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A Riemannian manifold $(M, g)$ has harmonic Weyl curvature iff its Schouten tensor is Codazzi, and if there exists $f: M \to \mathbb{R}$ such that $W(\bullet, \bullet, \bullet, \nabla f) = 0$, one say …

It is known that the existence of a function with hessian proportional to the metric implies that the metric is a warped product metric. Is the reciprocal true as well? I.e, if $(B \times N, g = g_B + …

In this paper, the authors characterize warped product metrics which are conformally flat (the fibers must have constant sectional curvature, on some cases there is a limitation on the number of possi …

Let $S$ be a symmetric $(0, 2)$ tensor on a Riemannian manifold $M$. Define $E_S : M \to \mathbb{Z}$ by $E_S(x) = \left(\text{the number of distinct eigenvalues of } S_x\right)$. I've seen the followi …

A Riemannian manifold $(M, g)$ is said to be an almost Ricci soliton if there exists a complete vector field $X \in \Gamma(TM)$ and a smooth function $\lambda: M \to \mathbb{R}$ such that $$\operatorn …

In dimension $4$, it is known that the curvature operator $\operatorname{Rm} : \Lambda^2(M) \to \Lambda^2(M)$ admits a block decomposition of the form $$\operatorname{Rm} = \begin{pmatrix} A & B \\ B^ …

After a lot more time thinking about it, I think I've figured it out. Let $\Gamma$ be a constant such that $ \Gamma \|g \odot g \| = 1$ (where $\odot$ denotes the Kulkarni Nomizu product, and the only …

In "The Ricci Flow: An Introduction", the authors define a $\delta$-necklike point of the Ricci flow as a point $(x, t)$ where $$\|\text{Rm} - R (\theta \otimes \theta)\| \leq \delta \|\text{Rm}\|$$ f …

In B. Chow and D. Knopf's book "The Ricci Flow: An Introduction", the authors claim that for any dimension $n$ and any Riemannian manifold $M^n$, there is a constant $C_n$ depending only on $n$ such t …

Context: A solution $(M^n, g(t))$ of the Ricci flow is said to encounter a Type III Singularity if $g(t)$ is defined for all $t \geq 0$ and: $$ \sup _{\mathcal{M}^{n} \times[0, \infty)} \|\operatorna …

Are there any known examples of Einstein manifolds $(M, g)$ such that $$\sup_{x \in M} \|\text{Rm}(x) \| = \infty$$ I'm looking for these examples because they might provide a counter-example to a pro …

I'm trying to prove the following identities (under the normalized Ricci flow on surfaces, on which $\partial_t g = (r-R)g$ holds true, where $r$ denotes the average scalar curvature and has the same …

I know studying the mean curvature flow is a very interesting area of research, I've fooled around with it a bit myself. But it honestly doesn't look like it has much applications within mathematics i …

In a couple articles I've read lately, I've seen it mentioned that the cigar soliton has linear volume growth. What does this mean? I thought maybe, if you compute the volume of geodesic balls and inc …

Let $\alpha: [0, L] \to \mathbb{R}^2$ be smooth and such that $\langle\alpha(L) - \alpha(0), (0,1) \rangle = 0$, $\alpha(0) \neq \alpha(L)$ and $\alpha'(0) = k\alpha'(L)$ for some $k$. We know that $ …