Question

Motivation for defining tangent vectors with derivations and why they should act on $f\in C^\infty(M)$

score 354 · Answer 1 · 2020-08-17 02:17:43

354

Views

Motivation for defining tangent vectors with derivations and why they should act on $f\in C^\infty(M)$

Published on 17 Aug 2020 - 2:17

score 173 · Answer 2 · 2020-08-28 19:19:28

173

Views

If $(T_t)_{t\ge0}$ is a flow on a submanifold $\Omega$ with boundary with velocity $v$ satisfying $\langle v,\nu_{∂Ω}\rangle=0$, then $T_t(∂Ω)⊆∂Ω$

Published on 28 Aug 2020 - 19:19

score 31 · Answer 3 · 2026-03-25 17:41:52

31

Views

If $ϕ$ is a boundary chart of a $k$-dimensional submanifold $M$ with boundary, then $(T_xϕ)v\in\mathbb R^{k-1}×\{0\}$ for all $v\in T_x\:\partial M$

Published on 25 Mar 2026 - 17:41

score 165 · Answer 4 · 2026-03-25 22:05:19

165

Views

A tangent vector of a manifold $M$ with boundary is either inward/outward pointing or tangential to $\partial M$

Published on 25 Mar 2026 - 22:05

score 217 · Answer 5 · 2020-09-01 05:53:33

217

Views

Tangent space of an upper half-space and the relation of $T_x\:M$ and $T_x\:\partial M$, for $x\in\partial M$, in general

Published on 01 Sep 2020 - 5:53

score 87 · Answer 6 · 2020-09-04 07:38:17

87

Views

Differential in terms of $T_pX \simeq \left(\mathfrak{m}_p/ \mathfrak{m}_p^2\right)^\ast$

Published on 04 Sep 2020 - 7:38

score 24 · Answer 7 · 2020-09-08 19:13:54

24

Views

If a function has a local $C^1$-extension, is it locally continuous?

Published on 08 Sep 2020 - 19:13

score 38 · Answer 8 · 2020-09-08 21:12:44

38

Views

Lipschitz function is differentiable at a point iff its tangent set is a k-dimensional plane

Published on 08 Sep 2020 - 21:12

score 279 · Answer 9 · 2020-09-10 16:03:19

279

Views

Is always possible define a coordinate basis for a smooth manifold?

Published on 10 Sep 2020 - 16:03

score 74 · Answer 10 · 2026-03-25 19:07:09

74

Views

If $M_i$ is a submanifold with $M_1\subseteq M_2$, what's the pushforward of the restriction $\left.f\right|_{M_1}$ of a map $f:M_2\to\mathbb R$?

Published on 25 Mar 2026 - 19:07

score 231 · Answer 11 · 2020-09-17 23:45:14

231

Views

How to show that $T_{(1,0)}\mathbb S^1 \cong \operatorname{span}(\{e_2\})$?

Published on 17 Sep 2020 - 23:45

score 209 · Answer 12 · 2020-09-18 19:29:17

209

Views

Is it possible to define a vector field of non-coordinate basis vectors?

Published on 18 Sep 2020 - 19:29

score 52 · Answer 13 · 2020-09-22 01:34:46

52

Views

Show that $T_{(p,q)}(M\times N)\cong T_p M\oplus T_q N$. Tangent space to the product manifold.

Published on 22 Sep 2020 - 1:34

score 480 · Answer 14 · 2020-09-23 17:39:12

480

Views

Characterizing the tangent space of a Lie group at an arbitrary point

Published on 23 Sep 2020 - 17:39

score 85 · Answer 15 · 2020-09-26 09:05:41

85

Views

"Unitary" extension of the normal field from the boundary of a manifold

Published on 26 Sep 2020 - 9:05

List Question

Motivation for defining tangent vectors with derivations and why they should act on $f\in C^\infty(M)$

If $(T_t)_{t\ge0}$ is a flow on a submanifold $\Omega$ with boundary with velocity $v$ satisfying $\langle v,\nu_{∂Ω}\rangle=0$, then $T_t(∂Ω)⊆∂Ω$

If $ϕ$ is a boundary chart of a $k$-dimensional submanifold $M$ with boundary, then $(T_xϕ)v\in\mathbb R^{k-1}×\{0\}$ for all $v\in T_x\:\partial M$

A tangent vector of a manifold $M$ with boundary is either inward/outward pointing or tangential to $\partial M$

Tangent space of an upper half-space and the relation of $T_x\:M$ and $T_x\:\partial M$, for $x\in\partial M$, in general

Differential in terms of $T_pX \simeq \left(\mathfrak{m}_p/ \mathfrak{m}_p^2\right)^\ast$

If a function has a local $C^1$-extension, is it locally continuous?

Lipschitz function is differentiable at a point iff its tangent set is a k-dimensional plane

Is always possible define a coordinate basis for a smooth manifold?

If $M_i$ is a submanifold with $M_1\subseteq M_2$, what's the pushforward of the restriction $\left.f\right|_{M_1}$ of a map $f:M_2\to\mathbb R$?

How to show that $T_{(1,0)}\mathbb S^1 \cong \operatorname{span}(\{e_2\})$?

Is it possible to define a vector field of non-coordinate basis vectors?

Show that $T_{(p,q)}(M\times N)\cong T_p M\oplus T_q N$. Tangent space to the product manifold.

Characterizing the tangent space of a Lie group at an arbitrary point

"Unitary" extension of the normal field from the boundary of a manifold

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