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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### Are conformal maps between Riemannian manifolds real-analytic?

This is a cross-post.
Let $M,N$ be oriented smooth ($C^{\infty}$) $n$-dimensional Riemannian manifolds, and let $f:M \to N$ be a smooth orientation-preserving weakly* conformal map.
Do there ...

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### Ricci flow with surgery without the “no locally separating $\Bbb RP^2$” assumption

In many places, Ricci flow with surgery is done with orientable manifolds. Morgan and Tian do not require orientability, but instead they impose the condition that $M^3$ have no embedded $\Bbb RP^2$ ...

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### Measure preserving coordinates of $S^2$ from $[0,1]^2$

Consider the unit sphere $S^2 = \left\{x\in\mathbb{R}^3 ~ {\large|} ~ |x|=1 \right\}$ and denote the uniform (Lebesgue) measures on the $S^2$ and $[0,1]^2$ by $m_S$ and $m_2$, respectively.
Question ...

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### Principal curvatures of $\mathbb{R}^{n^2}$-embedded SO(n)

It's well known that the sectional curvatures of a Lie group, endowed with a left-invariant metric have a nice closed-form formula $k(X,Y) = \frac{1}{4} \|[X Y]\|^2$.
I'm wondering if the following (...

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### Gaussian curvature of conformal transformations

Let $g$ be a smooth metric on a $2$-dimensional manifold and $g'=e^{v}g$, where $v$ is also a smooth function. Then it is well-known that
$$(*) -\Delta_g v +2k_g=2 k_{g'}e^{v}, $$
where $k_g$ ...

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### DIstance on a Riemannian manifold [on hold]

Given a Riemannian manifold $(M,g)$ is it possible to calculate the distance between two points on this manifold. Is it possible the inverse? That means: given a formula of the distance, for example:
...

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### Diffeomorphism type of Ricci-flat four manifolds

Let $(M,g)$ be an irreducible compact and simply connected Ricci-flat Riemannian four-manifold. My first questions are as follows:
A) Is there a classification of the possible homeomorphism types of ...

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### One-parameter group of nonvanishing vector field

Let $M$ be a smooth manifold( if necessary one can assume it is closed), $V$ be a non-vanishing vector field of $TM$.
Q: Under what condition, we can say that $\overline{\exp(tV)}$, i.e. the closure ...

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### Eigenvalue lower bounds for manifold with positive Ricci curvature

For closed $n$-manifold with Ricci curvature $\ge (n-1)$, it is known that the first eigenvalue $\lambda_1\ge n$ with equality holds if and only if $M$ is isometric to the Euclidean sphere $S^n$. My ...

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### Nash embedding for complete manifold

I, ask my question as a comment in this post. Without answer I post a more detailed version.
I am looking for a reference about $C^\infty$ Nash isometric embedding for non compact manifold.
My ...

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### Is there an explanation of analogies between the cross-ratio and the Riemann curvature tensor?

Define the cross-ratio of four real or complex numbers as follows:
$$[a,b,c,d] = \frac{(a-c)(b-d)}{(a-d)(b-c)}.$$
Then its logarithm has the same symmetries as the curvature tensor:
$$\log[a,b,c,d] = -...

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### Metric of non-negative scalar curvature

Let $(M,g)$ be a closed Riemannian manifold, if $Scal^g>0$, we know that in a metric space of $M$, there is a neighborhood of $g$, such that all metrics in this neighborhood have the positive ...

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### Hyperbolic manifolds containing totally geodesic hypersurfaces which themselves contain totally geodesic hypersurfaces

I will preface this by saying that while I am familiar with the general theory of (semi)-Riemannian manifolds, I am a complete novice when it comes to the specifics of hyperbolic manifolds (I am using ...

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### Realizing the cross product of $\mathbb{R}^3$ as the curvature tensor of a Riemannian metric on $\mathbb{R}^3$

Is there a Riemannian metric on $\mathbb{R}^3$ for which the corresponding curvature tensor $R$ satisfies $R(X,Y)Z=(X\wedge Y)\wedge Z$?
I have already discussed this question in the following post ...

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### Property of distance function with “smoothly” varying Riemannian metrics

Let $(M,g)$ be a smooth and compact Riemannian manifold. Suppose I have a "smoothly varying" (precisely formulating this is part of the question) one parameter family of Riemannian metrics $(M,g_t)$ ...

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### Does harmonic map heat flow of a curve always fully converge to a geodesic?

Consider a smooth closed curve $u_0$ in a compact Riemannian manifold $(M,g)$. Let $u_0$ evolve by harmonic map heat flow, $\partial_tu=\nabla_{\partial_su}\partial_su$, and call the result $u(t)$.
...

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### Nash isometric embedding for noncompact manifolds

It seems that the smooth isometric embedding theorem by Nash is true also for noncompact manifolds.
Is it true that any (complete, connected) Riemannian manifold $(M^n,g)$ admits a proper smooth ...

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### Clarification needed on vector field conditions in Smale's “On gradient dynamical systems”

I previously posted the question on MSE but I haven't received an answer. I'm now posting it here in a slightly revised form.
In S. Smale's, “On gradient dynamical systems,” Ann. of Math. (2), vol. ...

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### Nash isometric embedding theorem with keeping the symplectic structures of our ambient spaces

I apologize in advance if this question has an obvious answer.
Let $(M,g)$ be a Riemannian manifold.
Then the tangent bundle $TM$ carries a natural symplectic structure $\omega_g$. In fact $\omega_g$...

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### Existence of harmonic symplectic structure on symplectic Riemannian manifold

This post is an expanded version of this MSE post.
Assume that $(M, \omega)$ is a symplectic manifold which is equiped with a Riemannian metric.
Is there a symplectic structure $\...

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### Is the evaluation map from harmonic forms on the torus surjective on flat neighbourhoods?

In a nutshell:
Given a metric on the torus $\mathbb{T}^n$, can we extend any element $\sigma \in \bigwedge^k T_p^*\mathbb{T}^n$ to a global harmonic form?
Let $\mathbb{T}^n$ be the $n$-Torus. Fix ...

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### Special spheres: principal curvatures with different signs

For $\varepsilon > 0$, we say that a closed, connected and oriented immersed hypersurface $M^n$ of a riemannian manifold $(N^{n+1},g)$ is $\varepsilon$-convex whenever all principal curvatures of $...

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### Radial symmetry of the first eigenfunction

Let $M$ be a simply connected space form (i.e. $\mathbb R^n$, sphere, or hyperbolic space) and $B$ be a ball in $M$. Let $\phi$ be the first Laplacian eigenfunction on $B$, with respect to the ...

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### Hilbert space separability for spectral triples

A spectral triple $({\cal A},{\cal H},D)$ consists of a unital $*$-algebra ${\cal A}$ represented as bounded operators on a Hilbert space ${\cal H}$, together with an unbounded operator $D$ having ...

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### Some tensors in differential geometry

If we have a vector fiber bundle with a connection $D_X=D(X)$ and an endomorphism $e$; we can then define a new tensor by the following formula: $$E(e)=D_X e D_Y + D_Y e D_X - e D_Y D_X - D_X D_Y e - ...

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### The heat kernel in Hermitian bundles over Riemannian manifolds

In "Heat Kernels and Dirac Operators" by Berline, Getzler and Vergne, the authors construct a (unique) heat kernel associated to any generalized Laplacian in a vector bundle over a Riemannian manifold....

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### Integrand in equivariant Atiyah-Patodi-Singer index theorem

I am trying to determine the integrand in the equivariant Atiyah-Patodi-Singer index theorem by Donnelli ("Eta invariants for G-spaces") for spin manifolds if one only has isolated fixed points. In "...

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### A possible characterization of sphere or projective space

Is there a compact Riemanian manifold $M$ not diffeomorphic to sphere or real or complex or quaternion projective space which admit a diffeomorphism $f$ with the property that $$\forall x \in M, \...

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### The truth about an an analogy between prime ideals and prime geodesics

I've just learnt in this Unique factorisation of prime geodesics? question that there is an analogy between prime geodesics and prime ideals in number fields. You can read more about it here: https://...

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### Embedding Riemannian manifolds into some infinite dimensional manifolds?

First of all I am new to the field of embedding one manifold into another other.
I have recently come across with the paper "Embedding Riemannian manifolds by their heat kernel" by P. BERARD, G. ...

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### Submanifolds whose second fundamental form has constant rank in every direction

Let $M$ be a submanifold of a Riemannian manifold $\widetilde{M}$. Let $A$ be the second fundamental form of $M$.
Suppose that, for all $p \in M$, the linear map $A(v, \cdot)\: \colon T_{p}M \to N_{...

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### Heat kernel on Riemannian manifold

The idea to construct a heat kernel is first construct a parametric in a small neighbourhood. Then use a bump function to extend it. And do convolution iteratively. (Reference: Laplacian on a ...

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### Complex surfaces not admitting nonnegative sectional curvature metrics

Five simply connected closed 4-manifolds are known to admit Riemannian metrics with nonnegative sectional curvature:
$$\mathbb{S}^4,\,\mathbb{C}\mathbb{P}^2,\,\mathbb{S}^2\times\mathbb{S}^2,\,\mathbb{...

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### Is the conformal compactification of a convex-cocompact hyperbolic manifold $M$ conformally diffeomorphic to the convex core of $M$?

I am not a hyperbolic geometer, so I apologize if I get anything wrong here, and please correct me.
The conformal compactification $\overline {\mathbb{H}^n}$ of hyperbolic $n$-space $\mathbb{H}^n$ ...

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### The determinant curvature

Let $(M,g)$ be a riemannian manifold and $R(X,Y)$ the riemannian curvature as a two form with values in the endomorphisms of the tangent bundle. I define:
$$
D_g(X,Y)=det(R)(X,Y)
$$
with $det$ the ...

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### Does $\Lambda^2_{+}$ generate a differential ideal for a self-dual $4$-manifold?

Let $(X, g)$ be a smooth, oriented, Riemannian 4-diemnsional manifold. Let $\Lambda^2$ denote the bundle of 2-forms on $X$. Then the Hodge-star operator decomposes $\Lambda^2$ into the space of self-...

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### Does every manifold admit a Lagrangian Riemannian metric?

Let $(M,g)$ be a Riemannian manifold. The $LC$ connection associated to the metric gives an $n$ dimensional distribution $D$ for $TM$. Let $\omega$ be the symplectic structure of $TM$ which is ...

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### Elliptic regularity of harmonic forms in $L^1$

$\newcommand{\M}{M}$
This is a cross-post. I am looking for a reference for the regularity of harmonic forms which belong to $L^1(M)$.
Explicitly, let $\M$ be a smooth oriented Riemannian manifold.
...

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### How restrictive is $[\mathcal H,\mathcal V] =0$, or weaker, $[X,A_XY] = 0$?

Let $\pi :P \to M$ be a Riemannian submersion. Let us denote by $\mathcal H$ the horizontal space of $\pi$ and by $\mathcal V$ its vertical space. I want to know restrictions and consequences of the ...

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### Does the holomorphic curvature determine the connection?

Let $M$ be a simply connected complex manifold (of dimension greater than one), $L$ a line bundle, and $\nabla$ a connection on $L$ with possibly singularities along a divisor $D$. We define the ...

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### existence of riemannian metric on $\text{SL}_3(\mathbb{R})$ with special geodesics

Is there a left-invariant Riemannian metric on $\text{SL}_3(\mathbb{R})$ for which the geodesics (with respect to the corresponding Levi-Civita connection) through the identity are exactly the ...

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### Given a local metric which is $C^1$-close to another, can we extend it globally while preserving the approximation?

Let $M$ be a smooth closed manifold, and let $g_0$ be a Riemannian metric on $M$.
Let $U$ be a neighbourhood of $p \in M$, and suppose that we are given a metric $g$ on $U$, which satisfies $\| g-...

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### Definition of first normal space

Given an immersed submanifold $M$ of a Riemannian manifold $\overline{M}$, the first normal space of $M$ at a point $p \in M$ is defined as the linear subspace $N_{p}^{1}M$ of $N_{p}M$ spanned by the ...

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### Is every “higher-order” harmonic morphism conformal?

$\newcommand{\TM}{\operatorname{TM}}$
$\newcommand{\M}{\mathcal{M}}$
$\newcommand{\N}{\mathcal{N}}$
$\newcommand{\TM}{\operatorname{T\M}}$
$\newcommand{\TN}{\operatorname{T\N}}$
$\newcommand{\TstarM}{...

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### non-existence of global coordinates

Assume we have a smooth manifold, $M$, of dimension $n$. (An example of interest is the case when $M$ is a compact and orientable Riemann surface of genus $g$, but the question is intended to be broad....

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### Bishop-Gromov for Kähler metrics

Let $(M, g)$ be a (complete) Kähler manifold with Ricci curvature $\geq c$.
Is it true that the volume ratio of geodesic balls in $M$ with respect to balls in the corresponding (simply connected) ...

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### When can the metric be reconstructed (up to scaling) from knowing the conjugate points?

Let $M$ be a smooth manifold of dimension $\geq 2$. Let $g$ be a complete Riemannian metric on $M$. Let $C \subseteq M \times M$ be the set of pairs of $g$-conjugate points.
The set $C$ doesn't ...

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### Does the space of harmonic forms change continuously with the metric?

Let $(M,g_0)$ be a closed $n$-dimensional Riemannian manifold. Let $1<k<n$ be fixed, and let $\Delta_{g_0}:\Omega^k(M) \to \Omega^k(M)$ be the $g_0$-Laplacian. Let $H^k_{g_0}=\text{ker} \Delta_{...

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### Immersions of the hyperbolic plane

Is it possible to isometrically immerse the hyperbolic plane into a compact Riemannian manifold as a totally geodesic submanifold? Any nice examples?
Edit: Although I did not originally say so, I was ...

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### Flatness in a neighborhood of a point condition

Suppose that we have a Riemannian Manifold $(M,g)$ whose
curvature vanishes in an open neighborhood U of a point p.
When does this imply that the metric is Flat ?
In particular, does it happen ...