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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).

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Integral curves of the gradient

Let $f : M \rightarrow \mathbb{R}$ be a differentiable function defined on a riemannian manifold. Assume that $| \mathrm{grad}f | = 1$ over all $M$. Show that the integral curves of $\mathrm{grad}f$ are geodesics.
Sak
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Proof that angle-preserving map is conformal

Let $\phi: S \to \bar{S}$ be a diffeomorphism between two surfaces in $\mathbb{R^3}$. Such a map is called conformal if for all $p \in S$, and $v_1, v_2 \in T_p(S)$ (the tangent plane) we have $$\langle d\phi_p(v_1), d\phi_p(v_2) \rangle = \lambda^2…
koletenbert
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Volume of a geodesic ball

This may be embarassingly simple, but I can't see it. Let $M$ be a Riemannian manifold of dimension $n$; fix $x \in M$, and let $B(x,r)$ denote the geodesic ball in $M$ of radius $r$ centered at $x$. Let $V(r) = \operatorname{Vol}(B(x,r))$ be the…
Nate Eldredge
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Prove the curvature of a level set equals divergence of the normalized gradient

Suppose we have a function $\phi : \mathbb{R}^2 \to \mathbb{R}$, and a curve $\gamma:\mathbb{R}\to\mathbb{R}^2$ defined by a level set of $\phi$, ie. the codomain of $\gamma$ is $\{(x,y)\mid\phi(x,y)=C\}$ for a given constant $C$. Edit: assume that…
Garrett
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A question on closed convex plane curves (from Do Carmo)

Let $\alpha (s)$ , $s\in [0,l]$, be a closed convex plane curve positively oriented. The curve $\beta(s)=\alpha (s) -rn(s)$, where $r$ is a positive constant and $n(s)$ is the normal vector, is called a parallel curve to $\alpha$. Show that: $$…
MathematicalPhysicist
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Deriving calculation formulas for torsion and curvature

With much blood, sweat, and tears, I have managed to derive the formulas $$k = \frac{||a' \times a''||}{||a'||^3}, \quad \tau = \frac{\langle a' \times a'', a''' \rangle}{||a' \times a''||^2}$$ for the curvature and torsion of a smooth regular space…
Mr. Chip
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If $\| \alpha(s) \| \leq \| \alpha(s_0) \| = R$, then the curvature $k(s_0)$ is greater than $1/R$

This question is in Ted Shifrin's A first course in curves and surfaces, page 18, exercise 7: Suppose $\alpha$ is an arclength-parametrized space curve with the property that $\| \alpha(s) \| \leq \| \alpha(s_0) \| = R$ for all $s$ sufficiently…
Yagger
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What is a complete vector field?

I am currently studying a first course in manifolds and I have come across a definition that I don't really understand: Let $X$ be a vector field on a manifold $\mathcal{M}$. If all the integral curves of $X$ extend $\forall t\in\mathbb{R}$, then we…
user320140
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Explicitly proving invariance of curvatures under isometry

I would like to know how to explicitly prove that Riemann Curvature,Ricci Curvature, Sectional Curvature and Scalar Curvature are left invariant under an isometry. I can't see this explained in most books I have looked at. They atmost explain…
Student
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Weak tangent but not a strong tangent

Question: Show that $\alpha(t)=(t^3,t^2)$, $t\in \Bbb R$, has a weak tangent but not a strong tangent at $t=0$. Definitions from this answer: (Weak tangent) $\alpha: I \to \Bbb R^3$ has a weak tangent at $t_0 \in I$, if the line determined by…
Suzu Hirose
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"Natural" constructions of tensor fields from tensor fields on a manifold

This question begins is related to this question on physics.SE Uniqueness of Riemann Curvature Tensor, which asks roughly "what tensors can we make locally out of just the metric tensor? We can clearly make a (3,1) tensor out of just the metric, but…
BebopButUnsteady
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Difference between pushforward and differential

The pushforward of a map $F:M \to N$ at a point $P \in M$ is defined as $F_*:T_P(M) \to T_{F(P)}(N)$ where $$(F_*X)(f) = X(f \circ F)$$ where $X \in T_P(M)$. The differential of a function $f$ defined on $M$ at $P$ is $$df_P(X_P) = X_Pf.$$ What is…
hopo2
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Example of a surface where more than one coordinate patch is needed.

I find the sphere example underwhelming. Sure I can see that one open patch will not cover it, but it still manages to cover it mostly. So much so that you can go ahead and, say, calculate the area of a sphere using only one patch $$\sigma(u,v) = (r…
amcalde
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Product Manifold: Tangent Spaces

Problem Given a product manifold. How to prove that its tangent spaces split into direct sums: $$T_{(p,q)}(M\times N)\cong T_pM\oplus T_qN$$ Attempts One could try the geometric perspective: $$\Phi:T_{(p,q)}(M\times N)\to T_pM\oplus…
C-star-W-star
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About the second fundamental form

Let $U\subset\mathbb R^3$ be an open set, and $f:U\to \mathbb R$ be a smooth function. Suppose that the level set $S=f^{-1}(\{0\})$ is non-empty, and that at each $p\in S,$ the gradient $\overrightarrow \nabla f(p)$ is not the zero vector. Then $S$…
Peterson
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