Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such ...

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Need help on books on diff. equations/geometry and theoretical computer science

I am looking for recommendation of 3 different books on the following topics: 1.Differential Equations -Ordinary diff. equations -Vector field, transport equations -Equation of wave and heat -Use ...
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Geometric meaning of symmetric connection

If $(M, g)$ is Riemannian manifold, there is unique connection $\nabla$, called Levi-Civita connection, satisfying the following: 1) Compatibility with Riemannian metric, i.e. $\nabla(g)$=0 2) ...
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Questions about involute and evolute.

I have a couple of questions regarding differential geometry. Can two different curves have the same involute curve? It seems possible to me, but I can't be sure. In this lecture, at around 12:00, ...
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On an example of vector field

In the book Elementary Differential Geometry of Christian Bar, on page 153 there is an example as follows: Let $f: S\rightarrow \mathbb{R}$ be a smooth function. Since the first fundamental form is ...
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How to proof Frobenius Theorem in general?

The general Frobenius Theorem stating that Let $u_1,\dots,u_k$ be $k$ smooth linearly independent vector field on $M$. Let $$ W=\operatorname{Span}(u_1,\cdots,u_k) $$ Then $[u_i,u_j]\in W$ for ...
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Where does this expression of Gaussian curvature come from?

In my Differential Geometry course, we have seen a way to calculate the Gaussian curvature $K$ given a metric expressed as the sum of two Pfaff forms $Q = ω_1^2 + ω_2^2$: we find another Pfaff form ...
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Equivalent definitions of vector field

There are two definitions of a vector field on a smooth manifold $M$. A smooth map $V:M \rightarrow TM, \forall p \in M:V(p) \in T_p M$. A linear map $V:C^{\infty}(M) \rightarrow C^{\infty}(M), ...
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171 views

A differential topology lemma

Consider the following lemma (1) How come he talks about degrees here, after all he doesn't assume $X$ to be oriented? (2) Why is $\bar{v}|\partial X$ homotopic to $g$? (NOTE: we consider them as ...
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A question on self-adjointness of the shape operator

In the book Elementary Differential Geometry of Christian Bar, CUP, on the page 108, of the proof of theorem 3.5.5, the author wrote: Theorem: Let $S ⊂ \mathbb{R}^3$ be an orientable regular surface ...
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The completeness of Spatial Schwarzschild manifold

We know that the spatial Schwarzschild manifold is $\mathbb{R}^3/0$ with metric $g=(1+\frac{m}{2r})^4\delta$, where $\delta$ is the Euclidean matric. Is there anyone know how to prove the ...
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diffeomorphism of the bundle chart of a Tangent bundle

I am studying about tangent bundle from the book "J. M lee", on page 66 of the book a map $\tilde{\phi} : \pi^{-1}(U) \to \mathbb{R}^{2n}$ is defined by $$v^i \frac{\partial}{\partial x^i}|_p \mapsto ...
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Homology of manifolds with boundary

If $M$ is a compact topological manifold WITH boundary does it follow that its homology groups are finitely generated and zero almost all of them? I know it is true in case it has no boundary (i.e. is ...
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Levi-Civita connection compatible with Riemaniann and Pseudo-riemaniann metric

Given a Pseudo-riemaniann metric on ${\cal{M}}$, is it possible to find a Riemaniann metric on ${\cal{M}}$ with the same Levi-Civita connection? If in general this is not possible, what sufficient ...
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question on the summation of complete vector fields

Can anyone give an example of two vector fields $X_1$ and $X_2$ which are complete but their sum $X_1+X_2$ is not complete?
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Misuse of Tangent Vector

I am quite confused with the term tangent used in differential geometry books. It seems to me that people use this word quite loosely. For example, one definition about tangent space in my book is as ...
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Understand Cotangent Space as an Equivalence Class

In my differential geometry class, my teacher defined the co-tangent space as follows. Let $M$ be a smooth manifold and $p$ is a point on $M$. Now define two sets of $C^\infty$ real-valued functions ...
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Problem in proving the property of Lie bracket of vector fields

Let $M$ be a Riemannian manifold, $f \in C^{\infty}(M)$, $X,Y$ vector fields on $M$. Then i have to prove $[X,f\cdot Y]=f\cdot [X,Y]+X(f)\cdot Y$. First i use the definition of Lie bracket: $[X,f\cdot ...
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The Levi-Civita connection in infinite dimensions

Is there an analogue of the Fundamental Theorem of Riemannian Geometry for (some subclass of) infinite-dimensional manifolds?
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Proof: Force always perpendicular and motion in a plane implies that the trajectory is a circle

I am looking for a proof for a physics problem. Consider a particle which is subject to a force $\vec{F}(t)$ with $|\vec{F}(t)| = \text{const}$ which is always perpendicular to the velocity ...
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A question related to weierstrass function

Use Weierstrass function to determine how far back from the screen should a student sit in order to maximum the view range as given the floor is flat. The bottom of the screen is 1 meter above ...
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84 views

Regular Parametrization of a Sphere

Is there a function $f:U→ \mathbb{R^3}$, such that: (1) U is an open connected subset of $ \mathbb{R^2} $; (2) f is $ C^r , r≥1$; (3) the Jacobian of f is of maximal rank at all points of U; (4) ...
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Complex structure on $\mathbb{C}\mathbb{P}^2\# \dots \# \mathbb{C}\mathbb{P}^2$

I know the chern classes-related theorem that states that $\mathbb{C}\mathbb{P}^2\# \dots \# \mathbb{C}\mathbb{P}^2$ ($k$ times) has no almost complex structure (hence no complex structure) if and ...
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101 views

Integral Curves of a Vector Field

How do I find the integral curves of a vector field and what are they intuitively? eg. what are the integral curves of vector field $X=\frac{1}{x}\frac{\partial}{\partial ...
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Differential Geometry and Origami

Would anyone know how to relate origami with differential geometry? I mean clearly you can see how geometry plays into it but how would you describe it in terms of differential
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Actions of Weyl group

I get a feeling what I am going to ask is very standard and classic, but I am not able to find any reference. Any answer or reference would be appreciated. Let us assume that $G$ is a simply ...
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Chain rule for tensor of family of tensor fields

Let $f_\tau$ be a $\mathbb R$-family (parameter $\tau$) of diffeomorphisms that map from $\mathbb R^4$ to $\mathbb R^4$. $f^*_\tau$ is the corresponding pullback (I think that is the correct term). ...
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Are normal curvature and geodesic curvature independence of choice of curves?

By intuition, if the direction of tangent of a point $P$ is given, I think the curve passing through $P$ on the surface have only one choice (locally). So, does $T'(s)$ only depend on the given ...
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Special Family of metrics on transitive lie algebroids.

Let $\rho:E\longrightarrow TM$ is a transitive Lie Algebroid, then $L=ker\rho$ is bundle of lie algebras. Suppose $\Gamma:TM\longrightarrow E$ be a linear splitting. Define $$\nabla_X ...
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102 views

The completion of $C_c^\infty(M)$ with respect to $\lVert \nabla u\rVert_{L^2(M)}$ on a compact Riemannian manifold

Let $M$ be a compact smooth Riemannian manifold (eg. a smooth hypersurface) and let $X$ be the space given as the completion of $C_c^\infty(M)$ with respect to the norm $\left(\int_{M} |\nabla ...
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How to differentiate exponential map with parameter dependent basepoint

Let $(M,g)$ Riemannian manifold, $\gamma:I\rightarrow M$ a geodesic and $X$ a Jacobi field. For a proof $c:(-\varepsilon,\varepsilon)$ is defined to be another geodesic with $c^\prime(0)=X(0)$. ...
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The Morse complex of a manifold with boundary

For a smooth manifold with boundary $M$ and $\partial M = V_+ \cup V_-$ two disjoint sets of boundary components, one usually defines the Morse complex of $M$ using a Morse-Smale pair $(f,X)$ such ...
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Can anyone prove this identity without passing through the complexified tangent space?

Let $\rho: \mathbb{C} \to \mathbb{R}$ be a smooth function, $\Omega = \{ z : \rho(z) <0 \}$, and suppose $|\nabla \rho| = 1$ on $b\Omega$. It is true that $$\int_{b\Omega} f(z) d\bar{z} = -2i ...
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Isomorphism on a component of tangent space implies local diffeomorphism

Let $U$ open set in $\mathbb{C}^{2n}$, $p\in U$. Let $\phi : U \to \mathbb{C}^n$ be a submersion, i.e. $d\phi$ is surjective at every point. Assume that $\forall p \in\Re U$, $T_p U$ = $T_p (\Re U) ...
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Show that two intersecting curves on a regular surface with the same osculating plane that is not the tangent plane have the same curvature

Let $\bf{p}$ be a point on a regular surface $S$. Let $\boldsymbol{\alpha}(s)$ and $\boldsymbol{\beta}(s)$ be two curves parametrized by arc length on the surface $S$ such that ...
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Regularity of weakly harmonic map

Suppose $(M,g)$ is a smooth $n$-dimensional manifold with $C^k$-metric $g$ and let $U\subset M$ be an open subset. Does anyone have a reference for a statement about the regularity of a map ...
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“Bundle of metrics” on a principal bundle?

I've come across the term "bundle of metrics" on a principal bundle. In particular, my setting is that for $N\longrightarrow M$ a universal cover of a compact Riemann surface, $P\longrightarrow M$ a ...
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Tangent spaces at a point of an affine space

I have seen on wikipedia that an affine space in $\mathbb{R}^n$, with $V$ as its vector space of translation, is a smooth manifold but don't know the explanation. I have to know about the tangent ...
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Line Integrals in $\mathbb{R}^n$ and Differential 1-Forms

I am really lost in the notation of my book and am looking for some conceptual help. So... for a differential 1-form $\omega$, every value of $x\in \Omega$ maps to a linear function $\omega _x$, ...
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Gauss curvature of C^2 surfaces

In do Carmo's book on Differential Geometry of Curves and Surfaces, the proof of theorema egregium, that the Gauss curvature of a surface immersed in $\mathbb{R}^3$ is invariant under local ...
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Help explain why the proof works - Gradient Estimate Using the Maximum Principle

I found the following proposition and proof in a textbook for one of my classes, and the end of the proof seems a little "unfinished" to me. In other words, as the author left it, I'm not entirely ...
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Intuition for Integration of Differential Forms

In mathematics, we define $dx^i$ as linear functionals, when speaking of integration. However, in physics, we interpret $dx^i$ as very small quantities. There is nothing inherently small about a ...
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Are there closed manifolds that can be given multiple geometric structures of constant curvature

For example, is there a closed differentiable manifold that can be given both a Euclidean structure and a Hyperbolic structure? If not, is there a reasonably easy proof that no such manifold exists?
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Defining the integral of differential $1$-forms

Assume $M$ is a smooth manifold, $g:[0,1]\to M$ is a smooth curve on $M$, and $w$ is $1$-form on $M$. Definition: $$\int_gw=\lim_{\Delta\to 0}\sum_{i=1}^nw(x_i)$$ The tangent vectors $x_i$ are ...
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Tangent bundle of P^n and Euler exact sequence

I'm thinking about the Euler exact sequence for complex projective space, and I'm a bit confused. In the topological category, one has $$ T\mathbb{P}^n \oplus \mathbb{C} \cong L^{n+1} $$ where $L$ ...
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A question on the differential of the Gauss map

Let $S$ be a orientable, regular surface, locally parametrized by $(U, F, V)$. Let $N$ be the Gauss map. The Weingarten map is defined with a point $p$ in $U$ as $W_p: T_pS \rightarrow T_pS$ , ...
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Why is this statement about integral curves correct?

The following definitions and example are taken from John Lee's Smooth Manifolds, 2nd edition. Given a vector field $V$ on a smooth manifold $M$, we define an integral curve of $V$ to be a ...
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Alexander duality formulation + Jordan-Brouwer separation

In Davis & Kirk LNAT p.71 there is written: (1) How does this imply the Alexander duality $\tilde{H}^k(A)\cong \tilde{H}_{n-k-1}(\mathbb{S}^n\!\setminus\!A)$? (2) Is it assumed that the ...
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tangent/normal space to set of symmetric isospectral matrices

Let $\Lambda = \{\lambda_1, \ldots, \lambda_n\}$ be a set of $n$ distinct real numbers. $M_n(\mathbb{R})$ denotes the set of all $n \times n$ real matrices, and for $B\in M_n(\mathbb{R})$, $B^T$ ...
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Chern-Weil: why do we divide by $2\pi$?

So here's a somewhat incoherent question. To define characteristic classes in the Chern–Weil way, one takes a curvature form $\Omega$ on a vector bundle $E \to M$ and an invariant polynomial ...
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What does it mean for two hyper-surfaces to be tangent to each other?

In the book "Anathem" by Neal Stephenson, Part four begins: Six weeks after I joined the Edharian order, I became hopelessly stuck on a problem that one of Orolo's knee-huggers had set for me as a ...