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Results tagged with dg.differential-geometry
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user 156492
Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.
21
votes
Accepted
Riemannian manifold as a metric space
Isn't this the Myers-Steenrod theorem? "If $(M,g)$ and $(N,h)$ are connected Riemannian manifolds and $f:(M,d_g)\to(N,d_h)$ is an isometry, then $f:(M,g)\to(N,h)$ is a smooth isometry"
20
votes
0
answers
2k
views
Schoen and Yau's proof of the higher dimensional positive mass theorem
In April 2017 Schoen and Yau posted on the arxiv their solution of the time-symmetric positive mass theorem in all dimensions, which has been a significant conjecture since the 70s. As of now, July 20 …
17
votes
Accepted
Gromov's articles suitable for master students
I agree with Alexandre Eremenko in that most can be hard to read. But I think you can get a lot out of trying to understand them, as long as you're willing to black-box certain parts which may be inac …
10
votes
0
answers
408
views
Gromov's compactness theorem via Sacks-Uhlenbeck and Schoen-Uhlenbeck
Gromov's compactness theorem for pseudo-holomorphic curves, in section 1.5 of "Pseudoholomorphic curves in symplectic manifolds," is very well known. I'm aware of the following two proofs:
J.G. Wolfs …
10
votes
Accepted
Geometric definition of divergence using curvature mentioned in Tristan Needham
Given any hypersurface of a Riemannian manifold with a unit normal vector field $\nu$, extend $\nu$ to be unit length. Then
$$\operatorname{div}(u\nu)=u\operatorname{div}\nu+\text{d}u(\nu).$$
This pro …
10
votes
1
answer
660
views
Mean curvature flow and knot theory
I am wondering if the mean curvature flow of one-dimensional submanifolds of $\mathbb{R}^3$ is understood well enough to give some perspective on (and hopefully a proof of) something like the Fary-Mil …
6
votes
Proving an identity used in general relativity
This is the differential form of the Reilly formula. It holds for a function on any pseudo-Riemannian manifold. (Robert C. Reilly. Applications of the Hessian operator in a Riemannian manifold, Indian …
5
votes
0
answers
291
views
Chern-Weil theory in the cohomological Atiyah-Singer theorem
I am interested in the following piece of data appearing in the cohomological Atiyah-Singer theorem. My reference is "The index of elliptic operators. III" by Atiyah and Singer.
Let $D:\Gamma(E)\to\Ga …
5
votes
1
answer
222
views
Functions which are periodic along every geodesic
In an effort to understand some geometric rigidity theorems, I am curious about the following: let $(M,g)$ be a complete Riemannian manifold and suppose there is a nonconstant real-valued function on …
4
votes
0
answers
122
views
Umbilic points of minimal hypersurfaces and distributional Simons inequality
Let $\Sigma$ be a minimal hypersurface of a smooth Riemannian manifold $(M,g)$ with second fundamental form $h$. What can one say about the set $\{p\in\Sigma:h(p)=0\}$? Is each point isolated? (I feel …
4
votes
0
answers
812
views
History of Laplacian comparison theorem
The Laplacian comparison theorem says that if a $n$-dimensional Riemannian manifold has nonnegative Ricci curvature, then the distance function to any point satisfies $\Delta d\leq\frac{n-1}{d}$. Ther …
4
votes
Usage/Application of Raychaudhuri equation in Riemann geometry or pure maths
For any 1-form $\omega$ on a pseudo-Riemannian manifold, the product rule and commutation identity say that
$$\omega^p\nabla_p\nabla_q\omega^q-\nabla^q(\omega^p\nabla_p\omega_q)=-\omega^p\omega^qR_{pq …
3
votes
0
answers
71
views
Lorentzian cobordism through the dominant energy condition
Is the answer to the following problem, or some close variant thereof, known? Briefly:
Given two initial data sets $I_1=(M,g_1,k_1)$ and $I_2=(M,g_2,k_2)$, is there a time-oriented spacetime satisfyi …
3
votes
0
answers
523
views
Two possible meanings of "totally real" submanifold
It seems that there are two common meanings for a submanifold of an almost-complex Riemannnian manifold to be "totally real": one says that the almost-complex structure takes the tangent space into th …
2
votes
1
answer
180
views
Vector field along an immersion whose covariant derivative is the differential
Let $(M,g)$ be a Riemannnian manifold and let $f:\Sigma\to M$ be a smooth immersion. Then the vector bundle $f^\ast TM\to\Sigma$ has a natural bundle metric and metric-compatible connection. Can one c …