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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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Natural Coordinate Functions

I'm studying the "Elementary Differential Geometry" from O'Neil and he mentions what he calls the "Natural Coordinate Functions". In particular, he says that if $p = (p_1 ... p_n) \in \mathbb{R}^n$ is a point in the $n$-space then we define the…
Gold
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A smooth map regular at a point is regular near it

If a smooth $f:M\rightarrow N$ where $M, N$ are manifolds with boundary is regular at some point, is it regular near this point?
user380217
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If $\langle \nabla f (x), N(x) \rangle >0, \forall x \in S^2$, proof that $\exists p \in (S^2)^\mathrm{o}$ such that $\nabla f(p) = 0$.

Let $f: \mathbb R^3 \to \mathbb R$ a differentiable function and $N(x) = x$ the normal field of $S^2$. Suppose that $\langle \nabla f (x), N(x) \rangle >0, \forall x \in S^2$. Then $\exists p \in (S^2)^\mathrm{o}$ such that $\nabla f(p) = 0$. Where…
user 242964
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Tangent vector of $\frac{1}{f}$

On the smooth manifold $M$, suppose $v$ is a tangent vector to $x$. How can one show that for a differentiable function $f$ on $M$, non zero in a neighborhood of $x$, $v(\frac{1}{f})$= $- \frac{v(f)}{f(x)²}$ I just began studying tangent vector on…
guest
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Gauss map and the curvature of a regular curve on a surface

Let $X$ be a parametrization of a surface $S$, let $\alpha(t)=X(u(t),v(t))$, $\alpha(0)=p\in S$ and $N=\dfrac{X_{u}\times X_{v}}{\lVert X_{u}\times X_{v} \rVert}$ be the Gauss map. Do Carmo says in his differential geometry of curves and…
No One
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Regular surfaces in Differential Geometry

Question is from do carmo Differential Geometry of curves and surfaces Chapter 2.3 Parametrized surfaces are often useful to describe sets $\Sigma$ which are regular surfaces except for a finite number of points and a finite number of lines. For…
user376629
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What's the standard smooth structure on manifold of smooth functions

Let $M$ be a smooth manifold and $C^{\infty}\left(M\right)$ be the space of all smooth functions on $M$. My question is (a) What is the standard smooth structure that makes $C^{\infty}\left(M\right)$ a smooth manifold? That is, how can we define…
Binjiu
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Integrals on manifolds and pullbacks

Hi I have some questions regarding integrals on manifolds. 1) Let $M_n$ be differentiable orientable manifold. The integral of a differential $n$-form $w$ with compact support is: Let $(\Omega_i, \varphi_i)$ be an atlas compatible with the…
hopo2
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Reconciling the meaning of the codifferential in two contexts

I'm not a geometer, but I'm currently trying to relearn some basic geometry. My question is: What is exactly the difference (if there is any!) between the codifferential simply as the pullback of a smooth map, and the codifferential of forms on an…
Christopher A. Wong
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Differential of the height function

Consider the mapping $f:S^2\rightarrow \mathbb{R}$, with $f(x,y,z)=z$, giving the height of a point on the sphere. I am asked to find the points for which the differential $(df)_p$ is surjective and i have failed so far. It's an exercise on an…
kleinmeinpouts
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Finding integral submanifolds of non-involutive distribution

Let $D$ be the distribution in $\mathbb{R}^3$ generated by the vector fields $X=ye^x \ \partial_y-\partial_z$, $Y=\partial_x$. Find an integral submanifold for $D$ passing for $(1,0,0)$ Say whether $D$ is integrable or not. Say whether an integral…
DavideL
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Euler vector field and homogeneous function

In $\mathbb{R}^n$ consider the Euler vector field $V$ $$V=x^1\partial_1+\dots+x^n\partial_n$$ Let $f:\mathbb{R}^n \setminus \{0\} \rightarrow \mathbb{R}$ be a $c-homogeneous$ smooth function, i.e. $$\exists\ c \in \mathbb{R}: f(\lambda…
DavideL
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Existence of a convex covering on a Semi-Riemannian Manifold

A convex covering $R$ of a Semi-Riemannian manifold $M$ is a covering of $M$ by open geodesically convex sets, such that if elements $a$ and $b$ of $R$ meet, then the intersection is also convex. In the Book of O'Neill on Semi-Riemannian manifolds…
CKB
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Interpretation of Multilinear maps as tensors

Let M be a smooth manifold and $C^{\infty}(M)$ the smooth functions on it. Some authors are calling $C^{\infty}(M)$-multilinear mappings: $T:\mathcal{X}(M)^s\rightarrow\mathcal{X}$ Tensor fields of type (1,s). (One can think for instance about the…
Braten
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Vanishing order of metric relates to the one of its Weyl tensor

In Brendle's paper (S. Brendle, Convergence of the Yamabe flow in dimension 6 and higher, Invent. Math. 170 (3) (2007), 541-576.), let $p \in M$, he defined the following set $$\mathcal{Z}=\Big\{x \in M; \limsup_{x\to…
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