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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Centers of the osculating circles along an ellipse

Consider an ellipse on the plane $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. We will use the usual parametrization: $P(t)=(x(t),y(t))=(a\cos t,b\sin t)$. Then the tangent vector is $T(t)=(-a\sin t, b\cos ...
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How to tell whether a curve has a regular parametrization?

A parametrization of a 1-dimensional curve is called regular if its velocity is always positive. For example, the following parametrization: $$x(t)=t^3, y(t)=t^6$$ is not regular because its ...
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Showing $\dim T_0 ℝ^n = n$ using a derivation definition for the tangent space.

I’m trying to (re-)prove that $\dim_ℝ \mathrm{Der}_ℝ(C^1(ℝ^n)) = n$, where $$\mathrm{Der}_ℝ(C^1(ℝ^n)) = \{δ\colon C^1(ℝ^n) → ℝ;~\text{$δ$ is a $ℝ$-linear derivation}\},$$ and the ...
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50 views

Can we charachterize spheres by the symmetricity?

$n\geq2$. $c\in\mathbb{R}$. Given an orthogonal basis $(e_i)_{i=1}^n$ of $\mathbb{R}^n$ and a hyperplane $P=\{y^1e_1+\cdots+y^{n-1}e_{n-1}+ce_n\ |\ y^1,\cdots,y^{n-1}\in\mathbb{R}\}$ in it, we ...
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24 views

Curve shortening flow and strong maximum principle

I am in particular uncertain about how the strong maximum principle is used in the argument below. Could someone please clarify and add more detailed explanations. Thanks So assume we have a regular ...
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24 views

Hamiltonian vector field and symplectic geometry

I want to show the following theorem: For any Hamilton function $H : M \rightarrow \mathbb{R}$ on some symplectic manifold $M$ and symplectomorphism $f : M \rightarrow M$ we have $X_{H \circ f} = ...
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commuting property of connections and bundle homomorphisms

I have the following situation: $E,F$ are (smooth) vector bundles over a smooth manifold $M$. Assume we are given connections $\nabla^E,\nabla^F$ on $E,F$ and a homomorphism of $M$-bundles ...
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Morse functions are dense in $\mathcal{C}^\infty(X,\mathbb{R})$.

In Shastri's Elements of Differential Topology, p.210-211, there is written: Why do we get a Morse function $f_u$ on $X$? We know that for any $f\!\in\!\mathcal{C}^\infty(X,\mathbb{R})$, there is ...
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Differential at a point and differential (Differential Geometry)

Given $f\in C^\infty(U)$, $U$ open set of $\mathbb{R}^n$, we define the differential of $f$ at $p$ $$ (df)_p:T_p\mathbb{R}^n\to\mathbb{R}\\ (df)_p(v):=v(f) $$ and the differential of $f$ $$ df:U\to ...
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Are there noncontinuous derivations $C^1(X) → ℝ$?

I’m looking for an example of a Banach space $X$ and a derivation $δ \colon C^1(X) → ℝ$ which is noncontinuous with respect to the topology of uniform convergence on $C^1(X)$, that is a $ℝ$-linear map ...
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38 views

Local isometries preserve geodesics?

Question: It is well known that if $\varphi:M\to \tilde{M}$ is an isometry between Riemannian manifolds, then $\varphi$ maps geodesics of $M$ to geodesics of $\tilde{M}$. I am wondering if it is ...
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Completeness of the vector field $e^{-x} \frac{\partial}{\partial x} + \frac{\partial}{\partial y}$

I just want to bounce this off of the smart people on MSE to make sure I understand what's going on when we discuss complete vector fields. Consider the following field. $X = e^{-x} ...
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37 views

Is pointwise multiplication by a smooth non zero function a diffeomorphism

Say $f: \mathbb R \to \mathbb R$ is nowhere zero (like e.g. the constant map 1). Is the map $x \mapsto x f(x)$ a diffeomorphism? It seems to me that the answer is no because the derivative of a ...
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Topologically non trivial cycles

I am studying a Stiefel manifold $X$ which is topologically an $S^3$ bundle over an $S^4$ but is not a product space. I am not able to understand that why is it not the product space $S^3 \times S^4$? ...
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Hodge-Star-Operator on arbitrary oriented basis

Assume that $V$ is oriented finite dimensional vectorspace with dimension $n$, $g \in T^0_2(V)$ a given symmetric and nondegenerate tensor. Let $\mu$ be the corresponding volume element of $V$. ...
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Lie bracket is part of the intrinsic “geometry”? But I have seen it defined without a metric…?

I have seen two definitions of the Lie Bracket for a Riemannian manifold $(M,g)$. One is this : $[X,Y] = D_X Y - D_Y X$, where $D$ stands for covariant differentiation. When written out, this seems ...
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Stokes' theorem and symplectic geometry

Let $V = \mathbb{R}^2,$ as a vector space then the Poincaré invariant is an integral $\int_{\gamma} \theta$ where $\theta = p dx $ is the symplectic 1-form and $\gamma$ a closed curve. Now, it is ...
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1answer
55 views

Area form and surface area

I know how one can define the surface area via the charts of a surface in $\mathbb{R}^3.$ click here for instance Now, I read that the canonical surface area form for such a surface with surface ...
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Differential forms on fuzzy manifolds

This post will take a bit to set up properly, but it is an easy read (and most likely easy to answer); in any event, please bear with me. Question In the usual setting of open subsets of ...
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Tangent space of the space of compatible complex structures

Let $(M,\omega)$ be a symplectic manifold, and let $\mathcal{J}(M,\omega)$ denote the space of complex structures on $M$ which are compatible with $\omega$. I have been told the following fact: We ...
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Reciprocal relations in Roulette /glissette rollings

If a catenary rolls on a straight line its focus traces out a parabola and vice versa. Is it true? Are there more such examples and how are they co-related? In case of a circle rolling on a fixed ...
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95 views

Kovacic's algorithm

Is there any reference with some example, about how to solve a "riccati" equation in this (below) form :$$y'(x)+a(x)y^2(x)+b(x)y(x)+c(x)=0$$ by Kovacic's algorithm? Or can anybody help me to ...
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Question of well-definedness of the Levi-Civita connection?

On page $55$ of Do Carmo's Riemannian geometry, he proves that there is a unique symmetric affine connection compatible with a given metric on a manifold M. He defines it by a formula $\langle Z, ...
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TNB Frames (Principal Normal Vector)

From the way I'm looking at it, there should be two ways to find the principal normal vector of a plane curve $C$ given by vector equation $$ \pmb{r}(t) = x(t)\pmb{i} + y(t)\pmb{j} + 0\pmb{k}. $$ ...
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Riemannian Geometry notational tricks or alternatives

I am interested in learning tricks that people have developed to speed up / clean up calculations in Riemannian Geometry. I am hopeful about this question because there is often a lot of symmetry in ...
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1answer
85 views

What's wrong in my thinking about Bézout's theorem?

First, I know that every hypersurface of degree $d$ defined in $\mathbb{CP}^n$ is diffeomorphic. By using this fact, I wanted to calculate the Euler characteristic of hypersurface of degree $d$. To ...
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1answer
26 views

mean square displacement on the 3-sphere

I would like to compute the mean square displacement (MSD) for a particle moving on the surface of a 3-sphere of radius R. I see that I could eventually use the polar coordinates and get a polar ...
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1answer
43 views

Understanding $F \circ \phi^{-1}$ in differential geometry

I am struggling with a question in elementary differential geometry. I thought I understood the basics until I read page 20 of The Geometry of Physics by T. Frankel. Suppose we have a manifold of ...
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Natural Coordinate Functions

I'm studying the "Elementary Differential Geometry" from O'Neil and he mentions what he calls the "Natural Coordinate Functions". In particular, he says that if $p = (p_1 ... p_n) \in \mathbb{R}^n$ is ...
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Consequences of the Carathéodory conjecture

This is a very stupid question. What are consequences and applications of the Carathéodory conjecture? It seems to me interesting, but completely useless.
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Transitive Lie group actions and surjectivity of maps

I am reading a paper at the moment and I have come across two statements which I want to understand. Here is the setup: Suppose that $G$ is a Lie group which acts on a manifold $E$ differentiably and ...
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1answer
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Automorphism group of a lie algebra as a lie subgroup of $GL(\frak g)$

Let $G$ be a lie group with lie algebra $\frak{g}$. Let $Aut(\frak g)$ be the automorphism group of $\frak{g}$. Its clear to me that $Aut(\frak{g})$ $\subset GL(\frak{g})$ since any automorphism of ...
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Master's Exploration in General Relativity

just throwing a query out to the Math community. I'm about to embark on a master's in Gravitation, Cosmology and General Relativity and was looking for possible subjects to start researching. My main ...
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2answers
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Intuition about geodesic incompleteness

To state the context, I am familiar with the Hopf-Rinow theorem. My request is three fold, I would like to know of general classes of geodesically incomplete spaces. I basically want to see lots ...
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1answer
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Isometry from warped product onto the base.

Let $B$ and $F$ be semi-Riemannian manifolds with metric tensors $g_B$ and $g_F$, and consider the warped product $B \times_f F$ by a smooth map $f: B \to \Bbb R$, with metric tensor: $$g = \pi^\ast ...
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1answer
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finding tangent planes

Question: Determine the tangent planes of $x^2+y^2-z^2=1$ at the points $(x,y,0)$ and show that they are all parallel to the $z$ axis. Proof: Let the tangent plane of $f(x,y,z)=0$ at point $(x_{0}, ...
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Confused (disoriented?) by questions about orientation

I believe I have a reasonable basic understanding of orientation. Yet, i'm finding myself utterly confused when facing a specific question. Here are several examples: Exhibit an ordered basis ...
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Pullback of $1$-form in coordinates

Let $$\theta(p) = \sum_{i=1}^n f_i(p) \, dx_i$$ be a $1$-form in local coordinates. then we define $F^*(\omega(p))(X_1,\ldots,X_n) = \omega(F(p))(DF(p)(X_1),\ldots,DF(p)(X_n))$ as the pullback of a ...
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Mapping sphere surface to a vector space such that distances are preserved?

I have a unit radius sphere (say in 3D) centered on the origin. Thus the shortest distance between two points on the sphere is the geodesic. Is there a transformation (linear or non-linear) on the ...
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2answers
45 views

Integration over Riemannian Manifolds

Can we integrate over non-orientable riemannian manifold? If so, how do we do it? Some references would be nice. Thank you!
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1answer
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How to express curvature of a level set in terms of derivatives of a function?

Suppose I have a smooth function $u:\mathbb R^n\to\mathbb R$. Assume that its gradient doesn't vanish (near any point where we investigate it). Is there a list of different (intrinsic and extrinsic) ...
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Are $X=M \times [0,T]$ and $\partial X$ smooth compact manifolds when $M$ is smooth compact Riemannian manifold?

Let $X=M \times [0,T]$, where $M$ is a smooth and closed compact Riemannian manifold. I want to know if: $X$ is smooth compact manifold, and if $\partial X$ is smooth compact manifold? I am not ...
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SO(n) orientable?

I have to answer the question whether $SO(n)$ is orientable or not...Actually I have no idea - could someone help me? I already know that $SO(n)$ is a $n(n-1)/2$-dimensional manifold, but how can I ...
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Smooth map $S^1 \to S^2$ can not be surjective

Why cannot a smooth (or piecewise linear) map $S^1 \to S^2$ be surjective? There are space-filling curves, but the usual examples have very "twisty" definitions. UPD A bit of background for this ...
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Frenet-Serret formula proof

Prove that $$\textbf{r}''' = [s'''-\kappa^2(s')^3]\textbf{ T } + [3\kappa s's''+\kappa'(s')^2]\textbf{ N }+\kappa \hspace{1mm}\tau (s')^3\textbf{B}.$$ What is $\tau$, I can't figure that part out. ...
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Smooth isometric embeddings of Riemannian manifolds

The essence of this question is: Let $(M,g_M)$ and $(N,g_N)$ be Riemannian manifolds. How many different ways are there to embed $M$ isometrically in $N$? In this context, I say the embedding $i_1$ ...
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1answer
29 views

Some calculations with skew forms and wedge product

I have some problems with the language of multilinear forms. I have to prove that if $dim(V)\le 3$, then every $\omega\in\Lambda^q(V^\ast)$ is such that $\omega\wedge\omega=0$. I consider the case ...
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Why do we require differential manifolds to be Hausdorff? [duplicate]

Among the requirements for a differential manifold $M$ is that it be connected and Hausdorff. What fails if a manifold is not Hausdorff?
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35 views

Exercise about wedge product and multilinear forms

I'm considering $\omega\in \Lambda^{2q+1}(V^\ast)$, i.e. a multilinear skew-symmetric form. I want to prove that $\omega\wedge\omega=0$. How shall I proceed? Any suggestions? Do I have to write ...
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3answers
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Calculating euler number of disk

I'm trying to do exercise 3.1 from Polchinski, which should be a rather easy differential geometry problem. I have to calculate the euler number defined by $$\chi = \frac{1}{4\pi}\int_{M}d^{2}\sigma ...