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Questions tagged [differential-geometry]

Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses on "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such structures, use (differential-topology) instead. Use (symplectic-geometry), (riemannian-geometry), (complex-geometry), or (lie-groups) when more appropriate.

Differential geometry is a branch of mathematics that uses techniques in calculus and linear algebra to study geometry. It is closely related to differential topology and PDEs (geometric analysis).

32835 questions
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volume of geodesic balls for large Riemannian metrics

Let $M$ be a smooth compact manifold, let $p\in M$ and let $g_0$ be a fixed Riemannian metric on $M$. Does there exists a constant $C>0$ such that for any Riemannian metric $g\ge g_0$, the volume of the geodesic ball with respect to $g$ satisfies…
cantstop
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Confused about sections

From Wikipedia: "Let $F(U)$ be the set of all sections on $U$. $F(U)$ always contains at least one element, namely the zero section: the function $s$ that maps every element $x$ of $U$ to the zero element of the vector space $\pi^{−1}(\{x\})$." The…
James Moriarty
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Only one chart to parameterize a unit-cylinder

I want to parameterize a unit-cylinder $x^2+y^2=1$ with only one chart in a complete atlas (the sets must be open). The cylinder is in $\mathbb{R}^3$. One way to do the parametrization with two charts is: $$ \textbf{x}(u,v)=(\cos u, \sin u,…
Clare Francis
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Pre-image of submanifold under submersion

Let $F:M\to N$ be a submersion. I want to show that for any submanifold $S\subset N$, $F^{-1}(S)$ is a submanifold of $M$. We have not yet covered transversality theorems. However, we have regular value theorem that states the preimage of regular…
jack
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Subset of symmetric matrices is submanifold

Let $Sym(3)\cong\mathbb{R}$ denote the set of real symmetric $3\times 3$-matrices. Let $$M:=\{ P\in Sym(3)|\ P^2=P,\ \operatorname{tr}{P}=1 \}.$$ I asked to show that $M$ is a submanifold diffeomorphic to the projective space $\mathbb{R}P^2$. My…
Richard
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geodesics and unit speed curves

Say we have 2 surfaces $M$ and $\hat M$ that intersect perpendicularly --> $\left = 0$ along the curve of the intersection intersection, where $n$ is the unit normal to $M$ and $\hat n$ is the unit normal to $\hat M$. Assume the…
mary
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Quotient map $\pi\colon \mathbb{C}^{n+1}\backslash \{0\} \to \mathbb{CP}^n$ is a submersion

It's not to hard to see that the quotient map $\pi\colon \mathbb{C}^{n+1}\backslash \{0\} \to \mathbb{CP}^n$ is smooth and surjective. Does that imply that it is a submersion as well?
user43014
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Weingarten map given basis.

I have an exercise of several subsections, but I want to know how to do this subsection: Calculate the matrix of the Weingarten's operator with respect to the basis $(u,v)$ where $u=(1,0,g_x)$ and $v=(0,1,g_y)$. The manifold is an arbitrary one…
iam_agf
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Tangent Bundle and its (Isomorphic?) Dual Bundle

In general it is not true that a vector bundle $E$ is isomorphic to its dual bundle $E^*$. But it is true when the vector bundle is the tangnet space of a manifold (at least I think it's true). How does one prove this?
MikhailMatrix
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What does the symbol $f^*\langle v,w\rangle _p$ mean in DoCarmo's Differential Geometry?

I'm preparing the differential geometric exam, but I don't have the textbook on hand. My teacher's lecture note left a symbol $f^*\langle v,w\rangle _p$ without further explanation, so I don't know what it means. I think it is hard for me to google…
Eric
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Is there any loss in generality when I assume that a regular curve is arc-lenght parameterized?

My doubt is simple as that. When I have a smooth, regular curve (that is, its curvature is never zero), can I just assume that it is parameterized by arc-lenght, without any loss of generality? If not, is there any counter example?
Marra
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Smooth function $f\colon S^n\longrightarrow \mathbb{R}$ with $df_x=df_y=0$

How can I prove that for any smooth function $f\colon S^n\longrightarrow \mathbb{R}$ always exist $x,y\in S^n$ such that $df_x=df_y=0$? I have tried it by induction, but I don't know to prove it even with $n=1$.
Carlos Ramírez
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Product of immersions is immersion?

If $U\subseteq \mathbb{R}^n$ and $V\subseteq \mathbb{R}^m$, let's define immersions $f\colon U\longrightarrow \mathbb{R}^n$ and $g\colon V\longrightarrow \mathbb{R}^m$, if I define $$h\colon U\times V\longrightarrow \mathbb{R}^{m+n}$$ With…
iam_agf
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Hypercomplex structure as integrable $Gl(\mathbb{H},n)$-structure.

Although there is some mess with notation, we say that a manifold $M^{4n}$ has an almost hypercomplex structure if there are three almost complex structures $I,J,K$ satisfing quaternions relations. Now classical definition of hypercomplexity…
J.E.M.S
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Fréchet manifold structure of C(M, N)

Let $ F = C^{\infty}(M, N)$. I wish to give $F$ the structure of a Fréchet manifold. My plan was to emulate the construction of a smooth manifold. I know that for a finite dimensional smooth manifold M, $T_pM$ will be isomorphic to the model space…
Mathmonkey
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