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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 ...
Robert Bryant's user avatar
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, ...
leo monsaingeon's user avatar
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 ...
Anton Petrunin's user avatar
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 $T_pM$, ...
Willie Wong's user avatar
  • 37.4k
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 ...
Bernd Ammann's user avatar
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{...
Robert Bryant's user avatar
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,\...
Willie Wong's user avatar
  • 37.4k
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 ...
Keeley Hoek's user avatar
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 ...
Piotr Hajlasz's user avatar

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