Tagged Questions
2
votes
0answers
74 views
left-invariant n-form and metric on a Lie group
These two questions are from my exam practice question sets , which are quite similar. I got some problem understanding and solving both of them .
For (a) , I can only substite $dx\wedge dy\wedge ...
3
votes
1answer
75 views
What is the norm of the gradient of $f$ in normal coordinate?
Let $M$ be a Riemannian manifold and $f$ a smooth function on $M$. The Bochner formula proved in Schoen-Yau's book "lectures in Differential Geometry": (prop. 2.2)
$$
\Delta |\nabla f|^2(p)=2\sum ...
3
votes
1answer
109 views
Relationship beween Ricci curvature and sectional curvature
Let $(M,g)$ be a Riemannian manifold and assume that for all orthonormal $v,z$ the sectional curvatures is bounded from below i.e. $K(v,z) \geq C$, where $C > 0$. Is it in this case possible for ...
1
vote
1answer
62 views
Continuity of the orthogonal projection into tangent space.
Let $\mathcal M \subset \mathbb R^d$ be a smooth manifold, and for each $s \in \mathcal M$ let $T_s[\mathcal M]$ denote the tangent space of $\mathcal M$ at $s$. For each $s \in \mathcal M$ let $P_s$ ...
4
votes
1answer
133 views
The set of diffeomorphisms preserving some metric.
Let $M$ be a finite-dimensional, smooth manifold. Call a diffeomorphism $f : M \rightarrow M$ diagonalizable if there exists a Riemannian metric $g$ on $M$ such that $f : (M, g) \rightarrow (M, g)$ is ...
