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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Raising and Lowering Through Differentiation

I'm calculating the Christoffel symbols of the second kind which is of course defined as multiplying the symbol of the first kind multiplied by the contravariant metric. I was thinking of how to make ...
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What is the principal bundle structure of $O(n)$?

Consider the map $\pi:O(n)\rightarrow G(k,n)$ which maps $A\in O(n)$ to the subspace of $\mathtt{R}^n$ spanned by the first $k$ columns of $A$. Here $G(k,n)$ is the Grassmannian manifold. My question ...
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56 views

Why do we need to extend $\phi$?

Here is the proof of a lemma appeared in a text book: Lemma If $M\subset \mathbb R^k$ is a nonempty smooth m-manifold then $m\leq k$. Proof. Let $\phi:U\bigcap M\to \Omega$ be a coordinate chart ...
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47 views

How can a $k\times (k-m)$ matrix be multiplied by a $k\times m$ matrix?

While reading a book on differential geometry, I came across this line: Since the differential $d\psi_0(x_0):\mathbb R^m\to \mathbb R^k$ is injective, there is a matrix $B\in \mathbb R^{k\times ...
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Differential Geometry and classical mechanics basics.

This is so far my understanding of classical mechanics, Interspaced are a few questions where I am still not entirely sure what is going on. Thank you for your help !! A tangent space can be thought ...
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how to differential exponential of a matrix variable $f(X)=e^{X(t)\mathrm{d}t}$?

I have a function about a square matrix $X$ which depends also time: $f(X)=e^{X(t)}$, $t$ is time. So how to differential it about time to have $\frac{\mathrm{d}f(X)}{\mathrm{d}t}$? I know that ...
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1answer
22 views

Computing the differental of an orthogonal projection

I am having trouble computing the differental of a map. This is the context: Let $S\subset \mathbb R^3$ be a regular surface, and fix a point $p\in S$. Let $\pi:\mathbb R^3\to T_pS$ be the ...
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16 views

Finding parallels surface of revolution

$$X(u,v)=(f(u)\cos(v),f(u)\sin(v),g(u))$$ where $v$ is the rotation angle around $z$ axis. What is the parallel of this surface of revolution?
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group operations are smooth in $\text{SL}(n, \mathbb{R})$

I am told the following reason as to why group operations of multiplication and inversion are smooth on $\text{SL}(n, \mathbb{R})$. Multiplication is smooth because the matrix entries of a product ...
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58 views

Proof that this differential form is not exact

Please can someone verify my proof that $\psi = {x dy - y d x\over x^2 + y^2}$ is not exact? Here is my work: If $\psi$ was exact there would exist $f:\mathbb R^2 \setminus \{0\} \to \mathbb R^2 ...
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51 views

Immersion of $\mathbb{R}$ to $\mathbb{R}^2$

I have no idea how to prove that the set $\{(x, |x|): x\in\mathbb{R}\}$ is not the image of an immersion of $\mathbb{R}$ into $\mathbb{R}^2$ For example If $f(t)=(t^3, |t|^3)$ then $f'(0)=(0, 0)$.
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39 views

Symmetry of Killing Vectors in Covariant Derivative

Several times, I've seen statements along the lines of "$\nabla_X Y=\nabla_Y X$ because $X$ is a Killing vector field." One example I found on Stack Exchange is here. I have yet to understand why ...
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1answer
37 views

Proof of the fact that the set of (p,q) shuffles is a cross section of the subgroup $S_p\times S_q$

Definition Let $G$ be a group and $H$ its subgroup. We name a subset $K$ of $G$ a cross section if it has exactly one element from each left coset of $G/H$. Definition Let $n=p+q$ for some ...
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1answer
76 views

Differentiability of a map between manifolds

I have a question about this answer. Let $M$ be a a manifold and let $(U, (x^1, \dots, x^n))$ be a chart around a fixed point $m \in M$. The goal is to define a vector field $F$ such that $F(m) = v$ ...
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47 views

General linear group and special linear group

Consider the general linear group $$GL(n,\mathbb R)=\{g\in {\mathbb R}^{n\times n}\mid\det(g)\neq 0\}$$ Prove that the derivative of the function $f=\det:{\mathbb R}^{n\times n}\to\mathbb R$ is ...
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22 views

Differential Geometry concept: fit the X, Y, Z data and express X=F(Y,Z)

I've been given a project to fit the X, Y, Z data and express X=F(Y,Z); then compute the principle curvatures at any point on the surface (or at least the interior grid points). I know this has ...
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76 views

How to solve this questions about regular surfaces?

I'm trying to solve the following: $i)$ Show that if all normals to a connected surface pass trough a fixed point, the surface is contained in a sphere. $ii)$ Prove that if a regular surface $S$ ...
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34 views

Matrix representation of the derivative of a smooth function

Let $V:\mathbb R^n\to\mathbb R$ be a smooth function and define the Hamiltonian function $H:\mathbb R^n\times\mathbb R^n\to\mathbb R$ (kinetic plus potential energy) by $$H(x,y):=\frac ...
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1answer
16 views

Any reparametrisation of a regular curve is regular

So I'm having a little trouble algebraically showing this is true, the hint is that it is an exercise of the chain rule. From definition, a parametrised curve $\tilde\gamma : J \rightarrow ...
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39 views

Differentiable manifolds that allow isometric transition maps.

What is the class of differentiable n-dimensional manifolds that allow a differential structure, in which all transition maps are isometric? Note that isometric must be overlapping pieces of charts ...
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defining $C^\infty$ structure on finite-dim vector space, homeomeomorphism to tangent bundle, such that independent of choice of bases

If $V$ is a finite dimensional vector space over $\mathbb{R}$, how would I go about defining a $C^\infty$ structure on $V$ and a homeomorphism from $V \times V$ to $TV$ which is independent of bases? ...
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An example of regular singular points

I was reading a book on differential geometry and after the intro to the concepts of regular singular points I came across an example under it: The set $M:=\{(x^2,y^2,z^2,yz,zx,xy)|x,y,z\in ...
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Orientable manifold [duplicate]

I need help for this question: Let $M$, $N$ manifolds, $M$ orientable and $f: M \longrightarrow N$ local diffeomorphism, then $N$ too is orientable. I was trying by definition of orientable ...
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Quick question: Map being smooth vs Graph being submanifold of the product space [on hold]

Is $f:X\rightarrow Y$ smooth if and only if the graph $\Gamma_f$ is a closed submanifold of $X\times Y$? Thank you very much.
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1answer
79 views

Surface normal to point on displaced sphere

I want to calculate the surface normal to a point on a deformed sphere. The surface of the sphere is displaced along its (original) normals by a function $f(\vec x)$. In mathematical terms: Let ...
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1answer
72 views

Smoothly homotoping a sphere in $\mathbb{R}^3$

Start with the standard sphere $S^2$ and consider another (diffeomorphic) sphere $S$ such that there is a family of deformations of $S^2$ in $\mathbb{R}^3$ that ends in $S$. If $S$ is positively ...
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1answer
39 views

Local expression for a 1-form on a surface

Suppose that $\alpha$ is a non-vanishing 1-form on a 2-dimensional manifold. Why can $\alpha$ locally be written as $\alpha = f \ dg$ for some smooth functions $f$ and $g$?
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2answers
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Find the geodesics on the cylinder $x^2+y^2=r^2$ of radius $r>0$ in $\mathbb{R}^3$.

Find the geodesics on the cylinder $x^2+y^2=r^2$ of radius $r>0$ in $\mathbb{R}^3$. I know that the geodesics for cylinders are helices, circles, lines, and points, but i do not know how to ...
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38 views

Riemannian curvature tensor of product manifolds

Let $(M_{1},g_{1})$ and $(M_{2},g_{2})$ be two Riemannian manifolds. Let $% R_{1}$ and $R_{2}$ be the (1,3)-type Riemannian curvature tensors of $M_{1}$ and $M_{2}$, respectively. Finally, let $R$ be ...
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Every point in the codomain is a regular point

Let $0<r<1$ and define $f:\mathbb R^3\to\mathbb R$ by $$f(x,y,z)=(x^2+y^2+r^2-z^2-1)^2-4(x^2+y^2)(r^2-z^2).$$ Let's denote $x^2+y^2=a,r^2-z^2=b$. I don't know if this is allowed, but I just ...
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parallelizable manifolds

I know a differentiable manifold $M$ of dimension $n$ is parallelizable if there exist (smooth of course) vector fields $\{X_i\}_{j=1}^n$ which are linearly independent in $T_pM$ at each point $p \in ...
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1answer
29 views

Why then is $df(x)$ surjective if and only if $Ax\neq 0$?

$\forall x\in \mathbb R^k$, define the linear map $df(x):\mathbb R^k\to\mathbb R$ as follows:$\forall \xi\in \mathbb R^k, df(x)(\xi)=2x^TA\xi$, where $A$ is symmetric. Why then is $df(x)$ surjective ...
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2answers
54 views

derivative of a symmetric bilinear form (quadratic form version)

Let $A=A^T\in \mathbb R^{k\times k}$ be a nonzero symmetric matrix and define $F:\mathbb R^k\to\mathbb R$ by $$f(x):=x^TAx$$ Then why $df(x)\xi=2x^TA\xi$ for $x,\xi\in\mathbb R^k$?
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Completion of hyperbolic 3-manifold (Thurston's lecture note)

I need help with a part of Thurston's lecture note. p.55~56 in the 4th lecture note states that when obtaining hyperbolic manifold by gluing polyhedra having some ideal vertices, the set of ...
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Reparametrization of curve

I found this in a book: Let $\alpha(t) = (g(t),h(t),0)$ be a regular curve in the $xy$-plane. If $g'(t) \neq 0$ for all $t$, then $g$ is strictly increasing. Thus, it is one-to one and has an inverse ...
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59 views

Cohomology in Differential Geometry

Below is a communicative diagram: $$\begin{array}[c]{ccc} ...
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44 views

On a method to compute dimension of moduli space of Riemann surfaces

Consider genus $g$ Riemann surfaces, and thier moduli space $\mathcal M_g$. To determine dimension of $T\mathcal M_g$, start with a complex structure, which in some coordinates can be written ...
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What properties do isospectral Riemannian manifolds share?

I'm studying the Laplacian on (compact) Riemannian manifolds, and it turns out that if the Laplacian operators of two such spaces share their spectrum (the spaces are then called isospectral), then ...
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1answer
50 views

Jacobi field and the metric

I'm reading about Jacobi fields lately, and have noticed some features of it (and it's derivative) with respect to the metric. Thinking about that, I had an non-based, purely intuitive thought that ...
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2answers
29 views

conical surface, parametrization, immersion, Gaussian and mean curvatures

"Find the parametric form of a conical surface $S$ which is spanned by all rays starting $($but not including $)$ a fixed point $\gamma$ and passing through an arbitrary point on $\gamma$ and passing ...
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1answer
39 views

Surface area of a 2-sphere in Abstract Index Notation

I believe the following completely specify a 2-sphere of radius 1 in AIN: $$ R_{ijkl}=\epsilon_{ij}\epsilon_{kl} \\ R_{ij}=g_{ij}\\ R_{ii}=g_{ii}=2 $$ It is easy enough to determine the area by ...
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Covariant derivative for a covector field

In the lecture we had the immersion $f: U \subset \mathbb{R}^2 \rightarrow \mathbb{R}^3$. Now we said that a map $\lambda : U \rightarrow T^*f$ is a covector field. Okay, that's fine. Also, we ...
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Definition of roulette (curve)

I want to give a formal mathematical definition of a roulette, a curve described by a point attached to a given curve as that curve rolls without slipping, along a second given curve that is fixed. I ...
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Confusion about domains in PDE and “every $C^k$ manifold can be thought of as a smooth manifold”

I have read that every $C^k$ domain can be described instead by $C^\infty$ chart maps. So in this sense every $C^k$ domain is a $C^\infty$ domain. But consider standard PDE where people say thing like ...
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Reference request for studying on Fiber bundles

I am looking for some material (e.g. references, books, notes) to get started with Fiber bundles and vector bundles. Can someone help me? Thanks.
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Path of constant relative gradient to a cone.

The equiangular or logarithmic spiral has the property that the angle between any tangent and the radial line is a constant. I am looking for a curve with the same property with respect to a conical ...
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1answer
42 views

Geodesic equation

Assume that you have a parametrization of a surface $f:\Omega \subset \mathbb{R}^2 \rightarrow \mathbb{R}^3,(u,v) \mapsto f(u,v)$. Now if I have a curve defined by $g(t)=f(0,t)$. The geodesic ...
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Help needed in understanding a question to show that $M$ is a smooth manifold

Let $\rho : \mathbb{Z} \hookrightarrow GL(\mathbb{R}^r)$ be a representation. Consider $\mathbb{Z}$ as a subgroup of $(\mathbb{R},+)$ in the usual way. Define $M$ as the quotient of $\mathbb{R} \times ...
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Explicit formula for the (n-2)th derivative of the Jacobi equation

The $n-2$ order derivative of the Jacobi equation is given by: $$\frac{D^n}{dt^n} V_i+\sum\limits_{l=0}^{n-2} \binom{n}{k} (\nabla_{\gamma '}^{(n-2-l)}R)(\gamma ' ,\nabla_{\gamma '}^{(l)} V_i)\gamma ...
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Help in computing $\int \frac{\vec{r}}{r-\vec{r}\vec{n}} d\vec{S}$

Please help me integrate the function \begin{equation} \vec{v}(\vec{r}) = \frac{\vec{r}}{r-\vec{r}\vec{n}} \end{equation} on the surface of a sphere with radius R. $\vec{r}$ denotes a vector which is ...