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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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A question about orientation on a Manifold

Let $(U,x_1,x_2,\ldots , x_n)$ be a chart for a orientable manifold $M$, why $(U,-x_1,x_2,\ldots , x_n)$ is a chart for the Manifold $-M$, the same manifold with reversed orientation?
Jr.
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An elementary differential geometry proof

Given a differentiable function $k(s)$, $s\in I$, show that the parametrized plane curve having $k(s)=k$ as curvature is given by $$ \alpha (s) = \left( \int \cos\theta(s)ds + a, \int \sin\theta(s)ds + b \right) $$ where $$ \theta(s)= \int…
Карпатський
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Properly discontinuous action

This is exercise 9 of chapter 0 from Do Carmo's book in Riemannian Geometry: Let $G\times M \rightarrow M$ a properly discontinuous action from a Group $G$ on a smooth manifold $M$. Prove that $\frac{M}{G}$ is orientable if and only if there is a…
Jr.
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Proving that the coordinate basis is a basis of a tangent space

Given a differentiable manifold $M$ and some chart $(U, \psi)$ near $p$, we can consider the curve $\tilde{\beta}_i: t \mapsto \psi(p)+t e_i$, where $e_i$ denoted the standard basis in $\mathbb{R}^n$, $i \in \{1, \dots, n\}$. Now we can set $\beta_i…
Huy
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Scalar product on manifold.

Let $M$ be a closed Riemannian manifold and $\omega$ and $\eta$ two differential forms of the same degree. Then one can consider $\int_M \omega \wedge *\eta$, where $*$ denotes the Hodge star operator. Can you tell me, why this defines a scalar…
nick
  • 975
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Sanity check: smooth structure of tangent bundle

Let $M$ be a smooth $n$ manifold and let $TM$ denote its tangent bundle $$ TM = \bigsqcup_{x \in M} \{(x,T_x M)\}$$ I am trying to put a smooth structure (atlas) on $TM$ using the atlas on $M$. But I'm a bit confused and could do with some…
a student
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Did I integrate a differential form correctly?

I start with 1-form $\omega=f\,dx$ on $\left[0,1\right]$ where $f\left(0\right)=f\left(1\right)$ and a $g:\left[0,1\right]\to R$ with $g\left(0\right)=g\left(1\right)$ and I want to integrate $\omega-\lambda \, dx=dg$ on $\left[0,1\right]$. So I…
Martin Luciend
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Euclidean metric on a Riemannian manifold

Lets say we have a Euclidean configurations space $\mathbb E^n$ equipped with a smooth inner product $\langle \cdot ,\cdot \rangle$ with positive signature in the tangent space above each point. We have defined a Riemannian manifold. We can also…
AngusTheMan
  • 541
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About the definition of regular surface in do Carmo

According to do Carmo, the definition of regular surface requires us to check $X^{-1}$ to be continuous (where $X$ is a local parametrization). But doesn't it infer from other conditions (as shown in his Prop. 4 of Sec 2-2, see photo attached here…
John
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Twisted geodesics on circular torus

I am attempting to make a Mechanics of materials approach to describe non-linear deformations of thin lines/wires on a torus. A mathematical modeling of its probable geometry is required at start of formulation to bring in bending and twisting…
Narasimham
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Exterior derivative = infinitesimal change in differential form?

For simplicity I'll work in $M=\mathbf R^2$. Given $f\in C^\infty(M)=\Omega^0(M)$, its exterior derivative $df$ is a 1-form that eats a tangent vector and spits out the best linear approximation of (the change in) $f$ if we walk along the direction…
whippedcream
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If a mapping is bijective and regular, then the mapping is a diffeomorphism?

Let me ask a question which appears in the book 'Elementary Differential Geometry' written by O'Neil. The questions is: prove that if a one-to-one and onto mapping $f:\Bbb R ^n \to \Bbb R ^n$ is regular, then it is diffeomorphism. In the book, "$f$…
Mathcho
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Relation between jet and singularities of a curve

I am trying to understand the relationship between a jet and the singularities of a curve. Concretely I am trying to understand the following example: Let $\gamma(t) = (\cos t , b \sin t)$ where $b > 0$ be an ellipse. Let $f$ be the distance squared…
student
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Is this a normal form for $4$-forms on manifolds?

Starting from a $2$-form $\omega$ which is nondegenerate and closed on a $2n$-dimensional manifold, it is always possible to find local coordinates $x_1,y_1,\ldots,x_n,y_n$ so that the form $\omega$ can be written locally as $$\sum_{k:1}^n…
rubi
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Orthonormal frame on hyperbolic plane

I'm having trouble comprehending a question from Do Carmo's Differential Forms and Applications. The question (in its entirety) is as follows: (Exercise 5-2 in Do Carmo). Let $H^2$ be the upper half-plane, that is, …
Blake
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