1
vote
0answers
30 views

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 ...
2
votes
1answer
48 views

If $\Gamma^k_{ij}(p)=0$, then $\nabla_{E_i}E_j (p)=0?$

I'm having the same problem as it was questioned here. I can't get throught the step where I need to show that $\nabla_{E_i}E_j (p)=0$. It only leads to $$ \nabla_{E_i}E_j(p)=\sum_{lk}^n ...
0
votes
0answers
24 views

Variation on Fubini's Theorem

My attempt: Let $P_1$ be a regular partition of $R_1$ and $P_2$ a regular partition of $R_2$. Denote by $P$ the corresponding regular partition of $R_1\times R_2$. Given a generalized rectangle ...
8
votes
1answer
92 views

Using index notation to write $d^2=0$ in terms of a torsion free connection.

Let $(M,g)$ be a Riemannian manifold and let $\omega$ be a $1$-form on $M$. I want to rewrite $d^2\omega=0$ in terms of the Levi-Civita connection. I can show the following: $$d\omega(X,Y) = ...
0
votes
0answers
27 views

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 ...
1
vote
0answers
56 views

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 ...
1
vote
1answer
109 views

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 ...
2
votes
2answers
100 views

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 ...
2
votes
1answer
34 views

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 ...
2
votes
1answer
62 views

Volume of a complete, simply connected Riemannian manifold of constant negative curvature

Given an $n$-dimensional complete simply connected Riemannian manifold of constant negative curvature $-1$, I need to show that $$\operatorname{vol} B_{r}(p) = \alpha_{n-1} \int_{0}^{r} ...
1
vote
1answer
160 views

Homogeneous riemannian manifolds are complete. Trouble understanding proof.

I came across this proof while looking for hints on my homework, and I think it's only gotten me more confused. This is from Global Lorentzian Geometry. Lemma 5.4 If $(H,h)$ is a Riemannian ...
0
votes
0answers
98 views

About the riemannian metric on the torus

Iwas ask if the torus admits a $\mathbb{R}$-invariant riemannian metric, I think in use the fact that the torus is homeophorphic to $\mathbb{R}^2/\mathbb{Z}^2$ and use the action of $\mathbb{R}^2$ on ...
0
votes
0answers
64 views

Bi invariant Riemannian metric on a Lie Group

I'm trying to find an example when G is a lie group that admits a bi invariant riemannian metric and H a is closed subgroup wich the manifold $G/H$ does not admit a $G$-invariant riemannian metric. ...
4
votes
0answers
119 views

Shape Operators and Symmetric Linear Transformations

The exercise (from Sakai) is: Let $f: E\subseteq \mathbb{R}^{n-1} \rightarrow \mathbb{R}$ be smooth and let $M_f := \{p = (x, f(x)) \in \mathbb{R}^n\,;\,x \in E\}$ be the graph of $f$ considered ...
2
votes
1answer
73 views

About the symmetry of Riemann Tensor

It is a problem in my homework. First I was asked to show $$ \nabla_a\nabla_bA_c-\nabla_b\nabla_aA_c=R_{a,b,c}^{\;\;\;\;\;d}A_d $$ where $A$ is a (0,1)-tensor and $R_{a,b,c}^{\;\;\;\;\;d}$ is the ...
0
votes
1answer
51 views

Jacobi fields in polar coordinates.

This is from Sakai's Riemannian Geometry: Let $(r, \theta)$ be polar coordinates of the plane. We define a Riemannian metric $g$ on the plane by $g(\frac{\partial}{\partial r}, ...
0
votes
1answer
145 views

locally isometric is not a symmetric relation.

The relation of being locally isometric for Riemannian manifolds is reflexive and transitive. Is it symmetric? Can you give me an example?
5
votes
1answer
125 views

The Ricci flow and $\frac{\partial}{\partial t}g_{ij}=-2(R_{ij}+\nabla_i \nabla_j f)$ are equivalent up to diffeomorphism

Suppose $M$ is a Riemannian manifold. Consider flow $\frac{\partial}{\partial t}g_{ij}=-2(R_{ij}+\nabla_i \nabla_j f)$, where $f$ is a time-dependent function. I would like to prove that flows of this ...
4
votes
1answer
230 views

The divergence of the Weyl tensor

First, the Weyl tensor is given by $$W_{ijkl}=R_{ijkl}-\frac{1}{n-2}(g_{ik}A_{jl}-g_{il}A_{jk}-g_{jk}A_{il}+g_{jl}A_{ik})$$ where, $A_{ij}$ is the Schouten tensor, given by ...
1
vote
0answers
122 views

About Hessian of distance function

I'm studying the comparison Hessian theorem and I not understand the following: Let $(M, \langle\ ,\ \rangle)$ be a complete Riemannian manifold. Given $o\in M$, define $r=dist(o, \cdot)$. Then, for ...
2
votes
0answers
71 views

angle sum for triangle on helicoid

Given the helicoid $$ \boldsymbol{r} = (u\sin v, u\cos v,v)$$ in three-dimensional Euclidean space, consider the triangle $T$ defined by $$ 0 \leq u \leq \sinh v, \qquad 0 \leq v \leq v_0.$$ The ...
6
votes
1answer
304 views

Positive definiteness of Fubini-Study metric

Define the Fubini-Study metric $$g_{i\overline{j}} = \frac{\delta_{i\overline{j}}(1+|\boldsymbol{z}|^2)-\overline{z}^jz^i}{(1+|\boldsymbol{z}|^2)^2} $$ for $i,j=1,\ldots,n$ and $z_i$ complex variables ...
4
votes
1answer
180 views

local isometry for riemannian manifolds is not transitive

Let $(M_1,g_1)$ and $(M_2,g_2)$ be Riemannian manifolds of the same dimension, and let $\phi: M_1 \to M_2$ be a smooth map. We say that $\phi$ is a local isometry if $g_2 (\phi_* X, \phi_* Y ) = g_1 ...
1
vote
0answers
26 views

signature of pseudo-Riemannian metric made of Newton polynomials

Given a polynomial with roots $x_1,\ldots,x_n$ and real coefficients, it can be written $$ P(x)=\prod_{i=1}^n \left( x-x_i \right);$$ define Newton polynomials $$s_k(x_1,\ldots,x_n):=\sum_{i=1}^n ...
5
votes
2answers
243 views

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 ...
1
vote
1answer
197 views

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 ...
6
votes
1answer
326 views

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 $v\in T_pM$ ...
0
votes
1answer
42 views

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.
3
votes
1answer
131 views

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 $ ...
3
votes
2answers
304 views

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 $d(p_0,N)$ ...
8
votes
2answers
364 views

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 $\Delta_{M\times N}$ ...
2
votes
2answers
196 views

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 ...
5
votes
1answer
457 views

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 ...
2
votes
1answer
68 views

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 ...
3
votes
1answer
341 views

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 ...
2
votes
1answer
112 views

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 ...
0
votes
1answer
118 views

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 ...
1
vote
1answer
82 views

Question about curvatures of hypersurfaces

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 ...
2
votes
1answer
141 views

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 ...
2
votes
2answers
151 views

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 ...
1
vote
2answers
411 views

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: ...
4
votes
2answers
564 views

Riemannian metric in the projective space

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 ...
3
votes
1answer
395 views

Commutation formula for covariant derivative

Suppose $\nabla$ is the Levi-Civita connection on Riemannian manifold $M$. $X$ be a vector fields on $M$ defined by $X=\nabla r$ where $r$ is the distance function to a fixed point in $M$. $\{e_1, ...
2
votes
1answer
264 views

antipodal map of complex projective space

Let $CP(n)$ be the complex projective space with Fubini-Study metric with diameter $=\frac{\pi}{2}$. Fix a point say $p\in CP(n)$; my question is what is the set of points of maximum distance to the ...
8
votes
2answers
512 views

Why do horizontal curves have zero covariant derivative along their projection?

Background: Let $(M,g)$ be a Riemannian manifold. Let $(p,v) \in TM$ and $V, W \in T_{(p,v)}TM$. We can introduce a Riemannian metric on $TM$ via $$\langle V, W\rangle_{(p,v)} = \langle d\pi(V), ...
2
votes
2answers
2k views

Isometries of the sphere $\mathbb{S}^{n}$

Got this as homework and I don't know how to tackle this. Help please! Prove that the isometries of $\mathbb{S}^{n} \subset \mathbb{R}^{n+1}$, with the induced metric, are restrictions to ...
3
votes
1answer
261 views

Excercise in isometries of the half upper plane

This is one of Do Carmo's excersices and I got it as homework. Part (a) is easy and I include it here for the sake of completness. But I am entirely lost on part (b). A function $g:\mathbb{R} ...