Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such ...

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Connectedness of level sets

I have a $C^{1}$ real valued function $f$ defined on a connected manifold $M$, it doesn't have critical points, lets assume that $f^{-1}(0)$ is a (compact) connected submanifold of $M$, does that ...
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Expression for codifferential in terms of interior product

Let $(M^n,g)$ be a Riemannian manifold with local orthonormal frame $\{e_1,\ldots,e_n\}$ with dual basis $\{e^1,\ldots,e^n\}$ and with Levi-Civita connection $\nabla$. It can be checked on basis that ...
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Ambiguity in definition of $C^r$ maps between manifolds

Let $M$ and $N$ be smooth manifolds with corresponding maximal atlases $A_M$ and $A_N$. We say that a map $f : M \to N$ is of class $C^r$ (or $r$-times continuously differentiable) at $p \in M$ ...
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a question about finding umbilical points in an elipsoid.

Determine the umbilical points of the elipsoid $${x^2\over a^2}+{y^2\over b^2}+{z^2\over c^2}=1.$$ My thoughts: let $x=asin(\theta)cos(\phi),y=bsin(\theta)cos(\phi),$and $z=cos(\theta)$. Thus, I ...
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a question about differential geometry

Let $S$ be a surface and $x: U\to S$ be a parametrization of $S$. If $ac-b^2 <0$, show that $$a(u,v)(\dot u)^2+2b(u,v)\dot u\dot v+c(u,v)(\dot v)^2=0$$ can be factored into two distinct equations, ...
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Extending a vertical vector to a vertical vector field

Let's say $F: M \rightarrow N$ is a smooth submersion between manifolds. Then each fiber of $F$ is a properly embedded submanifold. If $S$ is such a fiber, and I take some derivation $D \in T_p S$, ...
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39 views

Extending the metric of a hyperbolic surface with boundary to its double

Let $M$ be a hyperbolic surface with totally geodesic boundary. Taking the double $DM$ of $M$, it is easy to see using Euler characteristic that $DM$ is itself a hyperbolic surface (without boundary). ...
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Maurer-Cartan Form on orthogonal group

I'm having trouble understanding an assignment regarding the Maurer-Cartan form on orthogonal matrices: Let $\text{O}(n)\subset\text{GL}(n,\mathbb{R})$ be the matrix Lie group of orthogonal ...
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Surface with $\nabla F=(0,0,0)$

In my geometry book there is a Proposition that says that: Lets M={ (x,y,z) | F(x,y,z)=0} be a $R^3$ set, and $p=(x_0, y_0, x_0)\in M$. If $\nabla F(p) \neq(0,0,0)\; \forall p \in M$ then M is a ...
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Heat semigroup estimate on complete Riemannian manifold

Consider a complete noncompact Riemannian manifold $M$ such that the heat kernel $h_t(x, y)$ satisfies $h_t(x, y) \leq Ct^{-n/2}$. Consider a function $u \in L^p(M)$. How can we prove that ...
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How can I prove the hypersurface $M$ is neither convex nor concave?

The question is so simple: how can I prove that $M$ (as defined below) is neither convex nor concave (in the real sense)? The point here is to define a new notion of convexity, the complex convexity, ...
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characterization of non compact surfaces in $\mathbb{R^3}$

Is there a way to characterize non compact surfaces with constant mean and gaussian curvature. I know that if $K=0=H$ then the surface is a plane. How can I know about the others? Just to add, for ...
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CY 3-folds are $T^2 \times \mathbb{R}$ fibrations over the base $\mathbb{R}^3$. What does it mean?

In this article at section 2. Toric geometry and Mirror Symmetry there is the statement that CY 3-folds are $T^2 \times \mathbb{R}$ fibrations over the base $\mathbb{R}^3$. Now, my questions refers to ...
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Coefficients of first and second fundamental form if gaussian and mean curvatures are constant.

I was solving a problem when at some point, I had this question. If $K$ and $H$ are constant for a surface. Can I say something about the coefficients of first and second fundamental form? I know that ...
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What is mean by “trace on any pair of indices”?

I am reading the book Riemannian Manifolds, written by John M. Lee. On page $53$, the author defines a connection on the tensor bundle $\text{T}_l^k(M)$ and says it satisfies (d) $\nabla$ commutes ...
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Left invariant vector field

Let $G$ be a Lie Group with $e$ as the neutral element. Taken $X_e\in T_e G$, define $$X(a)=(dL_a)_e X_e$$ Why this vector field is left invariant? I get confused with the notation. Thanks!
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Topological invariants by integrals

Some topological invariants that can be found e.g. in knot theory can be represented as integrals (Example: Integral for computing the Gauss linking number). Another example is the complex plane with ...
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Sets that are convex in two different metrics

Let $(M,g)$ be a complete Riemannian manifold, and let $C$ be a subset of $M$. We will say $C$ is convex if for any points $p,q \in C$, there exists a unique normal minimal geodesic $\gamma$ joining ...
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Is the analytic version of the Whitney Approximation Theorem true?

The Whitney Approximation Theorem states that any continuous map between smooth manifolds is homotopic to a smooth map. If the manifolds are real analytic, is every continuous map between them ...
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Three-Dimensional Metrics as Deformations of a Constant Curvature Metric?

I read the following paper Three-Dimensional Metrics as Deformations of a Constant Curvature Metric and discovered the following result: I have three questions: (1) Is $h$ also a conformally flat ...
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Is an hypersurface uniquely determined by an equation?

Consider $r:\Omega\to\Bbb R$ suffiently regular, $\Omega\subseteq\Bbb C^n$, $z_0\in\Omega$ s.t. $r(z_0)=0$. Then $r=0$ defines an hypersurface locally around $0$. My question is: is $r$ unique? By ...
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All rank two symmetric tensors are several conformally flat metrics summed together?

If given a rank two symmetric tensor $T_{mn}$ can I decompose it as \begin{align} T_{mn} = \sum_{i=1}^{M} \phi_ig_{mn}{}^{i}{} \end{align} where $\phi_i$ are the $i$th conformal factors and ...
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Notation in symmetric and alternating products of forms

In Lee's Riemannian Manifolds book, we see that a Riemannian metric $g$ can be expressed locally in coordinates $(x^1, \ldots, x^n)$ by $g = g_{ij} dx^i \otimes dx^j$. Introducing the symmetric ...
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$\Delta_L(\text{im}\,\delta^*_g)\subset\text{im}\,\delta^*_g$ and $\Delta_L\big(\text{ker}\,\text{Bian}(g)\big)\subset\text{ker}\,\text{Bian}(g)$?

Let $(M,g)$ be an Einstein manifold with Levi-Civita connection $\nabla$ and whose Ricci tensor $\text{Rc}(g)=g$, in components $R_{ij}=g_{ij}$. The Lichnerowicz Laplacian of $g$ is the map ...
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Can you have a “real” decomposition of differential forms on a symplectic manifold?

With a choice of (almost) complex structure on a symplectic (Kahler) manifold, has anyone thought of how or what it means to globally define "real" versions of the Dolbeault operators? i.e. Something ...
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integrating by parts on a manifold

Suppose $M$ is compact. Let $\phi$ be some smooth function, and $\beta$ an $n-1$-form. Then does integration by parts say that $$\int_M\phi d\beta=\int_Md\phi\wedge\beta?$$ If not, how does ...
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Motivation for the definition of an infinitesimal object

An infinitesimal object $D$ in a Cartesian closed category $\mathsf{C}$ is one for which the internal Hom functor $$(-)^D: \mathsf{C} \to \mathsf{C}$$ has a right adjoint. I am wondering what is the ...
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Kähler–Einstein condition

Let $(M,\omega)$ be a Kähler manifold, and $g$ the Riemannian metric such that $\omega(X,JY)=g(X,Y)$. If there is a function $f$ such that $\operatorname{Ric}=f\omega$ does this mean that ...
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Dilating a curved ball

Let $B$ be a ball sitting inside a manifold $(M^n, g)$. Now, let us dilate the metric $g$ to $\lambda g$, $\lambda$ being a positive number going to $\infty$. It seems intuitively true that the ...
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If $\Gamma\subseteq Diff(M)$ is finite dimensional, when is the evaluation $\Gamma\rightarrow M$ a submersion?

Let $M$ be a smooth manifold. Say it is compact and connected. Suppose that there exists a finite dimensional submanifold $\Gamma\subseteq\mbox{Diff}(M)$ such that the evaluation map ...
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Is a variety a CW-complex?

How to establish that any differentiable manifold and any complex algebraic variety is a CW-complex ? Thank you in advance for your help.
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Connection on Tangent Bundle of Group Manifolds

The thing puzzling me is about a transformation rule of connection on tangent bundle of group manifolds: Assume one has a compact and simply connected Lie group $K$. One can give a metric $g$ on $K$ ...
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differentiable structure on mobius strip

Define $M= \mathbb{R}^2/\sim$ where $(x,y)\sim(x',y')$ if $x-x'=2n$ for some integer $n$ and $y = (-1)^n y'$. Then how can I give a differentiable sturucture on $M$? Is there a general technique for ...
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Extending a function on a submanifold to the ambient manifold & proof of a property of a vector field.

$\newcommand{\wt}[1]{\widetilde{#1}}$ Hello, I just tried my hand at two exercises from John M Lee's book Riemannian Geometry and I would like to know whether my reasoning is sound or if I did ...
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Another Differential Geometry-Curve Theory

This is another problem that keeps arising year after year that none is able to solve. Any help is very appreciated. Let $r(s)$ be a regular closed curve which lays in sphere $S^2$. Prove that: ...
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Differential-Geometry question- Curve Theory

Let $r(s)$ be a curve parametrized by the natural parameter $s$ and for its curvature $k$ and torsion $t$ the following condition applies: $$k(s),t(s)\neq 0 $$ for every $s$. Prove that the curve ...
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Integration over manifolds

S is subset of $\Bbb R^3 $ consisting of the union of 1) $z$ axis 2) the unit circle $x^2+y^2=1,z=0$ 3) the points $(0,y,0)$ with $y \ge1 $ Let $A$ be the open set $\Bbb R^3-S$. Let $C_1, C_2, ...
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Is there a fibre bundle with fibre homeomorphic to $\mathbb R^k$ which cannot be given the structure of a vector bundle?

We define a (rank $k$) vector bundle to be a fibre bundle with fibre $\mathbb R^k$ such that the local trivializations are fibre-by-fibre vector space isomorphisms. I'm curious whether this linearity ...
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What is the geometric interpretation of the Koszul formula?

I saw this simple form of Koszul formula on a book: $$2\ g(\nabla_XY,Z) = \mathcal{L}_Yg(X,Z) + (d\theta_Y)(X,Z)$$ where $\theta_Y$ is the one-form $g(Y,\cdot)$. It is equivalent to the more ...
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Metric, torsion free connections on principal bundles

Let $F(M)$ be the frame bundle of a $n$-dimensional differentiable manifold $M$, and let $H\subset TF(M)$ be a connection. A Riemannian metric on $M$ can be equivalently written as an equivariant ...
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a tangent vector which belongs to intersection of a manifold and a subspace is tangent to their intersection?

I have the following situation: There is an embedded submanifold (in my particular case it is of codimension one but I do not think it matters here) $M\subseteq R^n$, and a vector subspace ...
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$SL(3,\mathbb{R})$ is a smooth manifold?

How do you show $SL(3,\mathbb{R})$ is a smooth manifold? I am thinking to use the preimage theorem, but what kind of thing I need to show first before I can apply the theorem?
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Trouble proving identity - Gauge theory/Maurer-Carton one-form/Adjoint representation

The Identity I am trying to prove is the one in this already asked question how to show that ${ad}_{g_{\alpha\beta}} \circ g_{\alpha\beta}^{\star}\theta=-g_{\beta\alpha}^{\star}\theta$? The author ...
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Geometric characterization of critical points of the Gauss map.

Let $\Sigma \subset \mathbb{R}^3$ an oriented surface by Gauss map $N: \Sigma \rightarrow S^2$. How can I find a geometric characterization of critical points of $N$?
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Intuitive meaning of immersions

I have a hard time understanding the concept of immersions. In my course, it was only introduced by the immersion theorem wich says: Let $f: U \subset \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}$ be ...
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Finding critical values of a function on an embedded surface

Prior to the problem, we have already shown that $\Sigma=\{x_1x_2^2+x_2x_3^2+x_3x_1^2=1\}\subset\mathbb{R}^3$ is an embedded hypersurface and that the function ...
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Is there a general coordinate transformation perserving the components of an Euclidean metric?

In the Euclidean space (or Lorentz spacetime, if you are interested in relativity), there is one orthonormal coordinate system $\{x^\mu\}$ such that the distance squared is given by ...
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Characterize the sphere using mean curvature.

We know the following result: if $\Sigma$ is a compact surface than $$ \int_{\Sigma}H^2 \ge 4 \pi, $$ where with $H= \frac{1}{2}(\kappa_1+\kappa_2)$ we denote the main curvature. I have to prove that ...
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Integration over a variety

If $ M $ is a differentiable manifold equipped with an Atlas $ \mathcal{A} = ( U_i , \varphi_i )_{ i \in I} $, we can then calculate the integral of a differential form $ \omega $ over $ M $ with the ...
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Generalize Gauss-Bonnet Formula to non-simple closed curves

According to the Classical Gauss-Bonnet Formula, I think it should can be generalized to non-simple closed curves in the following sense: For a domain $\Omega$ enclosed by an non-simple closed curve ...