# Tagged Questions

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.

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### Ricci flow and isometry group

It is known (via Kotschwar's uniqueness of backwards Ricci flows) that the isometry group of a Riemannian metric remains unchanged under the Ricci flow. But, one can easily observe that it can change ...
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### List of generic properties of Riemannian metrics

I am highly interested in compiling a list of generic properties of Riemannian metrics on a (may be compact) manifold in general, or under "relatively broad" assumptions, like generic properties of ...
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### Nearly length minimizing paths are close to geodesics? [closed]

Let $M$ be a Riemannian manifold which is geodesically convex. It's known that length minimizing curves are geodesics (after a possible reparametrization). Now fix* points $p,q \in M$ Is the ...
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### Faster (than normal) convergence of the normalized Ricci flow on surfaces

Consider a compact surface $M$ of genus $\gamma > 1$ (I am using the more usual letter "$g$" to denote metric), and the normalized Ricci flow on it. It is known that at time $t$, the scalar ...
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### Convex embedding with a positivity condition

We have a $n$-dimensional hypersurface $\Sigma$ embedded in the Euclidean $(n+1)$-space $\mathbb{R}^{n+1}$. We know that $\Sigma$ is compact without boundary, convex (not necessarly strictly convex), ...
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### ricci flow on surfaces

In Hamiltons paper "Ricci flow on surfaces" there is an estimate on $|\nabla R|^2$ which shows that $|\nabla R|^2 \leq C_1 \exp{\frac{rt}{2}}$ for some constant $C_1$. Actually for any solution of the ...
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### Is a manifold generically real analytic (with generic real analytic metric)?

I have heard it said in some differential geometry talks that "the generic situation in such and such case is real analytic". My question is, is the generic smooth manifold also real analytic in some ...
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### Calculating the Riemann Christoffel tensor for a diagonal metric

I am trying to calculate the entries of the Riemann curvature tensor $R^m_{\phantom{m}ijk}$ for the metric $g_{ij}$. The Riemann-Christoffel tensor is given as \begin{align} R^m_{\phantom{m}ijk} = \...
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### Flow on invariant Lagrangian tori

The most concrete version of the question is : A (necessarily) invariant Lagrangian torus $L$ on the unit cotangent of a Riemannian metric on the two-torus carries a periodic orbit with period $T$. ...
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### Does this PDE only have the trivial solution?

Let $(M,g)$ be a closed Einstein manifold of dimension $m>2$ and $$\mathrm{Ricc}(g)=\lambda g,$$ $h$ a symmetric $2$-covariant tensor, $\Delta=\nabla^*\nabla$ the Laplacian on functions as well ...
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### local definability of geodesics in an o-minimal structure

Let $R$ be an o-minimal expansion of the reals, and let $(M,g)$ be a Riemannian manifold, such that $M$ and $g$ are definable in $R$. Let $\gamma: [0,1] \to M$ be a geodesic, i.e. a curve such that ...
Let $X$ be a finite dimensional (possibly compact) Alexandrov space with curvature $\geq K$. Is it true that its boundary is again Alexandrov space with curvature bounded from below? If yes, is the ...