Tagged Questions

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Isometries and geodesics in projective plane using covering

We define a relation in the sphere by identifying the antipodal points, the quotient space obtained is the projective plane $\mathbb{P}^2$. Also, the quotient map ...
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Distance under Some Metric [duplicate]

It is my homework: Let $D^*=\{(x,y)\in\mathbb R^2|0<x^2+y^2<1\}$ be the punctured unit disc in the Euclidean plane. Let $g$ be the complete Riemannian metric on $D^*$ with the constant ...
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Inner product in projective plane

We define the projective plane as $P^2=\{[p]:\{p,-p\}\in S^2\}$ or as the set of all lines passing throught the origin in $R^3$. We define coordinates charts as page 10 in ...
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Show that the Lie derivative is equal to the commutator

Let $\Omega \subseteq \mathbb{R}^d$ be open. Let $\epsilon > 0$. Let $(\phi_t)_{t \in (-\epsilon , \epsilon)}$ be a family in $\mathrm{Diff}(\Omega)$ such that $\phi_0 = id_{\Omega}$ and ...
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Are these two 2-manifolds homeomorphic?

I have a 2-Sphere with a finite number $k$ of points removed (at least 3), and I want to equip it with a Riemannian metric of constant negative curvature. My first thought was to take a free ...
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Compact hypersurface of $\mathbb{R}^{n+1}$ with positive curvature is diffeomorphic to $S^n$.

I have a compact hypersurface $M$ of $\mathbb{R}^{n+1}$ with positive curvature. I need to show that it is diffeomorphic to $S^n$. The hint is to consider the shape operator $A_{\nu_p} x$, where ...
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Justification for this manipulation in a proof of the first variation of energy formula

As a part of my current homework assignment, I am to derive the first variation of energy identity. Working out the problem with my friends, we came to exactly the same argument as presented in these ...
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Curvature of a metric defined on an open disc in $\mathbb{R}^2$

Let $D$ be an open disc centred at the origin in $\mathbb{R}^2$. Give $D$ a Riemannian metric of the the form $(dx^2+dy^2)/f(r)^2$, where $r=\sqrt{x^2+y^2}$ and $f(r)>0$. Show that the curvature of ...
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Vector field and integral curve

Let M a riemaniann manifold, $V$ a vector field in M, and $\phi_t$ the respective flow Let $p\in M$, and $\gamma$ the integral curve of $V$. Show that for all $v\in T_pM$ ...
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Question about self homeomorphism of $\mathbb C\mathbb P^2$

Can anyone give me any idea about how to show that: any self homeomorphism of $\mathbb C\mathbb P^2$ is orientation preserving? Thanks.
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smoothness of hopf fibration projection with respect to standard differentiable structure on unit sphere

We know that steographic projection defines a differentiable structure on $S^n$ by sending points on $S^n$ to hyperplane $\{x^{n+1}\}=0$. In fact, stereographic projection $\sigma_P: S^n- P\to R^n$ ...
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Geodesic distance from point to manifold

This is question 1 of chapter 9 from Manfredo do Carmo's Riemmanian Geometry. $M$ is a complete Riemmanian manifold and $N\subset M$ a closed submanifold. $p_0\in M$ and $p_0\notin N$. Let $d(p_0,N)$ ...
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sign error proving product rule for the Laplacian on a product of Riemannian manifolds

Given two Riemannian manifolds $M$ and $N$, of dimension $m$ and $n$ respectively, the product manifold $M\times N$ has a Riemannian structure, and there is a Laplacian operator $\Delta_{M\times N}$ ...
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Upper half plane is complete with the Lobatchevski metric

How do I show that the Upper half plane is complete with the Lobatchevski metric? I tried to use the fact that $M$ is complete iff the lengh of any divegert curve is unbounded,but did not get any ...
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Question in do Carmo's book Riemannian geometry

This is a question on Do Carmo's book "Riemannian Geometry" (question 7 from chapter 7): Let $f:M\to \bar{M}$ be a diffeomorphism beetwen two riemannian manifolds. Suppose $\bar{M}$ complete and ...
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Differentiable map conserving geodetic lines which is no isometry

I am looking for a differentiable map $f: S^n\rightarrow S^n$, which conserves the geodetic lines of the standard metric on $S^n$, but is no isometry. The geodetic lines on $S^n$ should be the great ...
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Second Bianchi identity

This is q. 7 of ch. 4 from Do Carmo's book on Riemannian Geometry . Prove that: $$\nabla R(X,Y,Z,W,T) + \nabla R(X,Y,W,T,Z) + \nabla R(X,Y,T,Z,W)=0.$$ Let $\{e_i\}$ a geodesic frame on $p$ , it is ...
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Rotation of a catenary in $\mathbb{R}^5$

If you rotate a catenary in $\mathbb{R}^3$, then you get a catenoid. To show: If you rotate the same catenary in $\mathbb{R}^5$, then you get a 4-dimensional hypersurface. I'm not sure, if I got this ...
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A question about covariant derivative of a tensor

Let $R'$ be a tensor of order 4 in a riemannian manifold $M$ defined by: $R'(W,Z,X,Y)=\langle W,X \rangle \langle Z,Y\rangle - \langle Z,X\rangle \langle W,Y\rangle$ And let $R$ be the curvature ...
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Let $M^n\subseteq {\mathbb{R}}^{n+1}$ be a hypersurface. Compute the sectional curvatures in all planes which are spanned by two eigenvectors $X_i, X_j$ of the Weingarten map. Also compute the Ricci ...
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Does the value of the covariant derivative at a point of the metric tensor depend only on the involved tangent vectors?

Let $\nabla$ be an affine connection on a pseudo-Riemannian manifold $(M,g)$. Let $c:[0,1] \rightarrow M$ be a differentiable curve and consider vector fields $Y,Z$ along $c$. Is it true that the ...
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Tangential component of vector field

I'd like to know the definition of "tangential component" in this case, it is question 3 of page 57 of Do Carmo's book Riemannian Geometry: It says: Define $\nabla_XY(p) =$ tangential component ...
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Riemann tensor on a sphere

I was given this exercise but I don't even know where to start: to compute the Riemann tensor of the 2-dimensional sphere. The tensor acts on vector fields X,Y,Z like this: ...
Let $A: \mathbb{S}^n \rightarrow \mathbb{S}^n$ be the antipode map ($A(p)=-p$) it is easy to see that $A$ is a isometry, how to use this fact to induce a riemannian metric in the projective space such ...