**3**

votes

**0**answers

68 views

### Umbilic points on Euclidean hypersurfaces

Every smooth embedding of $S^2$ into $\mathbb{R}^3$ has at least one umbilic point (in fact, the recent proof of the Caratheodory conjecture yields two such points). The usual proof of this is to use ...

**1**

vote

**1**answer

127 views

### Zero set of eigenfunction along a sub manifold

Let $M$ be a 2-dimensional closed Riemannian manifold and let $$\phi:M\rightarrow M$$ be an isometry with $\phi^2=Id_M$. Consider the fixed point set $$F:=\lbrace x\in M: \phi(x)=x \rbrace\subset M,$$ ...

**0**

votes

**0**answers

95 views

### Some Confusion on Harmonic Map

I am stuck at Corollary $(3.4)$ on page $22$ in monograph Selected Topics in Harmonic Maps written by James Eells and Luc Lemaire.
Corollary Let $\phi:M,g\to N,h$ be harmonic and suppose $\text{Ricci}...

**6**

votes

**2**answers

206 views

### Epsilon regularity for minimal surfaces in arbitrary Riemannian manifolds

For experts in the analysis of minimal surfaces I will state the question first; then I will follow up with details.
Question: Does the $\varepsilon$-regularity theorem of Choi and Schoen (http://...

**3**

votes

**2**answers

166 views

### Examples of two-dimensional Riemannian manifolds that can't be isometrically embedded into $\mathbb{R}^4$

Can anyone give some examples of two-dimensional Riemannian manifolds $(M,g)$ that can't be isometrically embedded into $\mathbb{R}^4$? (Further more Globally)
What if it is smooth?

**2**

votes

**0**answers

70 views

### Does the tensor bundle of a compact manifold have a bounded geometry?

Let $M$ be a compact manifold. Let $S^2 T^*M $ be the vector bundle of all symmetric $(0,2)$ tensors and $S_+^2 T^*M$ be the open subset of all positive definite ones. Does $S_+^2 T^*M $ have bounded ...

**0**

votes

**0**answers

75 views

### Generic Identification of surfaces in a Riemannian Manifold

Let $(M,g)$ be a three dimensional Riemannian Manifold. Can we generically identify a class of embedded surfaces $\Gamma$ in $M$ such that the following property holds:
There exists a distance ...

**-1**

votes

**1**answer

113 views

### Construction of fibration over Riemannian Manifold

Let $\pi: E \rightarrow B$ be a fibration over a Riemannian manifold $B$, with $\pi^{-1}[b]$ homeomorphic to $\mathbb{R}$.
More precisely:
I want each fiber $\pi^{-1}[b]=Im(f_b)$ for some $C^{\...

**1**

vote

**0**answers

92 views

### Intuitive understanding of the mean curvature flow [closed]

I am trying to develop some intuition into the properties of mean curvature flow of a surface in $\mathbb{R}^3$. As an example, I am trying to understand what happens to a surface of revolution $S = (...

**2**

votes

**0**answers

93 views

### Darboux-like coordinates on a Kähler manifold

If $(M, g, J, \Omega)$ is a Kähler manifold, do there exist local coordinates in which 2 out of the 3 geometrical structures look nice? I have Darboux coordinates in which $\Omega$ looks nice, but ...

**2**

votes

**0**answers

58 views

### Christoffel symbols of a moduli of smooth curves

The Setting:
Let $H$ be the Hilbert space of all class $C^k$-curves into $\mathbb{R}$ with inner product: \begin{equation}
<f,g>:=\int_{\mathbb{R}} f'(x)g'(x) e^{-x}dx
\end{equation}
...

**9**

votes

**1**answer

826 views

### Solutions of equations characterizing a complex structure

Let $(S^n,g)$ denote the unit $n$-sphere endowed with its induced metric $g$ from its embedding into $\mathbb{R}^{n+1}$. The Levi-Civita connection of $g$ induces a splitting of the tangent bundle of $...

**5**

votes

**0**answers

107 views

### Some questions on the nodal geometry of Dirac operators

Let me begin by quoting a well-known result of Christian Baer (see here). The result goes as follows:
Theorem (Baer): Consider a connected $n$-dimensional Riemannian manifold with Dirac bundle $S$ ...

**5**

votes

**2**answers

188 views

### No normal coordinates on general Finsler manifolds

I recently read a footnote in Chern's article stating that a non-Riemmanian Finsler manifold does not possess normal coordinates.
As I'm still new to non-Riemmanian Finsler geometry I don't see why ...

**6**

votes

**0**answers

91 views

### Curvature Characterization of Homogeneous Spaces

A (Riemnnian) symmetric space is locally characterized by the first covariant derivative of its curvature tensor, namely, a manifold $M$ is locally a symmetric space if and only if $\nabla R=0$ (where ...

**0**

votes

**0**answers

63 views

### Weak $H^1$-limit of “almost conformal” maps

Let us consider a sequence of maps $\phi_n : (M, g) \to \mathbb{R}^k$, where $M$ is a surface. Let $\phi_n \to \phi$ weakly in the sense of $H^1$-norm, and let $\phi$ be non-trivial. Let $\mathcal{H}$ ...

**4**

votes

**2**answers

198 views

### Heat kernel asymptotics for small distances

I heard a talk where the speaker said that on a Riemannian manifold, for small values of $\text{dist }(x, y)$, the heat kernel $p_t(x, y)$ satisfies
$$p_t(x, y) = \frac{1}{(4\pi t)^{n/2}}e^{-\frac{\...

**3**

votes

**1**answer

93 views

### Parallel Transport on Hypersurface Spinor Bundle

So this has been driving me up a wall. I'm trying to digest parts of the Parker & Taubes paper, "On Witten's Proof of the Positive Energy Theorem." Here's a link:
https://projecteuclid.org/...

**1**

vote

**0**answers

68 views

### Euclidean metric in spherical coordinates [closed]

I apologize if this question is not research level, but it has been asked on MSE before, and not received really satisfactory answers, for example, see here, and googling does not reveal anything ...

**6**

votes

**2**answers

261 views

### Hodge map and the Cohomology Ring of a Riemannian Manifold

For a compact Riemannian manifold $M$, we know that the Hodge map $\ast$ and Laplacian $\Delta$ commute. From Hodge decomposition and its implied isomorphism between harmonic forms and cohomology ...

**3**

votes

**1**answer

198 views

### Totally geodesic subgroups in Lie groups

Let $G$ be a Lie group with a left invariant metric $g$.
Let $H$ be a (closed) Lie subgroup of $G$, and assume $g$ is right-$H$-invariant. (That is $d(R_h)_e:T_eG \to T_hG$ is an isometry for every $...

**3**

votes

**1**answer

109 views

### Carre du Champ, Subunit Paths and CC-metrics

Let the operator $L$ be given by $Lf(x):=\nabla\cdot (A\nabla f(x))$, where $f:\mathbb{R}^d\rightarrow \mathbb{R}$ belongs to a suitable class of functions $\mathcal{A}$. The carre du champ operator $\...

**0**

votes

**0**answers

36 views

### Obtaining Hessian of the embedding from an induced metric

Consider a hypersurface (not necessarily compact) smoothly embedded into $\mathbb{R}^n$ such that the Hessian is a positive definite bilinear form. Due to positivenes, Hessian can be taken as a metric ...

**4**

votes

**1**answer

220 views

### Is there an example of a Killing vector field on a complete Riemannian manifold with finite volume?

Is there a Killing vector field on a complete Riemannian manifold $M$ with finite volume that satisfies the condition
$$\displaystyle\liminf_{r\rightarrow +\infty} \displaystyle\frac{1}{r} \...

**5**

votes

**0**answers

98 views

### Regularity of the distance from the boundary in singular riemannian manifolds

I am looking for references related with the regularity of the distance from the boundary in singular Riemannian manifolds.
I assume the following setting. $(M,g)$ is a Riemannian manifold, with ...

**11**

votes

**1**answer

235 views

### Hausdorff convergence of submanifolds in Riemannian manifolds

Let $(M^n,g)$ be a smooth compact Riemannian manifold. It is well known that any sequence $\{X_i\}$ of compact subsets of $M$ has a subsequence which converges in the Hausdorff metric to a compact ...

**1**

vote

**0**answers

44 views

### Understanding the Exp map from a moduli of smooth curves

The setup:
Suppose I'm given a family of class $C^k$ curves $\{f_{z}\}_{z\in Z}$ with values in $\mathbb{R}$ parameter by a finite number of parameters $Z\subseteq \mathbb{R}^m$.
Let $\mathscr{M}$ be ...

**2**

votes

**1**answer

157 views

### Compact Riemann manifolds with constant injectivity radius

I'm interested in compact Riemann manifolds which have that property that the injectivity radius at a point $p$ doesn't depend on $p$. Another way to put this is that the function $$p \mapsto d(p, \...

**4**

votes

**0**answers

94 views

### Flows associated with Killing fields

Let $M$ be a Riemann manifold and $p, q$ two points on a geodesic $\sigma$ which are isotropically conjugate. That is, there is a Jacobi field along $\sigma$ vanishing at $p$ and $q$ which is the ...

**3**

votes

**2**answers

186 views

### Upper bound for Willmore energy

Good day to everyone! Does anybody know if there are upper bound estimates for Willmore energy for a given surface?

**1**

vote

**1**answer

56 views

### material derivative and relation to Riemannian metric

For each $n$ let $N_t$ be an embedded smooth hypersurface in $\mathbb{R}^n$ of dimension $n-1$. $\{N_t\}_t$ is a family of hypersurface that is evolving with some velocity $V$.
Smooth functions on $N$...

**3**

votes

**0**answers

92 views

### Differentiability of a map to the free loop space

While reading Morse theory, closed geodesics, and the homology
of free loop spaces, the author claims the following:
Given the $S^{n-1} \hookrightarrow Y_1 \rightarrow T^1S^n$ bundle over $T^1S^n$, ...

**4**

votes

**1**answer

80 views

### Uniform partitions of a compact Riemannian manifold

My question is a little bit vague. I want to know if an arbitrary compact Riemannian manifold (M^d,g) admits partitions that are uniform in some sense. To be more precise, I need for every eps > 0 a ...

**1**

vote

**0**answers

99 views

### Stability for open manifolds of finite volume under lower Ricci curvature bound

By a theorem of Cheeger-Colding if $N_i\to M$, where $\to$ means Gromov-Haussdorff convergence, $N_i$ and $M$ are of the same dimension, compact and smooth Riemannian manifold with finite volume, as $...

**2**

votes

**1**answer

68 views

### Constant Harmonic Mean surfaces

For surfaces embedded in $\mathbb R^3$ with principal curvatures $ \kappa_1, \kappa_2 $ we know bending/isometric mappings conserve $ K= \kappa_1 \kappa_2 $ and CMC DeLaunay type minimal surfaces ...

**0**

votes

**0**answers

91 views

### Conformally flat with zero scalar curvature 2 [duplicate]

First, i am the one who asked about the existence of compact manifolds of dimension $n\geq 4$ which are conformally flat, non-flat, with zero scalar. Due to the fact that i couldn't comment because i ...

**10**

votes

**2**answers

407 views

### A Converse to Cartan–Hadamard theorem?

Let $M$ be a complete Riemannian manifold, with the property that $\exp_p\colon T_pM \to M$ is a diffeomorphism for every $p \in M$.
Can we say something about it's curvature?
Is it true that its ...

**1**

vote

**1**answer

235 views

### Soliton equation and non-killing potential vector field

I am searching for a non-Killing vector field $\zeta \in\frak X\rm (M)$ where $(M,g)$ is a Riemannian manifold such that
$$\frac12 \frak L_\zeta \rm g+Ric=\lambda g$$
$$ \frak L_\zeta \rm Ric=\lambda \...

**6**

votes

**0**answers

144 views

### Is there any exotic smooth structure on open hyperbolic manifold?

I edited my post to clarify some confusions as suggested by Igor.
Let $M$ be an open hyperbolic manifold, with or without finite volume, Is there any manifold $N$ which is homeomorphic to $M$ but ...

**1**

vote

**0**answers

83 views

### Gaussian heat kernel bounds on Riemannian manifolds

I wish to know if we have Gaussian lower and upper bounds for the heat kernel,i.e. $$
t^{-n/2}e^{-\frac{\rho(x,y)^2}{C_1t}} \lesssim p_t(x,y) \lesssim t^{-n/2}e^{-\frac{\rho(x,y)^2}{C_2t}},
$$
on a ...

**3**

votes

**0**answers

62 views

### Minimal volume of manifold homotopic to a hyperbolic manifold with finite volume

Let $(X, g_0)$ be a $n$-dimensional open manifold with finite volume hyperbolic metric. Suppose $(Y, g)$ is another $n$-dimensional manifold and $f: Y\to X$ a proper degree one map. Then by Storm's ...

**8**

votes

**1**answer

161 views

### Does $S^n\times H^k$ have non-isometric conformal transformations?

Question: Consider the product Riemannian manifold $(M,g)=(S^n\times H^k,g_{round}\oplus g_{hyp})$ of a round sphere $(S^n,g_{round})$ of curvature $1$ and hyperbolic space $(H^k,g_{hyp})$ of ...

**4**

votes

**0**answers

112 views

### Geometric Characterization of Martingales

Recently I've read a paraphrasing from Ito saying that he sometimes thinks of martingales as Geodesics in a very large dimensional manifold.
My question is, is there any research studying this idea?
...

**8**

votes

**1**answer

127 views

### The scope of correspondence principle in quantum chaos

My understanding of the so-called correspondence principle in quantum chaos, is that it is a connection between the behaviour of a classical Hamiltonian system (chaotic/completely integrable) and the ...

**4**

votes

**0**answers

86 views

### Is positively curved Alexandrov surface isometrically embeddable in $\mathbb R^3$?

I guess it is not. The example I have in mind is: $X^2$ is the spherical suspension of a circle $S^1(t)$ of length $0<t<2\pi$. Then $X$ has constant curvature =1 except at two suspension points, ...

**5**

votes

**0**answers

190 views

### Aren't Riemannian geodesics also geodesics of the associated Cartan geometry?

I was inspired by R. W. Sharpe's book on doing differential geometry through Cartan connections. Unfortunately, the book is fairly thin in terms of specific examples in Riemannian geometry, so I ...

**1**

vote

**0**answers

61 views

### Isotropically conjugate points in reductive spaces

Is it the case that a naturally reductive Riemannian homogeneous space that contains a pair of conjugate points necessarily contains a pair of isotropically conjugate points? That is, there is a pair ...

**2**

votes

**0**answers

111 views

### Estimate for the first eigenvalue of the Laplacian

I was studying the paper of S. T. Yau - Seminar on Differential Geometry - and there asks if the first eigenvalue is equal to $ n $, if we have a embedded oriented Riemannian manifold and closed ...

**6**

votes

**1**answer

330 views

### Can a complete manifold have an uncountable number of ends?

Let $M$ be a complete and noncompact Riemannian manifold. Fix a point $p$ in $M$. Let $\gamma$: $[0, L]\rightarrow M$
(parametrized by its arc length) be a geodesic starting from $p$. Denote by $d(\...

**2**

votes

**2**answers

69 views

### Abelian isometry groups of codimension one

Good day.
Let (M,g) be an n-dimensional Riemannian manifold (complete, if you wish), and suppose that there exists an n-1 dimensional Abelian group acting by isometries on M. Or locally, near a point ...