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Results tagged with riemannian-geometry
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user 156492
Riemannian Geometry is a subfield of Differential Geometry, which specifically studies "Riemannian Manifolds", manifolds with "Riemannian Metrics", which means that they are equipped with continuous inner products.
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"
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20
votes
0
answers
2k
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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 …
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7
votes
Accepted
Ricci curvature : beyond heat-like flows
A small number of authors have considered hyperbolic versions of the standard flows, see e.g. "Wave character of metrics and hyperbolic geometric flow" by De-Xing Kong and Kefeng Liu and related artic …
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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 …
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4
votes
0
answers
122
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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 …
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4
votes
0
answers
812
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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 …
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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 …
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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 …
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2
votes
1
answer
180
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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 …
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1
vote
1
answer
158
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Smoothness of conformal transformations
Given a smooth pseudo-Riemannian manifold $(M,g)$ one can define the conformal group as the set of smooth diffeomorphisms $\varphi:M\to M$ such that there is a positive smooth function $u$ with $\varp …
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