Basis of tangent space of a sphere

Let S be the two-dimensional sphere in R3. For a given x∈S, i.e. ‖, I would like to find basis vectors b_1(x) and b_2(x) that span the tangent space of S a the point x. My first try was something like b_1(x) = (x_2, -x_1, 0)^T and b_2(x) = (0, x_3, -x_2)^T but that is … Read more

Prove that on a compact manifold, Laplacian can not take constant value

Prove that there is no function ϕ on a compact manifold without boundary, such that Δϕ=c, where Δ is the Laplacian operator. My intuition is that you can find a closed curve and after reparametrization, equation becomes ∂2ϕ∂θ2=c on the curve, which is impossible. But I don’t know if this is right or how to … Read more

Verification of the identity \langle \nabla _{[X,Y]}Z,Z \rangle = \frac{1}{2}[X,Y]\langle Z,Z \rangle\langle \nabla _{[X,Y]}Z,Z \rangle = \frac{1}{2}[X,Y]\langle Z,Z \rangle

In the book Riemannian Geometry, page 91, Do Carmo writes: \langle \nabla _{[X,Y]}Z,Z \rangle = \frac{1}{2}[X,Y]\langle Z,Z \rangle I could not understand how this happens. Can someone explain this to me please? Answer If (M,g) is Riemannian manifold, the Levi-Cevita connection is the unique affine connection on TM which is torsion-free and compatible w.r.t. g … Read more

Show H1dR(Sn)=0H_{dR}^1(S^n)=0 for n>1n>1 without de-Rham Theorem, and some similar questions.

Question Without using de Rham’s theorem, prove: (1) Show H1dR(Sn)=0 for n>1. (2) Use (1) to show H1dR(RPn)=0 (3) a n-form Ω is exact on Sn if and only if ∫SnΩ=0 (4) Use (3) to show that every smooth 2-form ω on RP2 has the form dα for some smooth 1-form α I know I … Read more

Compound map in manifolds

In the description of a manifold, we often start with the mathematical definition that M=∪Mi and if m∈Mi⊂M, where m is a point on the manifold, then it is mapped by a one-to-one map to Rn. Then another case where point m∈Mi∩Mj, then it is defined by an extra more map. Where I want to … Read more

Question about two definitions of Teichmuller space for a surface of genus gg

There are many equivalent definitions of Teichmuller space for a surface of genus g≥2. One of them concerns the complex structure: the Teichmuller space T(g) is the set of the couples (X,f) where X is a Riemann surface of genus g and f:S→X is the marking, i.e. an orientation preserving diffeomorphism up to the equivalence … Read more

Find the jacobian

I’m been struggling with the problem for a quite some time now. I need to find the jacobian for the following : u=x−y v=xy What I did : x=y+ux=vyy=x−uy=vx |dx/dudx/dvdy/dudy/dv| |11y−11x| 1x+1y But for some reason the answer say I should get : 1x+y The method they used in the book is to calculate J−1 … Read more

A closed surface in $\mathbb R^3$ has an elliptic point.

I know that compact set has maximum. My advisor said that solve this proposition by using the compactness property. I can’t understand what he said at all. Can anyone verify this proposition? Please teach me as easy as every undergraduate can understand. Answer Let $O \in \mathbb{R^3}$ be any point of the space and call … Read more

Unit sphere and Ricci curvature

Why is it that on the unit sphere the Ricci curvature Ric = g (where g is the metric defined on the unit sphere) ? Answer Here’s one way to view this: For any manifold, the Ricci curvature is an invariant of the metric, and so it must be invariant under the isometries (symmetries) of … Read more

On a scale of 1 to 10 how far is this manifold from being a normal bundle?

(DIS)-CLAIMER: All the manifolds considered in this post are completely humble and have no additional structure beyond the smooth structure. Let Y be a submanifold of M and let (−)0 denote the annihilator. I claim (and about to prove hopefully) that NY={(m,ω)∈T∗M:m∈Y,ω∈(TmY)0} can be given a natural structure as a submanifold of T∗M. (So before … Read more