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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+100

How to build smooth variations along arbitrary paths at the endpoint?

Let $M$ be a smooth manifold, $\gamma:[0,a] \to M$ a smooth path. Assume we are given another smooth path $\phi:(-\epsilon,\epsilon) \to M$ that starts from an endpoint of $\gamma$, i.e $\phi(0)=\...
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1answer
50 views

Difference between a manifold and a sub-manifold of the same dimension?

I appologize in advance in case this is a very trivial issue and for any mistakes due to translating stuff from my German lecture notes to English ... A subset $M \subset \mathbb{R}^n$ is defined to ...
5
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1answer
60 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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0answers
25 views

Partial Derivative of Line Integral as a Potential of F

Context to the question: Say $ \{F_{k} \} \to F$ uniformly on a compact subset $K \subset T$, for $ \{F_{k} \}$ a sequence of conservative vector fields and $T$ open and connected. I've shown that ...
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1answer
31 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 ...
3
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1answer
35 views

How is “expressing” a differential operator “in cylindrical coordinates” rigorously defined?

I'm a mathematician (with little knowledge of differential geometry) trying to study physics. One of the greatest problems is the language regarding coordinate transformations. I tend to think of such ...
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1answer
52 views

Cartan decomposition diffeomorphism at the level of (compact) groups

I am trying to understand the Cartan decomposition in the case of a compact group $G$ with subgroup $K$, such that their respective Lie algebras $(\mathfrak{g},\mathfrak{k})$ correspond to a Cartan ...
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1answer
25 views

Parallel transport

Suppose $M$ is a manifold with a connection $\nabla$. Fix $p\in M, v\in T_p M$ and let $B$ be a small neighborhood of $p$ such that every $x\in B$ can be joigned with $p$ by a unique geodesic $\gamma_{...
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19 views

a retraction map define by means of operator [on hold]

Let $B^n(0,R) \subset \mathbb{R}^n $ a ball of radus $R$ and center $0$ and $S^{n-1}=\partial B^n.$ and let $$r : B^n(0,R) \rightarrow S^{n-1} $$ a continuous retraction map, defined $r(x)=x \ \ \ \ \...
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29 views

Parallelizable open dense subset and integration

In Petersen's Riemannian Geometry (2016), it is stated on page 8 that any manifold $M^n$ has an open dense subset $O$ with $TO=O\times\Bbb R^n$. Thus it is orientable and one may define the integral ...
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1answer
49 views

Tensor fields on a manifold

Let $M$ be an $n$-dimensional smooth manifold. It is easily shown that the modules $\Gamma(TM)$ (the real vector space of vector fields on $M$) and $\Gamma(T^\ast M)$ (the real vector space of $1$-...
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2answers
417 views

How does one characterize surfaces with constant nonzero Gaussian and mean curvature

I know that for any surface, the Gaussian curvature $K$ and mean curvature $H$ satisfy the inequality $H^2 \geq K$ , and the sphere is a surface where that inequality becomes an equation. Thus, the ...
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1answer
100 views

Envelope of family of curves $x(u,v)=\cos^2(u)\cos(v)+\cos(u)\sin(u)\sin(v)$, $y(u,v)=\cos^2(u)\sin(v)-\cos (u)\sin(u)\cos(v)$

How to generally find singular solution or envelope of a two parameter family of curves $ x(u,v),y(u,v) $ in the plane? The parametric equations $$x(u,v) = \cos^2 (u) \cos (v) + \cos (u) \sin (u) \...
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2answers
56 views

How can I compute a presentation of the tangent bundle for a smooth manifold defined by a family of polynomials?

Consider a smooth manifold $M$ given by a system of polynomials $$ \begin{align*} f_1 = 0 \\ \cdots \\ f_k = 0 \end{align*} $$ in $n$ variables. This has the algebraic description as the $\mathbb{R}$-...
2
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1answer
51 views

Pipe-fitting conditions in 3D

Let's we have smooth (continuous and infinitely differential) curve $f(x(t), y(t), z(t)) = 0$ in 3D. Now I want to build a tube of diameter $D$ around it. Questions: What are the set of conditions ...
2
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1answer
28 views

Reductive homogeneous spaces

If $G$ is a connected Lie group and $K$ is a closed subgroup of $G$ then $G/K$ is a homogeneous space. If $\frak g,k$ are the lie subalgebras of $G,K$ resp. Then under the projection $\pi:G\rightarrow ...
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31 views

Proof of iff condition of harmonic map

How to compute the equation above the red line in the picture below ? Below picture is from the Harmonic maps and their heat flows. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~...
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1answer
194 views

Reducible to Separable First Order Differential Equation Word Problem in Analytic Geometry 1.4-29

I completed near all problems om a differential equations text chapter on reducing non-separable first order differential equations to separable by using an appropriate substitution for example $u = y/...
2
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0answers
37 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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26 views

tangent space in a moving coordinate frame

I've got a problem in some geometry of flow. For the sake of completeness I will give the complete derivation of the equation of interest, but I will seperate it into derivation part and question ...
2
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1answer
26 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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33 views

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 $$ (z,t)\cdot(w,s) =\left (z+w, \,t+s+\tfrac{1}{2}\Im m(z \bar{w})\right). $$ ...
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1answer
24 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, ...
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1answer
28 views

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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1answer
21 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 ...
2
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1answer
29 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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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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18 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?
4
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1answer
52 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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1answer
37 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-...
3
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2answers
104 views

Direct proof of the second Bianchi identity

Let $X$,$Y$,$Z$,$W$ be vector fields on a riemannian manifold, and let $R(X,Y)W$ be the riemannian curvature: $$ R(X,Y)W = \nabla_{X}\nabla_{Y}W - \nabla_{Y}\nabla_{X}W - \nabla_{[X,Y]}W $$ Let $g$ ...
2
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1answer
41 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) = ...
3
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1answer
858 views

What is ment by: “parallel transport preserves orientation”?

In my text its written that parallel transport on a Riemannian manifold preserves orientation. Can someone clarify what does that mean? I am confused about this notion.
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0answers
53 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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0answers
25 views

Introduction to flag manifolds

What is a good self-contained introduction to the geometry of complex flag manifolds?
2
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0answers
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
50 views

Show that $\gamma$ is a straight line [on hold]

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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0answers
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 ...
1
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0answers
49 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 ...
2
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1answer
86 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}...
2
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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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9 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)\...
4
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1answer
45 views
+50

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
37 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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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 $\...