# Fundamentals of differential geometry

@inproceedings{Lang1998FundamentalsOD, title={Fundamentals of differential geometry}, author={Serge Lang}, year={1998} }

This text provides an introduction to basic concepts in differential topology, differential geometry, and differential equations, and some of the main basic theorems in all three areas: for instance, the existence, uniqueness, and smoothness theorems for differential equations and the flow of a vector field; the basic theory of vector bundles including the existence of tubular neighborhoods for a submanifold; the calculus of differential forms; basic notions of symplectic manifolds, including…

## 426 Citations

### THE SINGLE-LEAF FROBENIUS THEOREM

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Using the notion of Levi form of a smooth distribution, we discuss the local and the global problem of existence of one horizontal section of a smooth vector bundle endowed with a horizontal…

### Background: Relevant Tools from Geometry

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- 2016

Shape spaces are typically nonlinear spaces where the rules of vector-space calculus do not apply directly. Instead, one uses tools from algebra, geometry, and functional analysis to develop…

### Differential Equations in a Tangent Category I: Complete Vector Fields, Flows, and Exponentials

- MathematicsAppl. Categorical Struct.
- 2021

This paper describes how to define and work with differential equations in the abstract setting of tangent categories. The key notion is that of a curve object which is, for differential geometry,…

### Semi-invariant Riemannian metrics in hydrodynamics

- MathematicsCalculus of Variations and Partial Differential Equations
- 2020

Many models in mathematical physics are given as non-linear partial differential equation of hydrodynamic type; the incompressible Euler, KdV, and Camassa–Holm equations are well-studied examples. A…

### Nonlinear generalized sections on manifolds

- Mathematics
- 2014

We present an extension of Colombeau's theory of nonlinear generalized functions to spaces of generalized sections of vector bundles. Our construction strongly builds on classical functional analytic…

### HILBERT STRATIFOLDS AND A QUILLEN TYPE GEOMETRIC DESCRIPTION OF COHOMOLOGY FOR HILBERT MANIFOLDS

- MathematicsForum of Mathematics, Sigma
- 2018

In this paper we use tools from differential topology to give a geometric description of cohomology for Hilbert manifolds. Our model is Quillen’s geometric description of cobordism groups for…

### Fenchel Duality Theory and a Primal-Dual Algorithm on Riemannian Manifolds

- MathematicsFound. Comput. Math.
- 2021

This paper introduces a new notion of a Fenchel conjugate, which generalizes the classical Fenchel conjugation to functions defined on Riemannian manifolds. We investigate its properties, e.g., the…

### TOPICAL REVIEW: On the geometric approach to the motion of inertial mechanical systems

- Mathematics
- 2002

According to the principle of least action, the spatially periodic motions of one-dimensional mechanical systems with no external forces are described in the Lagrangian formalism by geodesics on a…