Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such ...

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On the Use of the Topology on Tangent Bundles

On learning about how to define smooth vector fields on a manifold $M$, I learned that one should first define a tangent bundle , $T(M)$, as $\cup T_p(M)$ together with a topology(smooth structure). ...
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472 views

Diagonalization of Riemannian Metric and the Laplace Beltrami Operator

Consider the local representation of the Laplace Beltrami operator on a Riemannian n - dimensional manifold $(M,g)$: \begin{equation} \triangle_g = \frac{1}{\sqrt{\text{det}(g)}} \sum^n_{i,j = 1} ...
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418 views

Local diffeomorphism from $\mathbb R^2$ onto $S^2$

Is there any local diffeomorphism from $\mathbb R^2$ onto $S^2$?
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83 views

Question on moment maps.

I have some trouble in understanding the notion of a moment map for the Lie group $S^1$: \ In the book "Moment maps, cobordism and Hamiltonian group actions" it is said on page 15, which you can find ...
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71 views

Spin^c group as product.

Consider the Spin^c group, i.e. the elements of norm 1 in the complexified Clifford algebra $\mathbb{C}l^0$. Can you please tell me why this is isomorphic to $(SO(n) \times S^1)/\{1,-1\}$ where we ...
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247 views

Is the crossing number of a non-planar graph a function of the surface in which it is embeddable?

I know that a non-planar graph with one crossing can be embedded in a torus, and I expected that a graph with two crossings would require a double torus. This does not seem to be the case (cf. the ...
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1answer
127 views

How to work with Connections

I am currently reading a book which deals with complex manifolds. Since I am fairly new to the topic I don't know exactly the meaning of the followinig: Suppose we have a holomorphic vector bundle ...
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1answer
517 views

Complexification of Tangent Bundle

I am currently reading a book where the author says that the tangent and cotangent bundles $TM$ and $T^*M$ of a manifold $M$ are complexified. I am not familiar with Complex Manifolds so looked it ...
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2answers
134 views

Solution of a differential equation that would be a generalized mean?

I am trying to solve this differential equation on which I've been stuck for several days now. $$\frac{d X}{d t}=\frac{\int_{-\infty}^{\infty}\frac{\partial f}{\partial t}\frac{\partial f}{\partial ...
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1answer
114 views

On Tangent vectors as jets & submanifolds

Here is my second question on understanding jets better: For a smooth manifold $M$ the set of jets $J^1_0(\mathbb{R},M)$ is the same as the Tangent bundle $TM$. This implies that any equivalence ...
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171 views

On the definition of jets

I have some problems with the definition of jets and it would be great if someone could help me here: In many books it is written, that the $r-th$ order jet $j^r_xf$ of a smooth function $f:M ...
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1answer
84 views

This Integral should be zero

I am trying to evaluate the following integral: \begin{equation} \int_{M} \frac{(g')^2}{ g^{5/2}} - \frac{(g'')}{g^{3/2}} \ dx \end{equation} where $(M,g)$ is a one - dimensional closed and ...
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455 views

Geodesic on a plane

I guess that each geodesic on a plane is a straight line. Is it right? What can I use to prove it? I guess I have to use somehow Levi-Civita connection.
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698 views

Proof: Tangent space of the general linear group is the set of all squared matrices

Let us assume we have the following definition of a tangent space: Definition of smooth path Let $X\subset\mathbb{R}^n$. Let $I$ be a real interval. \begin{equation} P \text{ is a smooth path in } ...
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1answer
127 views

exercise on surfaces and geodesics

Maybe someone can verify my answers. The problem is as follows: Consider a line $l$ in $\mathbb{R}^3$ and rotate it around the $Oz$ axis. Denote by $A$ the set of all such obtained surfaces. ...
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351 views

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 ...
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925 views

The space of Riemannian metrics on a given manifold.

For a finite-dimensional smooth (Hausdorff, second-countable) manifold $M$, consider the set $$\mathcal{Met}(M) = \{ g : g \text{ is a Riemannian metric on }M \}.$$ I'd like to know about the typical ...
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1k views

Conformal transformation of the curvature and related quantities

Suppose we have a Riemannian manifold ${(M,g)}$, where ${g}$ is the metric of ${M}$. If ${f}$ ${\in}$ ${D(M)}$ (i.e. smooth function on ${M}$), and ${f}$ is positive. So, we can define a new metric ...
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Why are smooth manifolds defined to be paracompact?

The way I understand things, roughly speaking, the importance of smooth manifolds is that they form the category of topological spaces on which we can do calculus. The definition of smooth manifolds ...
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75 views

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 $$ ...
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155 views

hermitian distance functions and geometry in complex space

If we have fixed a hermitian positively definite form $h(.,.)$ in complex space $C^n$ and an analytic submanifold $M$ in $C^n$, then we may fix a point outside of $M$, say $P$, and consider distance ...
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998 views

Problems in do Carmo's Riemannian Geometry

I am reading do Carmo's Riemannian Geometry Chapter 7, and I want to do some exercises. I think that I need some hints to solve the following: Questions: How do I construct a counterexample that a ...
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236 views

Deriving an expression for minimum arc length along a 3D surface between any two points.

Consider a 3D surface, defined by the function $z = f(x, y)$. Assuming the surface is differentiable (no kinks), is there a function that expresses the minimum arc length traced along the surface ...
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462 views

What's the connection between derivatives and boundaries?

The (second) fundamental theorem of calculus says that $$\int_a^b f'(x) dx = f(b) - f(a)$$ which can also be stated, if one knows enough about what's coming next, as: The integral of the ...
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226 views

Intuition for induced bundle.

If $X$ is a manifold, $G$ a compact Lie group, $E$ a principal $G$-bundle over $X$ and $V$ be a vector space on which $G$ acts. Then one can form the vector bundle $E \times_{G} V$ over $X$. What is ...
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132 views

What does ad$f$ mean, for $f$ a smooth function?

I am currently reading Nicole Berline "Heat Kernels and Dirac Operators". On page 64 Differential Operators are introduced that are generalized from operators acting on scalar functions to vector ...
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1answer
173 views

How many terms in a series expansion

General: If $f \in C^1$ is a periodic function defined over some multi-dimensional space, then it should be possible to express $f$ as a FINITE fourier series. is this true of any periodic basis? is ...
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1answer
318 views

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 $F\left(z_1,0\right)=\left(1,0,0\right)$ ...
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1answer
117 views

Opposite Orientation of Boundary in Bordisms

In Lurie's "On the Classification of Topological Field Theories" (and certainly other places) he defines the category $\mathbf{Cob}(n)$ who objects are oriented $(n-1)$ manifolds. Given ...
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259 views

checking if a 2-form is exact

Consider the 2-form $$\sigma=\frac{x_1 dx_2 \wedge dx_3 + x_2dx_3\wedge dx_1+ x_3 dx_1 \wedge dx_2}{(x_1^2+x_2^2+x_3^2)^{3/2}}.$$ I need to show if it is exact or not. Suppose it is exact, then there ...
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295 views

elementary differential geometry question

Suppose $\Omega \subset {\mathbb R}^n$ be open and bounded with smooth boundary. Let $t_0 > 0$ be small enough so that for every $x \in \partial \Omega$, there exists a unique $y \in \Omega$ with ...
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370 views

Stokes for integration along the fiber

I want to use a version of Stokes theorem for integration along the fiber and I need some help in proving a general statement. Let $F$ be a $k$-manifold with boundary and let $E \to M$ be a smooth ...
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1answer
103 views

Computing $d\omega$ and $g^\ast \omega$ when $(x,y)=g(s,t)=(st,e^t)$ and $\omega= xdy$

Define $g:\mathbb{R^2} \rightarrow \mathbb{R^2}$ by $(x,y)=g(s,t)=(st,e^t)$ and let $\omega= xdy$. How can I compute $d\omega$ and $g^\ast \omega$? Actually, I computed $d\omega =tds \wedge e^tdt$. ...
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3answers
369 views

$\Omega=x\;dy \wedge dz+y\;dz\wedge dx+z\;dx \wedge dy$ is never zero when restricted to $\mathbb{S^2}$

We define a 2-form $\Omega$ on $\mathbb{R^3}$ by $\Omega=x\;dy \wedge dz+y\;dz\wedge dx+z\;dx \wedge dy$. How can I show that $\Omega|_\mathbb{S^2}$ is nowhere zero? Before proving that how can I ...
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1answer
699 views

Covectors $\omega^1, …, \omega^k$ are linearly dependent iff their wedge product is zero

How can I prove that covectors $\omega^1, ..., \omega^k$ are linearly independent iff their wedge product $\omega^1\wedge ...\wedge \omega^k$ is not zero?
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1answer
263 views

Knot with genus $1$ and trivial Alexander polynomial?

I would like to know whether there exists a knot $K$ with genus $g(K)=1$ and trivial Alexander polynomial $\Delta(K) \doteq 1$. A linked question could be: does there exist a Whitehead double with ...
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110 views

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 ...
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2answers
1k views

A question about Killing vector and Riemann curvature tensor

In Sean Carroll's Spacetime and Geometry, a formula is given as $${\nabla _\mu }{\nabla _\sigma }{K^\rho } = {R^\rho }_{\sigma \mu \nu }{K^\nu },$$ where $K^\mu$ is a Killing vector satisfying ...
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1answer
204 views

Preimage of submanifold under an embedding

Suppose we have two smooth manifolds $M_1$ and $M_2$ and a smooth map $i:M_1 \rightarrow M_2$ that is an embedding of $M_1$ into $M_2$. Moreover we have another submanifold $N \subset M_2$ that has a ...
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1answer
161 views

Image of smooth vector bundle morphism

Let $\pi_1:V_1 \rightarrow B_1$ and $\pi_2:V_2 \rightarrow B_2$ be smooth vector bundles and write $\phi_V: V_1 \rightarrow V_2$ as well as $\phi_B:B_1 \rightarrow B_2$ respectively for the total and ...
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2answers
409 views

Each point of $M$ has a smooth coordinate neighborhood in which the coordinate frame is orthonormal iff $g$ is flat

Let $(M,g)$ be a Riemannian manifold. Then I want to show that these are equivalent: (i) Each point of $M$ has a smooth coordinate neighborhood in which the coordinate frame is orthonormal. ...
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1answer
331 views

Heat kernel on a noncompact manifold

I'd like to understand some issues about the heat problem related to the Laplacian of a Riemannian manifold especially when the manifold is noncompact. So first recall the heat equation on a ...
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1answer
619 views

The Dimension of the Symmetric $k$-tensors

I want to compute the dimension of the symmetric $k$-tensors. I know that a covariant $k$-tensor $T$ is called symmetric if it is unchanged under permutation of arguments. Also, I know that the ...
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1answer
202 views

Riemannian metric

This is a very simple question that I got confused. Is Riemannian metric a symmetric 2-tensor or symmetric 2-tensor field? Wikipedia says that it is a (0,2) tensor but my book says it is a tensor ...
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2answers
205 views

Basis for the set of all covariant $k$-tensors on V

Here's a proposition from Lee's Smooth Manifolds: Let $V$ be a real vector space of dimension $n$, let $(E^i)$ be any basis for $V$, and let $(\epsilon^i)$ be the dual basis. The set of all ...
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100 views

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 ...
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1answer
565 views

Alternate definition of a 'geodesic ball'

Background: Let $M$ be a Riemannian manifold. Let $p \in M$ and $\epsilon \gt 0$. For sufficiently small $\epsilon$, the standard definition (correct me if I'm wrong) for the 'geodesic ball of ...
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445 views

De Rham cohomology for non-compact manifolds

Let $M$ be a non-compact differential manifold. It is true that in general $H^q_c(M) \neq H^q(M)$, where $H^q_c$ is the de Rham's cohomolgy with compact support group and $H^q$ is the usual de Rham's ...
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what is the domain of the Lagrangian of a surface embedding?

If we view our Lagrangian particle mechanics geometrically, then we describe a particle trajectory as a map from R to a manifold, and the Lagrangian $L(x,\dot{x})$ as a function on the tangent bundle ...
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175 views

Reference for densities and pseudoforms and non-tensorial representations of $\operatorname{GL}(n)$ and associated vector bundles

I'm looking for a reference that will set me straight on a few things. It started out with densities. In John Lee's book, "Introduction to Smooth Manifolds", densities on vector spaces are functions ...