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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.

3 votes
0 answers
488 views

kernel of the conformal Laplacian

Let $M$ be a smooth, closed manifold of dimension $n>2$. Let $L_g$ be the conformal Laplacian of the metric $g$. That is, $L_g=-\Delta_g + \frac{n-2}{4(n-1)}R_g$, where $R_g$ is the scalar curvature o …
Viktor Bundle's user avatar
1 vote
0 answers
271 views

Bound on $Q$ implies bound on $R$?

Let $(M,g)$ be a smooth, closed Riemannian manifold with dimension $n>4$. Define the $Q$-curvature through the formula $Q = \Delta R + \frac{n^3-4n^2+16n-16}{(n-1)(n-2)^2} R^2 - \frac{8(n-1)}{(n-2)^2 …
Viktor Bundle's user avatar
0 votes

Estimate on the size of flat balls where the Weyl tensor vanishes

I believe that the answer to the first question is the injectivity radius if the manifold is locally conformally flat. The injectivity radius provides us with an estimate on the size of a chart that c …
Viktor Bundle's user avatar
0 votes
2 answers
237 views

Estimate on the size of flat balls where the Weyl tensor vanishes

Let $(M,g)$ be a Riemannian manifold and suppose that the Weyl tensor of $g$ vanishes at a point $p \in M$. Can one estimate the size of the largest geodesic ball around $p$ that we can make $g$ flat …
Viktor Bundle's user avatar
12 votes
2 answers
1k views

Obstructions to Einstein metrics in high dimensions

It is well known that there exists three and four manifolds that do not admit an Einstein metric, but I wonder if this question is still open for manifolds of dimension higher than four. That is, does …
Viktor Bundle's user avatar
15 votes
1 answer
2k views

Geometric Interpretation of $Q$-curvature

Let $(M,g)$ be a Riemannian manifold of dimension $n>2$. Thanks to the late T.Branson we have the following definition of the so-called $Q$-curvature: $Q= \Delta R + \frac{n^3-4n^2+16n-16}{4(n-1)(n-2 …
Viktor Bundle's user avatar
10 votes
2 answers
1k views

Einstein metrics and conformal geometry

I recall reading somewhere that if a conformal class contains an Einstein metric then that metric is the unique metric with constant scalar curvature in its conformal class, with the exception of the …
Viktor Bundle's user avatar
3 votes

Some questions about scalar curvature

Also: when one is talking about positive scalar curvature, the Yamabe invariant is important. The Yamabe invariant is the supremum over all conformal classes of the Yamabe constants of a manifold. The …
Viktor Bundle's user avatar
4 votes
0 answers
343 views

Constant scalar curvature+Constant $\sigma_2(C_g)$ curvature = ?

Let $(M,g)$ be a closed, smooth, Riemannian manifold of dimension $n>4$. Suppose both the scalar curvature and norm of the Ricci tensor are constant. In addition suppose that $g$ satisfies the followi …
Viktor Bundle's user avatar
1 vote

Positivity of Second-Order Elliptic Differential Operators

I believe the answer is no if $n>2$. Let $g$ be a metric with a negative Yamabe constant. There will be a metric $h$ in the conformal class of $g$ such that $\int_M R_h dv_g> 0$. Let $L_h$ be the conf …
Viktor Bundle's user avatar
2 votes
2 answers
506 views

Positivity of Second-Order Elliptic Differential Operators

Let $(M,g)$ be a closed, smooth Riemannian manifold. Let $\Delta = -div\nabla$ be the Laplace-Beltrami operator. Let $h$ be a smooth function on $M$. Is there a condition on $h$ weaker than non-negati …
Viktor Bundle's user avatar
7 votes

Riemannian manifolds that are scalar flat but not Ricci flat

To generalize Anton's comment a little, I should add that with the appropriate choice of $l$ and $k$, the product manifold $S^l \times N^k$ will have the property that you are looking for, where $N^k$ …
Viktor Bundle's user avatar
3 votes
0 answers
234 views

Negative Paneitz constant on $n$-sphere

Let $\Pi$ be the Riemannian functional defined on the space of Riemannian metrics on $S^n$, $n>4$, as follows: $$ \Pi(g) = \int_M \frac{(n-4)(n^3-4n^2+16n-16)}{16(n-1)^2(n-2)^2} R_g^2 - \frac{2(n-4)}{ …
Viktor Bundle's user avatar
27 votes
0 answers
3k views

Ricci flat metric on $n$-sphere?

Can you put a Ricci flat metric on the $n$-sphere, $n>4$?
Viktor Bundle's user avatar