolic space and I am trying to understand its exponential map. However I do not see how this map is derived so I would appreciate either an explanation ...
- 11
1
vote
0
answers
24
views
What is a good way of defining an $A_k$ singularity for a discrete function?
Definition: Let $f$ be a smooth function defined on a neighborhood of some $t_0 \in \mathbb{R}$. Then for each integer $k \geq 0$, we say $f$ has an $A_k$ singularity at $t_0$ iff $\forall p, \, 1 \...
1
vote
1
answer
32
views
What is the result of a elementary $k$ alternating tensor acting on $k$ different vectors?
I just learn the concept of elementary k form: on an open set $A$. That is if $x\in A$ , then we have $$\begin{align*}
\phi_I(x)=\phi_{i_1}(x) \wedge \cdots \wedge \phi_{i_k}(x)
\end{align*}$$ ...
- 1,237
5
votes
0
answers
70
views
Differential Geometry's lecture notes by Will Merry
I have recently got to know these notes from this answer, i.e.,
Will Merry, Differential Geometry: beautifully written notes (with problems sheets!), where lectures 1-27 cover pretty much the same ...
- 13.6k
4
votes
1
answer
78
views
An analog of Jordan curve theorem for various type of smooth manifolds
I would like to know whether the following tentative generalization of the Jordan curve theorem in higher dimensions and for smooth manifolds are true, and in case I ask for references proving them.
...
- 101
1
vote
0
answers
34
views
About a Book to Study Differential Geometry: Jean Gallier's Differential Geometry and Lie Groups: A Computational Perspective
“Differential Geometry and Lie Groups: A Computational Perspective” by Jean Gallier and Jocelyn Quaintance is a relatively new book (2020) about differential geometry. It seems that this book is ...
- 108
1
vote
0
answers
17
views
How can we prove that the tangent plane in a point of a surface in $R^3$ is indepent of the surface parameterisation
For a surface in $R^3$, how can we show that the tangent plane is independent of how the surface is parameterised? I understand it intuitively, but cannot figure out how I would prove it rigorously.
...
- 11
3
votes
1
answer
84
views
How to exponentiate the vector field $v$?
i didn't succed to exponentiate these vector field
$$\boxed{v=\sqrt{x}\sqrt{y}\frac{\partial}{\partial x}-\sqrt{x}\sqrt{y}\frac{\partial}{\partial y}+2b\sqrt{x}\sqrt{y}u\frac{\partial}{\partial u}}$$
...
- 43
1
vote
0
answers
21
views
Equivalence problem and Maurer-Cartan structure equations
I am reading a proposition. The proposition is as follows:
Let $M$ and $\bar{M}$ be two manifold, with $\{\mu_i\}$ and $\{\bar{\mu}_i\}$ co-frame, respectively. The Maurer-Cartan equations for $M$ and ...
- 11
1
vote
0
answers
29
views
First chern class of canonical line bundle on $CP^n$
I am trying to calculate the first chern class $c_1(K)$ of the canonical bundle $K = \Lambda^n(T^*\mathbb{CP}^n)^{1,0}$, where my definition of the first chern class is $c_1(K)=\frac{i}{2\pi}[F(A)] \...
- 653
2
votes
1
answer
25
views
Showing that $df_p\neq 0$ for all $p\in f^{-1}(0)$?
I'm trying to solve the following problem:
Consider the function $f: S^2 \to \Bbb{R}$ given by $f(x,y,z)=x^{2023}+y^{2023}+z^{2023}$. Show that $df_p\neq 0$ for all $p\in f^{-1}(0)$.
I'm thinking ...
- 23.2k
3
votes
0
answers
46
views
Prove that Kahler form induced by Fubini-Study metric on $\Bbb{P}^n$ is the generator of the $\Bbb{Z}$ coefficient cohomology
I try to prove that the Kahler metric induced from the Fubini-Study metric on $\Bbb{P}^n$ is a generator of the cohomology $H^2(\Bbb{P}^n,\Bbb{Z})$
My attempt first by the Poincare duality we know ...
- 4,284
-1
votes
0
answers
55
views
characteristic classes in odd dimensions? [closed]
So Stiefel-Whitney, Pontryagin, Chern classes are all for even dimensions. There seem to be no characteristic classes for odd dimensions. Does that mean in odd dimensions everything is trivial?
- 231
0
votes
0
answers
52
views
How to tell the tangent bundle of $S^2$ from the bundle $S^2\times$ $y-z$ plane?
Intuitively, I would like to say that the tangent bundle of $S^2$ and the bundle ($S^2\otimes \rm{y-z\ plane}$) is different. By the latter I mean a product bundle, embedding in $R^3$ it is equivalent ...
- 231
4
votes
0
answers
104
views
Calculating parallel transport for a manifold which is a change of coordinates on the Euclidean space.
Let $\varphi:\bar{\mathcal{M}}=(\mathbb{R}^{2n},\text{usual metric})\to\mathcal{M}=(\mathbb{R}^{2n},G)$ be an isometry as follows:
$$\varphi(z_1,\dots,z_{2n})=(z_1,z^2_1-z_2,\dots,z_{2n-1},z^2_{2n-1}-...
- 41
5
votes
1
answer
96
views
How do we get $ J_{f \circ \psi} (x) = J_\psi(x) J_f(y) $ for $f$ on a submanifold, and $\psi$ is local coordinates?
The following is taken from Leon Simon Geometric Measure Theory: Let $f: M \to \mathbb{R}^P$ for $P \geq n$. Where $M$ is an $n$ dimensional smooth submanifold of $\mathbb{R}^{n+l}$, and $f$ is ...
- 763
0
votes
0
answers
45
views
Laplacian of standard hermitian inner product on $C^n$.
Treat $C^n$ as a complex manifold, and consider the standard Hermitian metric $h = \frac{1}{2} \sum_{i=1}^n dz_i d\bar{z}_i$, so that the corresponding J-invariant metric is $g = \sum dx_i^2+dy_i^2$, ...
- 653
1
vote
0
answers
27
views
Geodesic of a cone and arcsin function
I am trying to find the geodesic of a cone by using the Euler-Lagrange equation and the following parametrization:
\begin{align}
x &= \rho \cos \theta \\
y &= \rho \sin \theta \\
z &= \...
- 47
3
votes
1
answer
88
views
How can I compute the differential of this function between surfaces $C$ and $G$?
Let me consider the surface $S:=\{(x,y,z): x^2+y^2=1\}$ and $G:=\{(0,y,z): y,z\in \Bbb{R}\}$ and define $f:C\rightarrow G$ by $f(x,y,z)=(0,y,z)$. Let us take the following two patches for $C$ and $G$: ...
- 2,435
1
vote
1
answer
53
views
How to calculate the mean curvature of clifford torus immersed in the standard sphere $\mathbb{S}^3$?
We know that clifford torus is a minimal surface immersed in $\mathbb{S}^3$, but how to express this immersion in parametrization and how to calculate the mean curvature of this immersion? If anyone ...
0
votes
0
answers
35
views
Approximation of plane curve with tangent vector
Let $c:[-1,1]\to\mathbb{R}^2$ a $\mathcal{C}^1$ planar curve and suppose that $c(0)=(0,0)$ and $c'(0,0)=(a,0)$, $a>0$. I'm trying to prove the following statement (without any success): there exist ...
- 370
3
votes
0
answers
31
views
Schauder estimates on a punctured disk.
The following is related to my previous question [1], which I think can be resolved if this question is resolved. Is it true for $u\in C^{1,\alpha}(B-p)$ where $B=B_p(r)$ a function satisfying the ...
- 1,152
0
votes
0
answers
73
views
About the definition of pullback (in diff. geom.)
Consider $(d<\infty)$-dimensional ${ \Bbb R}$-vector spaces $V$ and $W$, and their dual spaces $V^*:=Hom(V,{\Bbb R})$ and $W^*:=Hom(W,{\Bbb R})$. Naturally, $V,W,V^*,W^*$ are all isomorphic. Now, ...
- 1
1
vote
0
answers
30
views
Free $S^1$-action on Seifert manifolds
It is certain that not every fixed point-free action is free.
I saw in an earlier post (https://mathoverflow.net/questions/77715/s1-action-in-three-dimensions) in MO regarding non-trivial $ S^1$-...
- 11
5
votes
0
answers
118
views
+50
Measurability of a classical topological surface and its measure
Let $\Sigma \subset \mathbb{R}^3$ be a set with the following property: Given any $p\in \Sigma$, $\exists$ $W_p \subset_{\text{open}} \mathbb{R}^3$, $U_p \subset_{\text{open}} \mathbb{R}^2$ such that $...
- 356
2
votes
0
answers
63
views
Classify U(1) bundle over $\mathbf{P}^3$, and its topological invariants
I am interested in knowing the classification of the U(1) bundle over the complex projective space $\mathbf{P}^3$. This is effectively a U(1) bundle over the 6-manifold $M^6$.
What are the possible ...
- 5,699
0
votes
0
answers
55
views
When is the complexification of a real vector bundle topologically trivial?
Suppose I have the complexification of a real vector bundle. When is this bundle topologically trivial?
In the application I have in mind, I actually have the additional information that the bundle ...
- 1
0
votes
0
answers
23
views
Are minimal area and mean curvature definitions of minimal surfaces really equivalent?
I am reading Lecture Notes on Minimal Surfaces, which can be accessed here and I am trying to find a proof of the following:
Mean curvature of a surface $S$ is zero everywhere if and only if
surface $...
- 1,552
1
vote
0
answers
47
views
Definition of the degree of a map
I have a question on the definition of the degree of a smooth map $f$ between two oriented smooth manifolds $M$ and $N$ with $M$ compact and $N$ connected. Then the degree is defined by
$$
\mbox{deg}(...
- 71
1
vote
1
answer
68
views
How to derive $F^{\mu\nu}$ from $F_{\mu\nu}$?
Let $\vec A = [\phi, A_x, A_y, A_z]$ be a vector field, and a covariant tensor field be defined as
$$ F_{ab} = \frac{\partial A_a}{\partial x_b} - \frac{\partial A_b}{\partial x_a}$$
Can the ...
- 285
2
votes
0
answers
45
views
Does $\partial\bar{\partial}+\bar{\partial}\partial=0$ imply integrability of the almost complex structure?
Let $X$ be an almost complex manifold. A well-known result says that $\bar{\partial}^{2}=0$ implies the integrability of the almost complex structure. My question is what about $\partial\bar{\partial}+...
- 21