All Questions
Tagged with reference-request dg.differential-geometry
800 questions
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Flows commuting with Anosov flows and further reference request
Hello respected members of Mathoverflow. I was reading the paper "Flots d’Anosov dont les feuilletages stables sont différentiables" by Etienne Ghys and there was a statement which he remarked was ...
- 229
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150
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A vector field over a complex riemannian manifold
Let be a complex riemannian manifold $(M,g,J)$. Is the following canonical vector field studied ?
$$
X_J = \sum_{i=1}^{2n} \nabla^{LC}_{e_i}e_i +\nabla^{LC}_{Je_i}Je_i+ J[e_i,Je_i],
$$
with the $(e_i)...
- 377
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162
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Gromov-Hausdorff relative compactness without curvature restrictions
A famous theorem of Gromov says that the set of compact Riemannian manifolds with $Ric \geq c$ and $\text{diam} \leq D$ is relatively compact in the Gromov-Hausdorff metric. Chapter 10 of the book by ...
- 1,407
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126
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What is an umbilic point of a convex polyhedron?
An umbilic point of a smooth ($C^2$) convex surface in Euclidean 3-space is a point where the principal curvatures are equal. Is there some good way to generalize this notion to convex polyhedra? See ...
- 7,159
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133
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Reference for example of gradient steady Ricci solitons
Recently I read a paper about Ricci solitons. I quote a paragraph of it here:
In dimension three, the classification of complete gradient steady Ricci solitons is still open. Known examples are ...
- 4,195
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156
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Proof of Berger in "Sur les variétés d’Einstein compactes"
I would appreciate any reference that contains either a translation or proof of the following interesting observation of Berger (Sur les variétés d’Einstein compactes, M. Berger paper (in French)).
...
- 4,195
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307
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Higher-dimensional version of Fenchel's theorem that total curvature is $\ge 2 \pi$
Q. Is there a higher-dimensional version of the theorem due to Fenchel that the
total curvature of a closed curve in $\mathbb{R}^3$ is $\ge 2\pi$,
with equality only if the curve is planar and convex?
...
- 151k
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82
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Scattering in (pseudo-)Riemannian spaces
I will ask my question in a broad way, leaving a lot of freedom for answers.
Suppose that we have a (pseudo-)Riemannian space $(M,g)$ and we fix some ball-like domain $B \subset M$. Suppose you are ...
- 1,461
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0
answers
35
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Measures for the Eccentricity of General Strictly Convex Smooth Closed Manifolds of Genus 0
Question:
Are there any measures for how much the shape of a strictly convex smooth closed manifold of genus 0 deviates from that of a hyper-sphere of equal dimension?
In euclidean 2-space and in ...
- 13.2k
1
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96
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Largest dimensional Lie subgroup of $SU(N)$ [duplicate]
What is the largest (Lie) subgroup of $SU(n)$ in the sense of its dimension.
I am aware of this potential duplicate subgroup of SU(N) with maximal manifold dimension , however, the title of this ...
- 2,099
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224
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Characterization of the Riemann curvature tensor [duplicate]
Let $(M^n,g)$ be a Riemannian manifold, $a\in M$ be a fixed point. It it well known that there exists a coordinate system near $a$ (e.g. the normal one) such that
$$g_{ij}(x)=\delta_{ij}+O(|x|^2).$$
...
- 21.8k
1
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0
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81
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Ring structure for $K^{-1}$?
My questions are
whether there exists a product structure for $K^{-1}(X)$? Here $K^{-1}$ is the odd topological $K$-group, and $X$ is a compact space (or a manifold), say.
If such a ring structure ...
- 1,173
1
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197
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Euler class and self-intersection number of a surface in a 4-manifold [duplicate]
In the first two paragraphs of Circle bundles over $CP^1$ and self-intersection number of $CP^1$ embeddings, it is claimed that
For a compact oriented surface $X$ in a 4-dimensional oriented ...
- 655
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0
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463
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Reference request for parallel transport
I am learning about parallel transport on a Riemannian manifold equipped with an affine connexion. It seems (if I understand it well) that, in general, we might not be able to compute the parallel ...
- 119
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119
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Orbits and indices of vector fields
I'm afraid this might be an exercise in differential topology (in which case a reference to a book where it is would be very much appreciated); apologies in advance. Given an analytic vector field (in ...
- 4,432
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382
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Question in the paper of Robert Bryant "Calibrated embeddings in the special Lagrangian and coassociative cases"
Hallo,
I am trying to read the paper "Calibrated embeddings in the special Lagrangian and coassociative cases" by Robert Bryant and I have a question. I hope that maybe one of you could give me some ...
- 339
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0
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371
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Simple development of simple curve on a cone
Let $\Lambda$ be a cone with apex $a$ and apex angle $\alpha$. Draw a simple (non-self-intersecting)
curve $C=(x,y)$ on $\Lambda$, and then develop it to a curve
$\overline{C}$ on a plane by rolling $...
- 151k
1
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0
answers
221
views
Co-normal bundle of orthogonal compliment
Is the following fact well known?
Let $X$ be a manifold and $V$ be a vector space. Let $E_1$ be a sub-bundle of the constant bundle $X \times V$. Let $E_2$ be its orthogonal compliment in $X \...
- 2,649
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362
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Archimedes’ and Galileo’s spirals in one equation.
The differential equation in polar coordinates
$r'^2+r^2=(kt)^2$, $r(t=0)=0$, k- Const, for large $t$ presents Archimedes’ Spiral and Galileo's spiral for $t \to 0$.
I find it surprisingly, however ...
1
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0
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318
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Where can I find an English translation of Grauert's paper?
The german title is : Holomorphe Funktionen mit Werten in komplexen Lieschen Gruppen.
- 778
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171
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Explicit construction of irreducible unitary connections
On a Riemann surface irreducible unitary connections on complex vector bundles of rank $n\geq 2$ are uniquley determined by their underlying holomorphic structure $\frac{1}{2}(\nabla+i*\nabla)$, where ...
- 6,825
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1
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201
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Ambient isotopy of the diagonal submanifold in product space
Given a closed manifold $M^n$ and its $k$-fold product space $M^n\times\cdots\times M^n$,Can the diagonal submanifold $\Delta:=\{(m,\cdots,m)\in (M^n)^k\mid m\in M\}$ be isotopied to the submanifold
$...
- 73
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1
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291
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Where is the paper "Theorie de Lie pour les groupoides differentielles (J. Pradines)"?
Can anyone help me finding the paper:
"Theorie de Lie pour les groupoides differentielles (J. Pradines)"
I'm researching Lie groupoids and I was refered to that paper several times but couldn't find ...
- 1
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1
answer
226
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Marcel Berger's "Sur les groupes d'holonomie homogènes de variétés à connexion affine et des variétés riemanniennes."
I would appreciate any reference that contains either a translation or proof of the main theorem in this paper. Thank you in advanced.
0
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1
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339
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Polarisation in a neighbourhood of a Lagrangian submanifold
Let $(X, \omega)$ be a symplectic manifold of dimension $2n$ and $\omega$ is an exact symplectic form i.e. $\omega = -d\alpha$. Let furthermore $M \subset X$ be a compact Lagrangian submanifold such ...
- 339
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1
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570
views
Is this a manifold of bounded geometry?
Let $M$ be a complete non-compact manifold (possibly with boundary). Let $E$ be an open proper connected non-precompact subset of $M$ with smooth topological boundary, so that $\overline{E}$ is a non-...
- 459
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1
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395
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Topology of manifolds and transition functions
let me start by describing some examples which may well demonstrate the motivation:
A manifold is obtained by glueing together Euclidean spaces, and there is a transition function on the overlap of ...
- 1
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1
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182
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Surjectivity of "nice maps" from local properties
What tools are available from real algebraic geometry, analysis and
topology to check surjectivity of a map $f:M_{1}\rightarrow\mathbb{R}^{d}$
from local properties and maybe function values?
...
- 1,256
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1
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153
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Entropy of Negatively pinched manifolds
Suppose $M$ is a compact negatively pinched Riemannian manifold of dimension $n$. We normalize the metric such that $-1\le K\le -a^2$ for some $0<a\le 1$. Let $G$ be the fundamental group of $M$. ...
- 1,101
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1
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108
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Intersection Grassmanian planes
I am reading a paper that used Grassmanian planes properties. In particular, they studied the intersection of Grassmanian planes; they check the intersection Grassmanian of $n-k$-planes and ...
- 1,043
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1
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289
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Estimate for Laplace equation with Neumann boundary on manifold with corner
Let $(M,g)$ be a compact Riemannian manifold with boundary and corner, i.e. locally modelled in $[0,\infty)^1\times \mathbb R^{n-1}$ or $[0,\infty)^k\times \mathbb R^{n-k}$, where $n=\dim(M)$.
...
- 1,915
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1
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124
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Relationship between the vortex filament equation and the cubic Schrödinger equation
How is the vortex filament equation
$$\partial_t \chi = \partial_s \chi \wedge \partial_{ss} \chi,$$
where $\chi(t,s)$ is a curve in $\mathbb R^3$,
related to the cubic Schrödinger equation?
Note 1. ...
- 277
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1
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166
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Non-trivial foliation (excluding the Reeb foliation) [closed]
Let $M$ be a closed oriented manifold, an oriented foliation $F$ is said non-trivial, if $F$ is not fibration of $M$, i.e. there does not exist a closed manifold $B$, such that $M\overset{F}{\to} B$.
...
- 1,915
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1
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119
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Analytic approach to geodesic connectedness in Semi-Riemannian manifolds
Can you point out a reference (or references) that deal with analytical methods (rather than methods from differential geometry) for the study of geodesic connectedness on Semi-Riemannian manifolds?
user60665
0
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2
answers
435
views
Isomorphism of connections on a complex line bundle
Reading an article I faced with the following theorem, please give me a reference to a proof of the fact which is stated without any reference in the article. Is it a well-known fact?
Theorem. Let $E ...
- 1,329
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1
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315
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G-structures and complete riemannian manifolds
what are possible fundamental and introductory texts about G-structures ?
and where i can find the proof of this proposition:
if G(group) acts properly discontinuously on a space X , then G is a ...
- 165
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0
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33
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Request for resources on directional derivative of the Riemannian distance function, and Berger's lemma about geodesics realizing the diameter
I've been recently interested in directional derivatives of the Riemannian distance function, and I came across this question, and its answer by Sergei Ivanov, where he stated an important result: (I ...
- 1,467
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0
answers
86
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Reference request for Poincare-Hopf theorem in a compact submanifold
I recently read the following question about the Poincare-Hopf theorem in a compact submanifold. All the answers were very satisfactory to me. Is there any reference where I can look for more details ...
- 1
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0
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70
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Normal vector to a level set and fractional Laplacian
Let $U=\{u\le0\}$ and $\partial U=\{u=0\}$. Suppose $\nabla u$ does not vanish on $\partial U$. Then the (canonical extension of the) normal vector field to $\partial U$ (pointing to the interior of $\...
- 99
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425
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Compact connected Riemannian manifolds are Ahlfors regular metric space
Let $(M,g)$ be a compact connected $n$-dimensional Riemannian manifold; let $(X,d)$ denote its associated metric (length) space. A comment on the original formulation of this post mentioned that $(X,...
- 5,405
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51
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References for local distance approximation over Riemannian manifolds [duplicate]
Over a complete Riemannian manifold $(M,g)$, in a neighborhood of $p \in M$, the local distance can be approximated as follows: $\forall v,u \text{ unit vectors in } T_pM, \text{ and small } s, t$
$$ ...
- 31
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126
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mean curvature for codimension $>1$?
The mean curvature of a hypersurface in a Riemannian manifold is defined to be the trace of the second fundamental form. I was curious, does the notion of mean curvature generalise to higher ...
- 3,625
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0
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84
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Prerequisites/Preparation for understanding a research paper - global solutions to Einstein field in Bondi Coordinates
I would like to read this paper:
João L. Costa, Filipe C. Mena, Global solutions to the spherically symmetric Einstein-scalar field system with a positive cosmological constant in Bondi ...
- 101
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0
answers
64
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Relationship between the vortex filament equation and the transport equation
Let us consider the vortex filament equation
$$\partial_t \chi = \partial_s \chi \wedge \partial_{ss} \chi,$$
where $\chi(t,s)$ is a curve in $\mathbb R^3$.
How is the Cauchy problem for the ...
- 277
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0
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139
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(Semi-)Riemannian geometry for working PDE analysts
What is a good reference on (semi-)Riemannian geometry written for PDE analysts (that is, with main focus on analytical problems and approaches)?
The closest thing I know to this, are two books by ...
user60665
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0
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61
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Heat trace asymptotic coefficients for conformal metrics $\widetilde{g}=e^{f}g$ surfaces
As is well known $\sum e^{-\lambda_{k}t}\approx(4\pi t)^{dim(M)/2}\sum a_{j}t^{j}$, where $a_{j}$ are geometric properties of manifold M.
Moreover, the arbitrary order coefficients don't have closed ...
- 5,474
0
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0
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134
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when is "fibering" preserved under homotopy equivalence
Suppose I have an oriented $F$ bundle over $B$ with total space $E$ (all of the three are closed manifolds) and i have a closed manifold $E'$ which is homotopy equivalent to $E$.Is there any condition ...
- 259
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0
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234
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What is the symplectic manifold whose Delzant polytope is a trapezoid?
What is the symplectic form on the manifold whose associated Delzant polytope is a trapezoid? I am trying to find it by using the Marsden–Weinstein theorem, but I have been unable to do so. If ...
- 1
0
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0
answers
289
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Third variation of area of a minimal surface
There is a formula for the third variation of area on page 96 of Nitsche's book,
Lectures on Minimal Surfaces, vol. 1 (English version). He says at the bottom of the
page it is good for normal ...
- 1,233
-1
votes
2
answers
1k
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Regarding understanding differential geometry [closed]
I am essentially looking for a book that would hold my hand through basic concepts to more complicated ones. I am coming from physics. I am looking to make some connections with Classical mechanics ...
user39066