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Results tagged with dg.differential-geometry
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user 504191
Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.
2
votes
0
answers
86
views
Measurability of the union of cut loci along a curve
Let $(M,g)$ be a Riemannian symmetric space and $\alpha(s)$ be a geodesic. Define
$$
U(t)=\cup_{s\in[0,t]}{\rm Cut}(\alpha(s))
$$
as the union of the cut loci ${\rm Cut}(\alpha(s))$ along the curve $\ …
- 125
1
vote
1
answer
157
views
Differentiability of an integral of geodesic distance
Let $(M,g)$ be an $m$-dimensional Riemannian symmetric space and $d(\cdot,\cdot)$ be the geodesic distance function. Fix any $\alpha\in M$ and $v\in T_\alpha M$ with $\|v\|=1$.
Q1: Define
$$
g(t)=\int …
- 125
1
vote
1
answer
101
views
Contraction and consensus on Hadamard manifolds
Let $\mathcal M$ be a Hadamard manifold and $\{x_i\}_{i=1}^n\subseteq{\cal M}$ be $n$ points. Define $\{y_i\}_{i=1}^n$ as the weighted Fréchet means:
$$
y_i=\arg\min_{y\in\mathcal M}\sum_jw_{ij}d^2(y, …
- 125
0
votes
Contraction and consensus on Hadamard manifolds
I have solved this question. One can refer to Lemma 3 in the paper Decentralized Online Riemannian Optimization with Dynamic Environments. arXiv:2410.05128v1
- 125