New answers tagged riemannian-geometry
4
votes
Systems of (hyperbolic) 2nd order PDEs with lower order constraints
Yes, there is a standard procedure to analyze such systems, essentially, it is Cartan's method of prolongation combined with his theory of involutive systems. There are other approaches as well, but ...
0
votes
Gradient flows: evolution of geodesics
As currently asked the answer is NO, because your desired upper bound already fails for $t=0$ (or equivalently, $t=1$). Indeed, it is well understood that the small-time deviation along the heat flow, ...
3
votes
Accepted
Does a Riemannian submersion map horizontal geodesics to geodesics, and a relevant question?
Yes, this is true.
There is a DG-style proof, but let me do it metrically.
(The following proof works only in the Riemannian world; the DG-style proof should work in pseudo-Riemannian as well.)
Choose ...
2
votes
Accepted
Handling degenerate planes in pseudo-Riemannian geometry: impact on sectional curvature and comparison theorems
How should this situation be interpreted?
Why should there be any interpretation? The geometric intuition behind sectional curvature in the Riemannian setting is this: given a plane $\Pi$ in $T_pM$, ...
4
votes
Accepted
For a closed Riemannian manifold $M$, must the set of points with non-unique closest points to a closed submanifold $S$ of $M$ be of 0 volume measure?
My answers:
yes
does not apply, see 1.
I did not think sufficiently about it in order to make a precise statement, but I would guess, the answer is positive as well, probably with the same proof as ...
5
votes
Frobenius theorem and the size of integral manifold
Your equations are equivalent to the $1$-form equations
$$
\mathrm{d}f = X_0(f,g)\,\mathrm{d}s + Y_0(f,g)\,\mathrm{d}t
\quad \text{and}\quad
\mathrm{d}g = X_1(f,g)\,\mathrm{d}s + Y_1(f,g)\,\mathrm{...
5
votes
Accepted
Bochner Laplacian in coordinates
Example 10.1.32 (which starts on page 456) does not consider $\nabla$ the Levi-Civita for a Riemannian metric. It is considering a general vector bundle $E$ equipped with a Hermitian metric $\langle,\...
3
votes
Tangent bundle of a tensor product bundle
The other answer is very helpful, but I believe it has some subtle problems. Let me expand on a few of the details because I think they can be confusing.
The tangent bundle $T E$ of a vector bundle $p ...
7
votes
Elliptic regularity on manifolds: Is this true?
Using a partition of unity as pointed out by Deane Yang you can reduce the problem to the local coordiante system. Then you can reduce it to an elliptic equation on a torus and you can prove ...
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