1
vote
0answers
39 views

Why is the matrix of a Riemannian metric positive definie?

Maybe I could post this as a linear algebra problem but I'll give some context. I know that if $(U, x_1, \ldots, x_n)$ is a local chart of a smooth manifold $M$ I can write locally a Riemannian ...
2
votes
1answer
75 views

relative sign in Hodge star of tensor product

Let $V$ be a vector space of arbitrary (finite) dimension and let $(V, \langle \ ,\ \rangle, I) = (W_1, \langle\ ,\ \rangle_1, I_1) \oplus (W_2, \langle\ ,\ \rangle_2, I_2)$ be a direct sum ...
2
votes
0answers
36 views

Why Riemannian metrics defining the same angles are conformal [duplicate]

Suppose $g_1$ and $g_2$ are two metrics defining the same angles, which means $g_1(X,Y)/(g_1(X,X)g_1(Y,Y))^{0.5}=g_2(X,Y)/(g_2(X,X)g_2(Y,Y))^{0.5}$ for all pairs of vector $X,Y$.I want to prove that ...
1
vote
1answer
43 views

Local coordinates for two riemannian metrics

Let $(M,g)$ be a Riemannian manifold, $g' = g + f$ be another metric. Is it possible to get local coordinates such that at a point $P \in M$, $g_{ij} = \delta_{ij}$ and $f_{ij} = 0$ for all $i \not = ...
3
votes
1answer
65 views

Block Decomposition of a linear map on $\Lambda^2TM$

I'm trying a exercise from Peter Petersen's book, and I did the following: Let $*$ be the Hodge star operator, I know that $\Lambda^2TM$ decompose into $+1$ and $-1$ eigenspaces $\Lambda^+TM$ and ...
7
votes
1answer
319 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 ...
0
votes
1answer
573 views

Vector perpendicular to timelike vector must be spacelike?

Given $\mathbb{R}^4$, we define the Minkowski inner product on it by $$ \langle v,w \rangle = -v_1w_1 + v_2w_2 + v_3w_3 + v_4w_4$$ We say a vector is spacelike if $ \langle v,v\rangle >0 $, and it ...
3
votes
0answers
120 views

Curvature of particular Riemannian metric

Let $U = \{ (x_1, \dots, x_n) \mid x_j > 0 \text{ for all } j\}$ and let $\|x\|^2 = \sum_j x_j^2$. The function $x \mapsto -\log \|x\|^2$ is strictly convex on $U$ and thus defines a Riemannian ...
7
votes
1answer
166 views

A scalar product in the space of oriented volumes?

Let $L\colon \mathbb{R}^n \to \mathbb{R}^N$ be an injective linear map. By the Cauchy-Binet formula, $\det(L^TL)$ equals the sum of the squares of all minors of $L$ of order $n$: this looks just like ...
1
vote
1answer
89 views

Complexification of Metric

If we have given an inner product space $(V,g)$, where $V$ is vector space and $g$ is inner product. What will be corresponding bi-linear form $g'$ on $C\otimes V$.
14
votes
1answer
207 views

Can a continuous map $S^2 \rightarrow S^2$ preserve orthogonality without being an isometry?

Suppose I have a map $\phi: S^2 \rightarrow S^2$ and I know that a) $\phi$ is continuous and bijective b) If $a$ and $b$ subtend an angle of $\pi / 2$ at the center of the sphere, then so do ...