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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There is no smooth retraction from an oriented compact manifold to its boundary

That there is no $C^1$ retraction from a compact, oriented manifold to its boundary is a common lemma in proving a weaker version of the Brouwer fixed point theorem. I recall seeing in class a simple ...
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28 views

Need a very simple example of coordinate functions and parameterization of a manifold

This is a very simple question from introductory differential geometry. Suppose I have an 2-dimensional manifold $M^2$ that is, for simplicity, a subset of $\mathbb{R}^2$. Now suppose $(U,\phi)$ is a ...
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1answer
31 views

The projective space as a homogeneous space

I want to understand why the projective space $\mathbb RP^n$ is diffeomorophic to $SO(n+1)/O(n)$? and why we can write the latter as $O(n+1)/O(n)\times O(1)$?
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homogeneous dimension of the Heisenberg group [on hold]

How to compute the homogeneous dimension of the Heisenberg group $\mathbb C \times \mathbb R $ endowed with the group law $$ (z,t)\cdot(w,s) =\left (z+w, \,t+s+\tfrac{1}{2}\Im m(z \bar{w})\right). $$ ...
2
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1answer
27 views

Convex combination of projection operators

If $P_1, P_2: V \to V$ are linear projection operators on the vector space $V$ with $R := P_1(V) = P_2(V)$, is it true that any convex combination of $P_1$ and $P_2$ is again a projection operator $...
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1answer
34 views

Is integration with respect to spherical measure equivalent to manifold integration over sphere?

Let $S$ be an $n$--sphere $\mathcal{S}^n(0,R)\subset\mathbb{R}^{n+1}$, $\Theta\subset\mathbb{R}^n$ an open subset and let $\phi:\Theta\subset\mathbb{R}^n\longrightarrow S\subset\mathbb{R}^{n+1}$ be a ...
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20 views

Planar intersections of constant Gauss curvature K surfaces

Have they been studied? It appears they did not generate enough interest except the conic sections $K=0$. Do they give rise to curves of fourth order?
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1answer
25 views

About transformations of the metric: should we use the old or the new one to raise/lower indices?

Let $(M,g)$ be a (Pseudo-)Riemannian manifold. If I perform a transformation on the metric, getting a new metric $\tilde{g}$, which metric should I use to raise and lower indices? As I understand, ...
2
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1answer
43 views

Confused About Indices in Deriving Curvature

Asking about a step regarding indices in deriving the Curvature tensor from the geodesic equation. Starting from $$ \frac{d v^a}{du} = - \Gamma^a_{bc}v^b \frac{dx^c}{du}$$ we integrate $$v^a(u) = ...
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20 views

I want to show that $ \mid D pr(m) \mid \leq \dfrac{1}{1-\Vert pr(m)-m \Vert \Vert h_{pr(m)}\Vert}.$

Let M be a smooth surface, and $U$ a neighborhood where the orthogonal projection pr is well defined. I would like to show that $ \forall m \in U$, if $ m \in S$, pr is differentiable on m and we ...
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26 views

Introduction to flag manifolds

What is a good self-contained introduction to the geometry of complex flag manifolds?
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21 views

Fiber bundle $U(n-1)\rightarrow U(n)\rightarrow S^{2n-1}$

Can someone help me to visualize geometrically the fiber bundle $U(n-1)\rightarrow U(n) \rightarrow S^{2n-1}$, what are the open sets where it trivializes?
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1answer
33 views

existence of smooth vector field

I am trying to solve the following problem: Let $M^n$ be a smooth orientable maniofold. Suppose $f\colon M \rightarrow \mathbb{R}$ is a smooth function such that $df\neq 0$ at each point of $M$. ...
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1answer
38 views

Area of a domain with Stokes' Theorem

This question came up on a preliminary exam: Define $$g(s,t)=(x(s,t),y(s,t))=(\cos(s)+\cos(t),\sin(s)+\sin(t)),$$ on the region $-\pi<s<\pi$, $s<t<s+\pi$. (The function $g$ is one-...
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1answer
50 views

Show that $\gamma$ is a straight line [closed]

Let $\gamma : I \rightarrow \mathbb{R}^3$ be a parametretrized smooth curve with unit speed. Assume there exist a fixed vector $q$ such that $\gamma ''(s)=q, \ \forall s \in I$. Show that $\gamma$ is ...
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23 views

Family of (closed) parametric 2D-curves with bounded curvature

Let's consider the set of (closed) parametric 2D-curves $(x(s),y(s))$ such that the curvature and its derivative are bounded at any point, i.e., $|\kappa(s)|\leq b_1$, $|\kappa'(s)|\leq b_2$. Do you ...
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55 views

Is the contraction of an harmonic form harmonic?

As I'm still a beginner in complex differential geometry, as soon as I tried reading an article I got stuck on what (I think) should be a minor detail. I hope someone can help me a bit. Let $M$ be a ...
2
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1answer
88 views

Second Differential

Let $(x,y,z)$ a coordinate system, $M=\mathbb{R}^3$ and we also denote by $x$ the first coordinate function : $x:M \rightarrow \mathbb{R},\; q=(a,b,c) \mapsto a$. We have $dx:TM \rightarrow \mathbb{R}...
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11 views

Example of nonuniqueness of asymptotes of a ray

Let $(M, g)$ be a complete Riemannian manifold and let $\gamma : [0, \infty) \to \mathbb{R}$ be a ray, i.e. a unit speed geodesic such that for every $s, t \ge 0$ : $$ dist\big(\gamma(s), \gamma(t)\...
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57 views

Cauchy Sequence of Differentials and Point-Wise Limits

Let $D\subseteq R^2$ be an open and connected subset, and $\{f_{n} | D\to R^2\}$ a sequence of differentiable functions. Suppose that $\{(Df_{n}) | D\to Hom(R^2,R^2)\}$, the sequence of Jacobians, is ...
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1answer
38 views

Riemannian connection on Lie groups

Let $G$ be a Lie group with a bi-invariant metric. Then, the Riemannian connection is given by $\nabla_XY=\frac1 2 [X,Y]$ for all $X,Y\in \mathfrak g$. In the proof: Since $\langle X,Y\rangle$ is ...
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33 views

Integration on manifolds with boundary

How can I define integral on manifolds with boundary? To use unity partition don't have I to deal with open sets of the same type, I mean, how can I be sure that there is a unity partition on my ...
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1answer
56 views

“Barred” Tensor Indices in Complex Manifolds

I'm having an embarrassingly hard time straightening out how to work with the "barred" indices that show up in tensors on complex manifolds. For example, the Kahler form $\omega = \frac{i}{2}g_{i \...
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7 views

From the given information, what is $\tau_u^{'}(0,0)\times\tau_v^{'}(0,0)$? Also does the reparametrization preserve orientation?

Observe the parametrization $\sigma(U)$ where $U=\{(u,v) \in \mathbb{R}^2\ |\ u^2+v^2<1\}$ and a reparametrization $\tau=\sigma\circ\psi$ where $\psi$ is a diffeomorphism of $U$ onto itself and $\...
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1answer
63 views

Is exponential map locally a diffeomorphism w.r.t. base point?

Let $M$ be a riemannian manifold and $\exp_p: T_pM \rightarrow M$ the exponential map at $p \in M$. At each point $p\in M$, $\exp_p$ can be restricted to a neighborhood $V$ of $0\in T_pM$ so that $\...
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11 views

linear surjective map is locally open on embedded submanifolds

Let $f \colon V \to W$ be a linear surjective map between two vector spaces. Assuming $M \subset V$ is a embedded submanifold of $V$ and $f(M)=N$ is an embedded submanifold of $W$. Is $f \colon M \...
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0answers
40 views

Show that $\operatorname{grad} f(p)=\sum_{i=1}^{n}{(E_{i}(f))E_{i}(p)}$

Let $M$ a Riemannian manifold.Let $X\in\chi(M)$ and $f\in\mathcal{D}(M)$. Define the gradient of $f$ as the vector field $\operatorname{grad} f$ in $M$ define by $$\langle\operatorname{grad} f(p),v\...
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1answer
35 views

The group $\mathrm{Diff}(F)$ and transition functions of a fibre bundle.

Let $M$ and $F$ be differentiable manifolds, and let $F\to E\to M$ be a differentiable fibre bundle over $M$. A trivialising cover $\{(U_i,\phi_i)\,|\,i\in I\}$ of $M$ determines a set $\{t_{ij}:U_{ij}...
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20 views

geomatrical problem to solve using differential equation.. [closed]

How to get the eqution for this using differential equations..Find the curve in which the portion of the tangent included between the cordinate axes is bisected at the point of contact....
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Equation for plugging in right-invariant vector fields in canonical connection?

Consider a matrix Lie group $G$ with Lie algebra $\frak g$ identified with left-invariant vector fields $\mathcal L(G)$. The $0$-connection is given by: $$ \nabla_{X^l}{Y^l}=\frac{1}{2}[X^l,Y^l]=\frac{...
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1answer
21 views

Constant Binormal Implies Curve is Planar (Frenet-Serret)

Using the standard Frenet-Serret notations (https://en.wikipedia.org/wiki/Frenet%E2%80%93Serret_formulas) We have a claim: If $\gamma(s)$ is a normalized curve in $R^3$ and $B(s) = b_{0} \in R^3$ ...
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2answers
44 views

$3$-dimensional shape

What $3$-dimensional shape is represented by graph of the set of pints $ (rcos\theta, rsin\theta,z)$ where $r$ is a constant real value, $\theta$ range from $0$ to $2\pi$ radiant and $z$ range over ...
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0answers
9 views

Cusp edges of constant and infinite Gauss curvatures K

What difference is there between singular cusped boundary edges of $C_1$ continuous parametric surface one with constant $K$ and another with infinite $K$? One radius of curvature is infinite in ...
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24 views

Tangent and normal vectors for parameterization of a straight line.

Hi everyone just a simple question. Suppose the curve $$r(t) = (x_0, t)\\ 0 \leq t \leq 1 $$ $x_0$ is some real number. We see that the unit tangent $T(t)= (0,1) = \dfrac{r'(t)}{\|r'(t)\|} $ ...
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66 views

Can one define 'geodesic' solely in terms of the betweenness relations among the points on that geodesic?

In the Euclidean plane (though I assume the following result can be generalized to any Euclidean n-space), Tarski showed that one can define what it is to be a straight line solely in terms of the ...
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1answer
50 views

Vanish christoffel symbols implies $g_{ij,k}=0$

Let $(M,g)$ be a pseudo-riemann manifold and $(U,\psi=(x^1,\ldots,x^n))$ a local chart around some point $p$ in $M$. It is easy to show that if $\partial g_{ij}/\partial x^k=0$ in $p$ for all $i,j,k$ ...
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Lorentz manifolds - Change of co-ordinates

Let $(M,g)$ be a lorentz 4-dimensional manifold. Let $p\in M$ and $(U,\psi=(x^1,x^2,x^3,x^4))$ a local chart around $p$ such that $x^i(0)=0$ for all $i$. Let new coordinates $\bar x^i$ that verify: $$...
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1answer
78 views

What does it take for a smooth homeomorphism to be a diffeomorphism?

I have an open subset $A$ of $\mathbb{R}^k$ and a subset $B$ of $\mathbb{R}^n$, $n>k$, that are homeomorphic and $f:A\longrightarrow B$ is a smooth homeomorphism between two sets. I'm wondering if ...
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1answer
47 views

Frobenius condition in terms of Lie brackets

Let $\alpha$ be a $1$-form and $\xi = \ker \alpha$. Frobenius theorem tells us that $\xi$ is integrable iff $\alpha\wedge{\rm d}\alpha = 0 .$ In the book "Introduction to Contact Topology" from ...
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Sectional curvature of 2-manifold

This is problem7, chapter 5 of Do Carmo's Riemannian geometry. Let $M$ be a Riemannian two manifold, $p\in M$ , $exp_p$ is a diffeomorphism on a neighbourhood of origin $V\in T_pM$. Let $S_r(0)\...
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How is the Euclidean mean curvature of a minimal submanifold of $ \mathbb{S^{n-1}} $ is equal to the metric Laplacian of the position vector?

I am reading about minimal cones from the book "A Course in Minimal Surfaces, T.H Colding, W.P Minicozzi II". It says that if $N^{k-1} \subset \mathbb{S^{n-1}}$ is $k-1$ dimensional minimal ...
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28 views

Limit of the commutator of two elements?

Given a Lie group $G$ such the $\mathfrak{g}$ denoted its Lie algebra. Let $[g,g']_{G}$ the commutator of two elements $g,g' \in G$ and denoted by $[X,X']_{\mathfrak{g}}$ the Lie bracket of two ...
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42 views

What will happen if evolve metric under Ricci flow on general manifold?

Because the scalar curvature under Ricci flow evolve by $$ \partial_t R=\Delta R+ 2|Ric|^2 $$ I treat it as heat equation with heat source . So, no matter the heat source is very hot or not , the ...
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1answer
38 views

Is the flow of an analytic vector field also analytic?

Let $X$ be an analytic vector field on a smooth manifold. Is it true that the flow $\Phi_t:M\to M$ associated to that vector field is also analytic?
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1answer
31 views

Half strip neighbourhoods for regular surfaces

Let $S$ be a regular connected and compact surface in $\mathbb{R}^3$. It is well known that such surfaces admit a global tubular neighborhood, of thickness $\epsilon>0$. In particular, by ...
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53 views

Schwarzschild metric, speed of ball as measured by observer who catches the ball, just before ball is caught?

The Schwarzschild metric, describing the exterior gravitational field of a planet of mass $M$ and radius $R$, is given by$$ds^2 = -(1 - 2M/r)\,dt^2 + (1 - 2M/r)^{-1}\,dr^2 + r^2(d\theta^2 + \sin^2\...
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1answer
47 views

compute tangent map

Let $f:O(n)\rightarrow O(n), f(M)=M^3$ be a map, $O(n)$ are the orthogonal matrices. Calculate the tangent map at $I$. My idea would be to firstly calculate the tangent space at $I$, it is the kernel ...
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1answer
21 views

normal connection on immersed hypersurface vanishing

I am studying Riemannian geometry using Do Carmo's book. I am learning about isometric immersions right now, and I got stuck with the following claim about Codazzi's equation. Let $f:M^n \to \...
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1answer
21 views

Holonomy of a curve in case of principal $U(1)$ bundle

Suppose $\pi : P\rightarrow M$ is principal $U(1)$ bundle. Let $\gamma$ be a loop in $M$ based at $x_0$ and write $iA$ as connection 1-form on $P$ where $A\in \Omega(P)$. Now define $hol_{\gamma}(A)\...
3
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1answer
18 views

Connection after a metric rescaling and compatibility

It's known (see here for example) that after a rescaling of the metric $\tilde{g}=e^{2\omega}g$, we can find a new connection $\tilde\nabla$ associated to the new metric: $ \tilde\nabla _X Y = \nabla ...