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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### Does an 8 dimensional compact Riemannian manifold contain an embedded minimal hypersurface?

It is well know that if $(M^{n+1},g)$ is a compact Riemannian manifold and $n \leq 6$ then there exists a smooth, embedded minimal hypersurface $\Sigma^n$ in $M$ (infinitely many such $\Sigma$, even)
...

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### Problem about the homology groups of the complex projective space $\mathbb{C}P_n$ [on hold]

My question is how we can compute the homology groups of the complex projective space $\mathbb{C}P_n$ by the following Corollary5.4 at page 31 in Milnor's book：
Corollary5.4
If $c_{\lambda+1}=c_{\...

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### A geometric rank of Riemannian manifolds

There are various ranks which have been assigned to a smooth manifold. The following ranks are two examples of such ranks:
The maximum number of global independent vector fields which can be defined ...

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### Jacobi equation in saddle Finsler surface

Setting : Consider a two dimensional surface in $ (\mathbb{R}^n,\|\ \|)$.
Here we define a function $f: \mathbb{R}^n\rightarrow
\mathbb{R}^n$ s.t. $L(v)(X)=\langle f(v),X\rangle$ where $\langle\ ,\...

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### Convexity of curves in Riemannian surfaces

It is known that a curve $f:[0,2\pi]\to \mathbf{R}^2$ is convex if $\partial_t (\arg f'(t))\ge 0$. My question is: does this statement have an analogue in the setting of Riemannian surfaces instead of ...

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### geodesic balls in the conformal change

Consider a compact smooth Riemannian manifold $(\mathcal{M}, g)$. We consider a conformal change of the metric. Let $\tilde{g}=\phi g$, where $\phi$ is smooth and positive. Moreover, it satisfies $...

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### Sectional curvatures under simple maps

Suppose that we have a submanifold $X$ of $\mathbb{R}^n$ with the induced Euclidean metric, whose sectional curvatures we have a handle of (say, they are lower bounded by some $\kappa$).
Is there a ...

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### When $\int_M \exp(-d_M(x,y)^2/t) dvol(y)$ becomes constant for a Riemannian manifold $M$?

Let $(M,g)$ be a closed and connected Riemannian manifold. $d_M$ is its geodesic metric and $dvol_M$ is its standard volume measure. For each $t>0$, define a map $f:M\rightarrow\mathbb{R}_{>0}$ ...

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### Log-concavity of areas of level sets

Suppose $f: \mathbb{R}^d \to \mathbb{R}$ is a smooth convex function.
Consider the level sets of the function, namely $M_s = \{x: f(x) = s\}$.
Is it true/known that the surface areas of $M_s$ are ...

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### Solutions to $\Delta u\ge u^2$

Let $(M,g)$ be a complete Riemannian manifold. Suppose that $u$ is a nonnegative solution to $\Delta_gu\ge u^2$. Does it follow that $u$ must be identically 0?
I know that the answer to above ...

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### Polynomial structure on differentiable manifold [closed]

I am a research scholar. My topic of research is
$ \text {Polynomial structure on differentiable manifold}$.
I want to know some good books and good sources on this topics where I can get help. ...

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### Master thesis about Myers-Steenrod theorem

I hope here is the best place to ask this, anyway this year I have to write a master thesis in pure mathematics and since I'm interested in geometry and topology normally I will look for a topic about ...

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### Some hypersurface has a positive second fundamental form potentially

Notation : $r^2=x^2+y^2$.
Exercise : Define $$F_\sigma (x,y)= (f_\sigma
(x,y),\frac{x^2-y^2}{2},xy)$$ where $$f_\sigma (x,y)=(1- \sigma^2
r^2)(x-\sigma x^3,y-\sigma y^3)$$
Define $ G_\sigma: \mathbb{...

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### Why might the Lawson minimal surface $\xi_{1,2}$ have a Morse index 9?

In page 15 of the article New Applications of Min-max Theory, Andre Neves said that: a wishful thinking would suggest the Lawson minimal surface $\xi_{1,2}$ in the 3-sphere to have a Morse index 9, ...

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### Weak elliptic maximum principle on manifolds without strict ellipticity

This question is not to be confused with the similarly titled question here.
In the above lined question, I gave a complete answer, but noticed that things are apparently not so simple in the ...